EUROPEAN SYMPOSIUM ON C O M P U T E R AIDED PROCESS ENGINEERING - 10
COMPUTER-AIDED CHEMICAL ENGINEERING Advisory Editor: L.M. Rose Volume Volume Volume Volume
1: 2: 3: 4:
Volume 5:
Volume 6: Volume 7: Volume 8:
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)
COMPUTER-AIDED CHEMICAL ENGINEERING, 8
EUROPEAN SYMPOSIUM ON COMPUTER AIDED PROCESS ENGINEERING- 10 33~aEuropean Symposium of the Working Party on Computer Aided Process Engineering 619 ~ Event of the European Federation of Chemical Engineering (EFCE) Organized by AIDIC, the Italian Association of Chemical Engineering ESCAPE-10, 7-10 May, 2000, Florence, Italy
Edited by
Sauro Pierucci CIIC, Politecnico di Milano, Piazza L. da Vinci, 32, 1-20133 Milan, Italy
2000 Elsevier Amsterdam
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Oxford - Shannon
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Tokyo
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PREFACE This book includes papers presented at ESCAPE10, the 10th European Symposium on Computer Aided Process Engineering, held in Florence, Italy, from the 7th to the 10th May, 2000. ESCAPE 10 was the tenth event of a series, started in Elsinore Denmark in 1992, of annual Symposia promoted by the Working Party 'Computer Aided Process Engineering (CAPE)' established in 1966 by the 'European Federation of Chemical Engineering (EFCE)'. However, it must be acknowledged that the ESCAPE series emerged from a strong tradition of the Working Party dating back to 1968 when the first event on computer applications was organized in Tutzing, Germany. Twenty three such symposia were then organized in almost a dozen European countries before a new departure of the Working Party with the ESCAPE series. Therefore, ESCAPE-10 was the 33 rd event of the CAPE Working Party, and the 609 th event of the EFCE. The most recent symposia were organized in Hungary, Budapest 1999 (ESCAPE-9), Bruges, Belgium 1998 (ESCAPE-8), Trondheim, Norway 1997 (ESCAPE-7) and in Rhodes, Greece 1996 (ESCAPE-6). ESCAPE-10 was organized by AIDIC, the Italian Association of Chemical Engineering, a member society of the European Federation of Chemical Engineering. The ESCAPE-10 scientific program reflected two complementary strategic objectives of the CAPE Working Party: the former checked the status of historically consolidated topics by means of their industrial application and their emerging issues, while the latter was addressed to opening new windows to the CAPE audience by inviting adjacent Working Parties to co-operate in the creation of the technical program. The former CAPE strategic objective was covered by the topics:
Numerical Methods, Process Design and Synthesis, Dynamics & Control Process Modeling, Simulation and Optimization. The latter CAPE strategic objective derived from the EFCE promotion of scientific activities which autonomously and transversely work across the Working Parties terms of references. These activities should enhance the exchange of the know-how and knowledge acquired by different Working Parties in homologous fields. They also aim to discover complementary facets useful to the dissemination of WP's tools and of their novel procedures. As a consequence the WP's 'Environmental Protection', 'Loss Prevention and Safety Promotion' and 'Multiphase Fluid Flow' were invited to assist in the organization of sessions in the area of:
A Process Integrated approach for." Environmental Benefit, Loss Prevention and Safety, Computational Fluid Dynamics. A total of 473 abstracts from all over the World were evaluated by the International Scientific Committee. Out of them 197 have been finally selected for the presentation and reported into this book. Their Authors come from 30 different Countries. The Selection of the papers was carried out by 28 international reviewers. We hope that these proceedings will be a reference document to the scientific and industrial community and will contribute to the progress in Computer Aided Process Engineering. Sauro Pierucci Guido Buzzi Ferraris
vi
SCIENTIFIC COMMITTEE PIERUCCI Sauro BUZZI FERRARIS Guido AZZOPARDI Barry BISTOLFI Marco BOGLE David BRIDGES Steven DONAT1 Gianni ESPUNA Antonio FRAGA Eric GANI Rafiqul GLAVIC Peter GREGERSEN Lars HEYEN George JORGENSEN John B. JOULIA Xavier KRAVANJA Zdravko Le LANN Jan Marc MACCHIETTO Sandro MEYER Michel MEYER Xuan-Mi PASMAN Hans J. PASTORINO Renato PERRIS Tony PUIGJANER-CORBELLA Luis ROVAGLIO Maurizio VENSELAAR Jan ZANELLI Severino ZULLO Luca ORGANIZING COMMITTEE Del ROSSO Renato BALLO Giuliana
BENINCORI Carlo POLI Giulio WOLF MACIEL Maria Regina
(Italy) co-Chairman (Italy) co-Chairman (England) (Italy) (England) (Denmark) (Italy) (Spain) (England) (Denmark) (Slovenia) (Denmark) (Belgium) (Denmark) (France) (Slovenia) (France) (England) (France) (France) (The Netherlands) (Italy) (England) (Spain) (Italy) (The Netherlands) (Italy) (U.S.A.) (Italy) Chairman (Italy) Secretary (Italy) (Italy) (Brasil)
SYMPOSIUM SECRETARIAT ESCAPE 10 c/o AIDIC Piazzalr Morandi 2 1-20100 Milano (Italy) Tel. +39-02-76021175 Fax +39-02-799644 e-mail: escape
[email protected] web : http://www.aidic.it/escape- 10/escape 10.html VENUE
Palazzo Congressi, Florence, Italy May 7-10, 2000
vii CONTENTS
Mixed Integer Non-Linear Programming Using Cutting Plane Techniques R. PSrn and T. Westerlund A Novel Interactive MINLP Solver for CAPE Applications J.P. Henriksen, S.F. Stoy, B.M. Russel and R. Gani An MILP-Based Reordering Algorithm for Complex Industrial Scheduling and Rescheduling J. Rosl6f I. Harjunkoski, J. BjSrkqvist, S. Karlsson and T. Westerlund Non-Linear Partial Least Squares through Data Transformations B. Li, E.B. Martin and A.J. Morris Optimisation of an Industrial Cogeneration System by means of a MultiObjective Genetic Algorithm G.A. Efthimeros, D.I. Photeinos, I.G. Katsipou, Z.G. Diamantis and D. T. Tsahalis Grid refinement in multiscale dynamic optimization T. Binder, L. Blank, W. Dahmen and W. Marquardt Numerical Strategies for Optimal Experimental Design for Parameter Identification of Non-Linear Dynamic (Bio-)Chemical Processes J.R. Banga, K.J. Versyck and J.F. Van Impe Solution of population balance equations for prediction of particle size distribution in emulsion polymerization: comparison and evaluation of different numerical methods A.H. Alexopoulos and C. Kiparissides Solution of the hyperbolic model for heat and mass transfer in packed bed reactors A.A. Iordanidi, A.E. Kronberg, J.A.M. Kuipers and K.R. Westerterp Moving finite difference method for tracking a shock or steep moving front Y.1. Lim, S.H. Jeong, J.M. Le Lann and X. Joulia Neural Network in Physical Equilibria Prediction S. Ore~ki, J. Zupan and P. Glavi( Novel Methods for the Efficient Evaluation of Stored Mathematical Expressions on Vector Computers B.R. Keeping and C.C. Pantelides Global Optimization of Nonconvex Problems with Differential-Algebraic Constraints W.R. Esposito and C.A. Floudas Scheduling to Minimize Expected Completion Time in Flowshop Plants with Uncertain Processing Times J. Balasubramanian and I.E. Grossmann
13 19
25 31
37
43
49 55 61
67
73
79
viii
Automatic Re-Weighting of Maximum Likelihood Functions for Parameter Regression Y. Xin, V.R. Vasquez and W.B. Whiting Energy cost minimization in an energy intensive industrial plant: an MINLP approach J. Vin, M.G. Ierapetritou, P. Sweeney and M. Chigirinskiy Generic object-oriented modelling, simulation and optimization of dynamical systems T. Wack, G. Deerberg and S. Schlfiter Detecting outliers in multivariate process data by using convex hulls J.P. Barnard and C. Aldrich MINLP Optimization of Several Process Structures for the Separation of Azeotropic Ternary Mixtures D. Brusis, T. Frey, J. Stichlmair, L Wagner, R. Duessel and F.-F. Kuppinger MINLP Optimization of Reactive Distillation Columns T. Frey and J. Stichlmair Batch Distillation Optimization with a Multiple Time-Scale Sequential Approach for Strong Nonlinear Processes M. Wendt, P. Li and G. Wozny Non-Linear Constrained GRG Optimization under Parallel-Distributed Computing Environments G.E. Vazquez, R. Rainoldi and N.B. Brignole A Bi-Index Continuous Time MILP Model for Short-Term Scheduling of Single-Stage Multi-Product Batch Plants with Parallel Units C.- W. Hui and A. Gupta Importance of parameter selection in classification systems using neural networks J. Ordieres and F. Ortega A two dimensional conceptual model to support data integration in process plant operations A.D. Yang, H.-S. Li and M.-L. Lu Feedforward Control Based on Online Concentration Calculation of a Heatand Mass-Integrated Distillation System K. LSwe and G. Wozny Analysis of Input-Output Controllability in Reactive Distillation Using the Element model A.D. Estrada- Villagrana, L D.L. Bogle, E.S. Fraga and R. Gani Hybrid Simulation of Continuous Discrete Systems V. Bahl and A.A. Linninger Interaction between Design and Control of Heat-Integrated PFR C.S. Bildea, A.C. Dimian and P.D. Iedema
85
91
97 103
109 115
121
127
133
139
145
151
157 163
169
ix
Optimal control of batch reactors using Generic Model Control (GMC) and Neural Network N. Aziz, M.A. Hussain and I.M. Mujtaba Stability analysis of delayed chemical systems L. Pellegrini, M. Ratto and M. Schanz Nonlinear model based control of optimal temperature profiles in polystyrene polymerization reactor G. ~)zkan, S. Ozen, S. Erdogan, H. Hapoglu and M. Alpbaz Experimental Verification and Optimisation of a Detailed Dynamic High Performance Liquid Chromatographic Column Model H.K. Teoh, M. Turner, N. Titchener-Hooker and E. Sorensen Expert Control of DO in the Aerobic Reactor of an Activated Sludge Process M. Galluzzo, R. Ducato, V. Bartolozzi and A. Picciotto Dynamic Behavior of a Counter-Current Fixed-Bed Reactor with Sustained Oscillations M. Mangold, F. Klose and E.D. Gilles Use of Gap Metric for Model Selection in Multi-Model Based Control Design: An Experimental Case Study of pH Control O. Galdn, J. Romagnoli, Y. Arkun and A. Palazoglu Dynamic and Control of High Purity Heterogeneous Azeotropic Distillation Process C J. G. Vasconcelos and M. R. Wolf-Maciel Training a Recurrent Neural Network by the Extended Kalman Filter as an Identification Tool R. Scheffer and R. Maciel Filho An Algorithm for Efficient Specification Analysis in Large-Scale Dynamic Process Simulation J. R. Paloschi Adaptive Neural Network Model Based Nonlinear Predictive Control of a Fluid Catalytic Cracking Unit Z. Nagy, S. Agachi and L. Bodizs Computer Design of a System of Predictive Control for a Continuous Process Purification of Bioproducts A. Mattedi and R. Maciel Filho Knowledge Based Modular Networks for Process Modelling and Control J. Peres, R. Oliveira and S. Feyo De Azevedo Computer aided and control of a rotary kiln incinerator E.T.I. de Souza, R. Maciel Filho and E. Tomas The use of process dynamic simulation for learning to design digital controllers M.S. Basualdo, J. Salcedo B and D. Ruiz Model Based Control of Batch Chromatography G. Diinnebier and K.-U. Klatt
175 181
187
193 199
205
211
217
223
229
235
241 247 253 259 265
Model Predictive Control of an Industrial Dryer V.M. Cristea, M. Baldea and ~;.P. Agachi Approximate Dynamic Models Using Coarse Grid Parameter Relaxation V.J. Law Analysis of different control possibilities for the Divided Wall Column: feedback diagonal and Dynamic Matrix Control M. Serra, M. Perrier, A. Espuna and L. Puigjaner Control Strategies for Brine Electrolysis by Ion Exchange Membrane Cell Process S. P. A gachi and fl. Imre-Lucaci A new methodology for the active control of the heat transfer in Autoclave technology V. Antonucci, M. Giordano, S. Inserra and L. Nicolais Model Predictive Control: A Multi-Parametric Programming Approach A. Bemporad, N.A. Bozinis, V. Dua, M. Morari and E.N. Pistikopoulos Flowsheet Simulation for the Steel Industry- Using Experiences from Chemical Engineering and Modern Software Approaches H. Miiller, T. Peuker and G. Wozny Some aspects of rate-based modelling and simulation of three-phase distillation columns E. Eckert and T. VanOk Modeling and Simulation Tools for Supercritical Fluid Processes S. Diaz, S. Espinosa and E.A. Brignole A computer aided tool for heavy oil thermal cracking process simulation R. Maciel Filho and M.F. Sugaya Natural Gas Fired Power Plants with CO2-Capture- Process Integration for High Fuel-to-Electricity Conversion Efficiency H.M. Kvamsdal, T. Andersen and O. Bolland Simulation of convective drying of multicomponent moisture in a computer code MultidryPAK Z. Pakowski An Algorithm for Analysis of Eletrolytic Liquid-Liquid Extraction Process for Concentration of Organic Acids R.T.P. Pinto, L. Lintomen, A.J.A. Meirelles and M.R. Wolf-Maciel Estimation of the heat released by chemical reactions: Application to control of a simulated batch reactor F. Xaumier, M.-V. Le Lann, M. Cabassud and G. Casamatta Modelling and Simulation of Biotechnological Processes: BIOSIMA Package suitable for Integration in Process Engineering Tools U. Bergstedt, H.J. K6rner, S. Kabasci and G. Deerberg Simulation and Optimisation of Atmospheric and Vacuum Distillations of a Lube Plant F.G. Martins, M.A.N. Coelho, C.A.V. da Costa, M.A.S. Jer6nimo,
271 277
283
289
295 301
307
313 319 325
331
337
343
349
355
xi
C. Martins and A.S. Braga A coalescence and breakup module for implementation in CFD-codes L. Hagesaether, H.A. Jakobsen, K. Hjarbo and H.F. Svendsen Fluid Dynamics and Thermochemical Simulation of a Smelting Cyclone M. Modigell and M. Weng Computational Fluid Dynamics Modelling of Multiphase Reactors M. Bistolfi, N. Mancini and F. Podenzani Simulation of silica deposition in an Atmospheric Pressure Chemical Vapour Deposition reactor, using a modified CFD software J.P. Nieto, B. Caussat, J.P. Couderc, C. Artufel, S. Coletti, L. Jeannerot and O. Simonin Validation of a CFD model of a novel recycle axial flow cyclone for droplets removal from gas streams D. Stanbridge, R. Swanborn, C.P. Heijckers and Z. Olujic Simulating Flow and Heat Transfer in Tubes Using a Fast CFD Formulation E. R. L. Mercado, V. C. Souza, R. Guirardello and J. R. Nunhez Improving the Flow of Stirred Vessels with Anchor Type Impellers S. M. C. P. Pedrosa, C. G: Duarte and J.R. Nunhez Influence of turbulence modelling and grid discretization on the simulation of flow-forces on tubes in cross-flow K. SchrSder and H. Gelbe A CFD - Finite Volume Method to Generate Deterministic Model: Application to Stirred Tank Reactors R. Maciel Filho and V.M.F. Bezerra Simulation of Nox formation of glass melting furnaces by an integrated computational approach: CFD+Reactor Network Analysis D. Benedetto, M. Falcitelli, S. Pasini and L. Tognotti CFD-Analysis of Heat Transfer and Initiator Mixing Performance in LDPE High Pressure Tubular Reactors F.O. Miihling, A. Daifi, N. Kolhapure and R.O. Fox Dynamic Simulation of Complex Reaction Schemes and Biochemical Applications in Stirred Tank Reactors with Respect to Imperfect Mixing U. Boltersdorf G. Deerberg and S. Schliiter The steady state analysis of the twin helix heat exchanger E.D. Lavric and V. Lavric Simulation of the bubble formation dynamics in rheologically complex fluids H.Z. Li and Y. Mouline Coarse-grained formulation for the time evolution of intermaterial contact area density in mixing systems A. Adrover, M. Fidaleo and M. Giona Dynamic Optimization of Semicontinuous Emulsion Copolymerization Reactions: Composition and Molecular Weight Distribution C. Sayer, G. Arzamendi, J.M. Asua, E.L. Lima and J.C. Pinto
361 367 373 379
385
391 397 403
409
415
421
427
433 439 445
451
457
xii
Optimizing the Operation of a Sequential-Simulated Moving-Bed Separation Process Using MINLP S. Karlsson, F. Pettersson, H. Skrifvars and T. Westerlund Multiperiod Planning for a Utility System Considering Emergency Situation by New Approach J.H. Kim, S. Ju, C. Han and S.H. You Minimization of Natural Gas and Water Consumption in the Operation of Utility Plants S. M. Corvaldn and A. M. Eliceche Dynamic optimization of chemical and biochemical processes using restricted second order information E. Balsa-Canto, J.R. Banga, A.A. Alonso and V.S. Vassiliadis Interaction Between Process Plant Operation and Cracking Furnaces Maintenance Policy in an Ethylene Plant E. Schulz, S. Diaz and A. Bandoni Convergence Refinement of Stochastic Optimization by Coupling a Genetic Algorithm and a Simulated Annealing Procedure A. Davin, C. Azzaro-Pantel, P. Floquet, L. Pibouleau and S. Domenech Fuzzy Modeling of Catalytic Multi-phase Reactor B.B. Freitas Jr. and R. Maciel Filho Strategy and Mathematical Development for Scale-Up of Molecular Distillators for Recovering Carotenoids from Palm Oil C.B. Batistella, E.B. Moraes, M.R.W. Maciel and R. Maciel Filho Characterization and quantification of liquid distribution in a packed column on a pilot scale M.S. Kobayasi, M.R. Wolf-Maciel, F.A.N. Fernandes, D. Moraes Jr. and S. M. Pizzo Sensitivity in Optimization of a Reactor System with Deactivating Catalyst I. Lovik, M. Hillestad and T. Hertzberg Detailed Mathematical Modelling of Membrane Modules J.I. Marriott, E. Sorensen and I.D.L. Bogle A novel approach to the analysis of distillation columns for multicomponent mixtures A.R. Giona, M. Giona and L. L.M. Lombardi ROME: A Repository to Support the Integration of Models over the Lifecycle of Model-based Engineering Processes L. Von Wedel and W. Marquardt Increase business benefits by using on-line models. D. Dempf and T. List Symbolic Discretization of Population Models for Process Simulation M. Brahmadatta, R. KShler, A. Mitrovid, E.D. Gilles and M. Zeitz
463
469
475
481
487
493 499
505
511 517 523
529
535 541 547
xiii
Heat Integration in Process Design and Retrofit- Software Tools and Data Interchange E. Aust, S. Scholl and C. Ubler
553
Modelling and optimisation of polymerisation reactors in gPROMS M. Asteasuain, S.M. Tonelli, A. Brandolin and J.A. Bandoni
559
Modeling Particle Size Distribution (PSD) in Emulsion Copolymerization Reactions in a Continuous Loop Reactor P.H.H. Arat~jo, J.C. de la Cal, J.M. Asua and J.C. Pinto
565
Process modelling of metallurgical processes- software tool and modelling concept M. Modigell, A. Traebert, P. Monheim, S. Petersen and U. Pickartz
571
Modelling High Pressure Extraction Processes M. Skerget and Z. Knez
577
Waterless wool cleaning process with supercritical carbon dioxide: extractor modeling and optimisation F. Trabelsi, J-C. Luc, J. Miquel, M-A. Larrayoz, M. Capilla and F. Recasens
583
Equation Based SPYRO | Model and Solver for the Simulation of the Steam Cracking Process M.W.M. van Goethem, F.I. Kleinendorst, C. van Leeuwen and N. van Velzen
589
A Shortcut Method for Design and Synthesis of Multicomponent Thermally Coupled Distillation Flowsheets B.G. Rong, A. Kraslawski and L. Nystrdm
595
A heating-cooling management to improve controllability of batch reactor equipped with a mono-fluid heating-cooling system H. Bouhenchir, M. Cabassud, M.V. Le Lann and G. Casamatta
601
Evaluation of time varying parameters in polymerization reactors by means of Temperature Oscillation Calorimetry P. Guerrini De Luca, C. Scali and G. Maschio
607
Integer-Programming Based Algorithms and Computational Performance for Terminal-Drop Zone Assignment Problems M- T. Kong and N. Shah
613
Automatic Generation of Switching Start-Up Schemes for Chemical Processes E. Klein, A. Itigin, J. Raisch and A. Kienle
619
Creative Design of Distillation Flowsheets Based on Theory of Solving Inventive Problems B.G. Rong, A. Kraslawski and L. Nystrdm
625
Technological change by system design- the industrial production of aromatics G.P.J. Dijkema and J. Grievink
631
Symmetric multiprocessing algorithm for conceptual process design E.S. Fraga
637
xiv
Optimisation of distillation and pervaporation system for ethanol dehydration Z. Szitkai, Z. Lelkes, E. Rev and Z. Fonyo Shape and Terminal Velocity of Single Bubble Motion: a Novel Approach G. Bozzano and M. Dente The Myth of Decomposition P. Kesavan and P.I. Barton Parameter Analysis and Optimization of Ideal Heat Integrated Distillation Columns (HIDiC) M. Nakaiwa, K. Huang, K. Naito, A. Endo, M. Owa, T. Akiya, T. Nakane and T. Takamatsu Computer-aided screening of adsorbents and porous catalyst carriers F. St@dnek, M. Marek, M. Kubi~ek and P.M. Adler A Hierarchical Framework for Modelling Biopharmaceutical Manufacture to Address Process and Business Needs S. Farid, J. Washbrook, J. Birch and N. Titchener-Hooker Study of the insertion of partial oxidation gas turbine to satisfy high temperature requirements of industrial processes using energy integration techniques F. Marechal and B. Kalitventzeff Abstract design in the development of pharmaceutical processes M. Sharif N.J. Samsatli and N. Shah Batch Distillation of Azeotropic Mixtures in a Column with a Middle Vessel M. Warter and J. Stichlmair Development and design of a forced unsteady-state reactor through numerical simulation M. Cittadini, M. Vanni, A.A. Barresi and G. Baldi Intent and Rationale in the Design of Chemical Processes: A Case Study A. Guzmdn-Reyna and R. Bafiares-Alcdntara Energy Efficient Distillation by Optimal Distribution of Heating and Cooling Requirements T.R. Andersen, G. Siragusa, B. Andresen, P. Salamon and S.B. Jorgensen Optimal Design of Heat-Integrated Multipurpose Batch Facilities A.P.F.D. Barbosa-P6voa, T. Pinto and A.Q. Novais Plant-independent Process Representation K. Wall, P.N. Sharratt, N. Sadr-Kazemi and J.N. Borland The design and management of material cycles towards a functional specification for an awareness-tool E. V. Verhoef G.P.J. Dijkema and M.A. Reuter A Strategy for the Generation of Robust Accident Scenarios in Quantitative Risk Assessment Using Multi-Component Analysis K.H. Kim, J.H. Song, D. Shin and E.S. Yoon
643 649 655
661 667
673
679 685 691
697 703
709 715 721
727
733
XV
Simulation of Blowout Events: Integration of Different Modelling Approaches Within Industrial Risk Assessment and Management Tools N. Mancini, F. Podenzani, M. Bonuccelli, P. Andreussi, P. Blotto and R. Galinetto
739
Fault diagnosis system support for reactive scheduling in multipurpose batch chemical plants D. Ruiz, J. Cant6n, J.M. Nougu~s, A. Espu~a and L. Puigjaner
745
Improving on chemical process safety through distributed computer assisted knowledge analysis of preliminary design B.A. Schupp, S.M. Lemkowitz, L. Goossens, H.J. Pasman and A.R. Hale
751
Plant Monitoring and Fault Detection:Synergy between Data Reconciliation and Principal Component Analysis T. Amand, G. Heyen and B. Kalitventzeff
757
Note on vapour disengagement dynamics modelling A. Sogaro, M.L. Caldi, D. Franchi and G. Biardi
763
Computer aided transportation risk assessment R. Bubbico , S. Di Cave and B. Mazzarotta
769
Using PHA Results for Real Time Operator Support during ASM S. Dash and V. Venkatasubramanian
775
Leak Detection and Localisation in Pipes and Pipelines G. Geiger, W. Gregoritza and D. Matko
781
Industrial Applications of Intelligent Systems for Operating Procedure Synthesis and Hazards Analysis for Batch Process Plants J. Zhao, S. Viswanathan and V. Venkatasubramanian
787
Model-based safety verification under uncertainty H. Huang, C.S. A djiman and N. Shah
793
Computerized Screening of Chemicals for Energy Release Hazards B. K. Harrison
799
A Hybrid Modular Hierarchical Approach for Fault Diagnosis in Complex Transient Processes N. Scenna, B. Drozdowicz, S.J. Benz and E.J. Lamas
805
Dynamic Simulation of the Behaviour of Pressure Relief Systems J-P. Pokki, J. Aittamaa and M. Hurme
811
From Environmental Care to Sustainability: the role of CAPE tools and methods J. Venselaar
817
Accounting for sustainability requirements in process design M. P. C. Weijnen, P.M. Herder and H.D. Goel
823
An Intelligent System for Identifying Waste Minimization Opportunities in Chemical Processes I. Halim and R. Srinivasan
829
xvi
A CAPE tool for evaluation of adsorber-reactor systems for treatment of exhausts from mobile sources J. Jirdt, M. Kubigek and M. Marek Quick identification of the wastewater biological treatment process by using shortcut techniques and previous plant operation data G. Maria, C. Constantinescu and P. Ozil Implementation of Flue Gas Cleaning Systems into an Object-Oriented Process Simulator for Practical Use G. Schuster, K. Weigl and A. Friedl Dynamic optimisation of small size wastewater treatment plants including nitrification and denitrification processes B. Chachuat, N. Roche and M.A. Latifi A New Procedure for Predicting NOx Emissions from Furnaces T. Faravelli, L. Bua, A. Frassoldati, A. Antifora, L. Tognotti and E. Ranzi Liquid Effluent Properties Prediction from an Industrial Wastewater Treatment Plant Using Artificial Neural Networks C.A. Gontarski, P.R. Rodrigues, M. Mori and L.F. Prenem Incorporating Production Scheduling in the Optimal Operation of Wastewater Treatment Plants R. Gouveia and J. M. Pinto Analysis of the Operation of a NSDX Pilot Plant for Cr(VI) Recovery A.M. Eliceche, S.M. CorvalCm, A.I. Alonso and I. Ortiz Optimum deNOx performance using inferential feedforward reductant flow control H.C. Kr~insen, J. C.M. van Leeuwen, R. Bakker, H.P.A. Calis and C. M. van den Bleek Software tool for waste treatment selection using economic and ecological assessments L. Cavin, O. Jankowitsch, U. Fischer and K. Hungerbiihler Distributed Information System For Environmentally Conscious Process Design Y. Fukushima and M. Hirao Decision Making for Batch Manufacturing Sites under Uncertainty A.A. Linninger and A. Chakraborty Minimization of Water Consumption and Wastewater Discharge in the Sugar Cane Industry R. Pastor, L. Abreu, A. Espu~a and L. Puigjaner Hydrodynamics and Chemical Model to Evaluate Environmental Risks in Proximity of River Mouth M. Di Natale, G. Merola and D. Musmarra Simulation and optimization of the reactive absorption of HF/HNO3 during pickling acid regeneration W. Wukovits, W. Karner, A. Lebl, M. Harasek and A. Friedl
835
841
847
853 859
865
871 877
883
889
895 901
907
913
919
xvii
Trend recognition of process data of a refinery using wavelets B. Bitzer and J. Richters Comparison of methods for assessing human health and the environmental impact in early phases of chemical process development G. Koller, U. Fischer and K. Hungerbiihler An integrated framework of process and environmental models, and EHS constraints for retrofit targeting F. Nourai, D. Rashtchian and J. Shayegan Soft sensor development and experimental application to wastewater treatment process D. Zyngier, O.Q.F. Arahjo and E.L. Lima Computer Aided Technique for Pollution Prevention and Treatment P. M. Harper and R. Gani Using driving force based separation efficiency curves within an integrated system for process synthesis/design E. Bek-Pedersen, M. Hostrup and R. Gani Pairing Criteria and Tuning Methods Towards Integration for Heat Exchanger Networks S. Chetty and T.K. Zhelev A Conceptual Programming Approach for the Design of Flexible HENs L. Tantimuratha, G. Asteris, D.K. Antonopoulos and A.C. Kokossis Mass Exchange Network Synthesis by Coupling a Genetic Algorithm and a SQP Procedure S. Shafiei, A. Davin, L. Pibouleau, S. Domenech and P. Floquet Synthesis of Reactor Networks in Overall Process Flowsheets within Multilevel MINLP Approach B. Pahor and Z. Kravanja Synthesis of reactive and extractive Distillation Units Using Distillation Line Diagrams L. Jimdnez, O.M. Wanhschafft and V. Julka Separation System Synthesis of Fractional Crystallisation Processes With Heat Integration L.A. Cisternas, C.P. Guerrero and R.E. Swaney Optimization of bleed streams in evaporation systems based on pinch analysis: new approach D.L. Westphalen and M.R. Wolf-Maciel Knowledge Based Models for the Analysis of Complex Separation Processes P.B. Shah and A.C. Kokossis Synthesis of Solvent Extraction Separation Schemes in Hydrometallurgy L.A. Cisternas and E.D. Gdlvez Synthesis of Separation Processes by Using Case-based Reasoning E. Pajula, T. Seuranen, T. Koiranen and M. Hurme
925
931
937
943 949
955
961 967
973
979
985
991
997 1003 1009 1015
xviii
An analytical process performance model for batch distillations S.D. Zamar, S. Xu and O.A. Iribarren Synthesis of Heat Exchanger Networks Considering Stream Splitting and the Rigorous Calculation of the Heat Transfer Coefficient According to the Bell Delaware Method M. C. Roque and L. M.F Lona Using Conceptual Models for the Synthesis and Design of Batch Distillations J. Espinosa, E. Salomone and S. Xu Mixed Integer Linear Programming and Constrained Based Search Approaches in a Multipurpose Batch Plant Short Term Scheduling Problem L. Gimeno, M. T.M. Rodrigues, L.C.A. Rodrigues and W. Alvarenga A Continuous-Time Approach to Short-Term Scheduling of ResourceConstrained Multistage Batch Facilities C.A. Mdndez, G.P. Henning and J. Cerdd A Comparative Study of Bi-linear and Tri-linear Approaches for the Monitoring of an Industrial Batch Process X. Meng, E.B. Martin and A.J. Morris Planning and Scheduling in a Pharmaceutical Research and Development L. Mockus, J. Vinson and R.B. Houston Mixed Integer programming techniques for the scheduling of fuel oil and asphalt production J. M. Pinto and M. Joly A Time-Windows Approach for Enhancing the Capabilities of Batch Scheduling Systems: An Application to Simulated Annealing Search L.C.A. Rodrigues, M. Graells, J. Canton, L. Gimeno, M.T.M. Rodrigues, A. EspuBa and L. Puigjaner A Hierarchical Approach for Real-Time Scheduling of a Multiproduct Batch Plant with Uncertainties G. Sand, S. Engell, C. Schulz, A. Markert and R. Schulz Closed-Loop Implementation of Optimal Operating Policies in Batch Distillation M. Barolo and P. Dal Cengio Development of an efficient system for scheduling large-scale industrial process and consumer goods factories with multipurpose and flexible storage C. Charalambous, T. Tahmassebi and K.S. Hindi Optimal Cyclic Operation of Biomass Production B.H.L. Betlem, P. Mulder and B. Roffel Short-term Scheduling and Recipe Optimization of Blending Processes K. Glismann and G. Gruhn Planning and Maintenance Optimization for Multipurpose Plants C.G. Vassiliadis, J. Arvela, E.N. Pistikopoulos and L.G. Papageorgiou
1021
1027 1033
1039
1045
1051 1057
1063
1069
1075
1081
1087 1093 1099 1105
xix
A Mathematical Programming Approach for the Optimal Scheduling of HeatIntegrated Multipurpose Plants under Fouling Conditions M. C. Georgiadis and L. G. Papageorgiou
1111
Development of Batch Process Operation Management Platform A. A oyama, I. Yamada, R. Batres and Y. Naka
1117
Synthesis, experiments and simulation of a heterogeneous batch distillation process I. Rodriguez-Donis, E. Pardillo-Fontdevila, V. Gerbaud and X. Joulia
1123
Robust Mixed Stochastic Enumerative Search Technique for Batch Sequencing Problems M. Graells, J. Cant6n and L. Puigjaner
1129
Systematic Assessments of Uncertain Demand Effects on Multiproduct Batch Process Design Using a Multi-Objective Optimization Technique H.I. Park, C. Azzaro-Pantel, P. Floquet and LB. Lee
1135
A Mixed Integer Model for LPG Scheduling J.M. Pinto and L.F.L. Moro
1141
Simulation-aided Implementation of Supervisory Control for Industrial Batch Reactors K. Preufi, M.V. Le Lann, M. Cabassud, G. Anne-Archard and G. Casamatta
1147
Dynamic modeling of batch distillation: comparison between commercial software L. Jimbnez, M.S. Basualdo, L. Toselli and M. Rosa
1153
Design, Synthesis and Scheduling of Multipurpose Batch Plants via an Effective Continuous-Time Formulation X. Lin and C.A. Floudas
1159
A Novel Superstructure and Optimisation Scheme for the Synthesis of Reaction-Separation Processes P. Linke, V. Mehta and A. C. Kokossis
1165
Short term Product distribution plan for multisite batch production, warehousing and distribution operations: solutions through Supply-Demand network and Resource-Task network optimisation approaches B.P. Das, N. Shah, A.D. Dimitriadis and B. Cao
1171
A New Generation in ERM Systems: The Tic Tac Toe Algorithm M. Badell and L. Puigjaner
1177
AUTHOR INDEX 1183
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European Symposiumon Computer Aided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
MIXED INTEGER NON-LINEAR PROGRAMMING USING CUTTING PLANE TECHNIQUES Ray P 6 m a and Tapio Westerlund b
a Department of Mathematics, * b o Akademi University F~nriksgatan 3B FIN-20500 ,~bo, F I N L A N D (email"
[email protected]) b Process Design Laboratory, Abo Akademi University Biskopsgatan 8, FIN-20500 * b o , F I N L A N D (email" twesterl @abo.fi) In the present paper a modification of the extended cutting plane (ECP) method is described and illustrated. It is shown how it is possible to solve general MINLP (Mixed Integer Non-Linear Programming) problems with pseudo-convex objective as well as constraints to global optimality by a sophisticated cutting plane approach. The method relies on the ability to construct valid cutting planes for the entire feasible region of the problem. The method is illustrated on a simple test example and on some demanding practical scheduling problems. A comparison with a recently developped branch-and-bound approach is also given. 1. INTRODUCTION The extended cutting plane method (ECP) originally introduced in Westerlund and Pettersson (1995) is an extension to the mixed integer case of the classical cutting plane method for convex NLPs by Kelley. In Westerlund et al. (1998) the ECP method was extended to cover general MINLP problems with pseudoconvex inequality constraints. The convergence properties of this method is more rigorously analyzed in Still and Westerlund (2000). In the present paper the problem domain is even further enlarged to also include the case of a pseudo-convex objective function. It is shown how MINLP problems with pseudoconvex objective and inequality constraints can be solved to global optimality by a modification of the cutting plane approach used in the ctECP method. The method is illustrated through a simple example and four different instances of a cyclic scheduling problem. There exist only a few methods in the literature which are able to solve pseudo-convex MINLP problems to global optimality. To the authors knowledge, the only MINLP method previously published in the literature which directly addresses the problem of minimizing a pseudo-convex function (maximizing a pseudo-concave) is a branch-and-bound approach recently given in Jain and Grossmann (1998). Other common MINLP methods (e.g. Geoffrion (1972), Duran et al. (1986) and Fletcher et al. (1994)) have, in general, global convergence properties for convex problems only. In the area of deterministic global optimization there exist methods which are applicable to general non-convex MINLPs. This can be done by considering a binary variable in an explicit manner (e.g. Smith et al. (1999)) as a 0-1-variable or implicitly (e.g. Ryoo et al. (1995)) where a binary variable is considered continuous and modeled, for example, using concave equalities. Although, global optimization methods are applicable to general MINLPs the computational effort of these approaches increases rapidly with the number of variables (especially binary/integer) and constraints in the model. Therefore, these methods are only applicable to MINLP problems with moderate size.
b corresponding author
2. THE MINLP P R O B L E M
The MINLP problem considered in this paper can be formulated as follows
min
{ f (z) }
z~ N n L
N = {zl g(z) < 0}
L={zlAz
This variant is globally convergent if f ( z ) is a linear function, i.e. f ( z ) - c T z . The case with a convex objective is also included since after an introduction of a dummy variable /t the problem (P) is equivalent to minimizing /1 subject to the original constraints plus the convex constraint f ( z ) - l t
< O.
This rewriting procedure is, of course, also valid if f ( z ) is pseudo-convex but in this case the additional constraint will no longer, in general, be pseudo-convex and the global convergence properties of the method may be lost. The problem with a linear objective and pseudo-convex inequality constraints is solved by the 0tECP method by a sequence of MILP sub-problems of the form
min { crz } z~f~t
a k =La{zllj(z)
.... Jk}
k = 0,1,2 .... K The sub-problem solved in iteration k is called (Pk). The procedure is initialized with g~0 = L. The linear functions lt(z) added in iteration k are of the form lj(z)=fft +a tk .Vg~t r (z-zk), j~ {Jk-, +1 ..... J~} where each fit corresponds to the function value of a particular non-linear constraint g i(z), i = 1..... n, with positive value at zk. The vector V~t is the corresponding gradient of the non-linear constraint at zk. If only one cutting plane is added in each iteration then gt =rnax{gi(zk)} and Jk =Jk-I -I-1. Initially t
ark = 1 , but these values are sequentially increased in order to make all linearizations valid cutting planes at termination. For convex constraints
o~tk _ 1 is enough. In order to ensure global convergence three
essential steps are required: I.
If zk is infeasible in (P) then this point is excluded from all subsequent iterations by each of the constraints lj (z) < 0 added at the point zk (since lt(z ~) = ~t > 0 ).
II. Every lt(z) added to the sub-problems must, at termination, be a valid cutting plane (not cutting off parts of N n L). In order to ensure this an up-dating strategy for the o~k -parameters is required. The up-dating strategy is activated before termination and after detection of infeasible MILP sub-problems.
III.A solution z, to (pk) is optimal in (P) if z, ~ N n L and N n L c f ~ . That is z, is feasible in (P) and all linearizations lj(z) are valid cutting planes. In step II an up-dating procedure is mentioned. The up-dating is usually of the form tzj.,+1 = ft. o~k j where fl > 1. Both feasibility and under-estimation must be checked. feasibility: max{gi(z k )}< eg
under-estimation: o~jk _>--~-, j = l , 2 .... Jk
t
where h = ffj for a linearization that corresponds to a convex constraint and h = e h if it is obtained from a pseudo-convex. Both
F_.g and e h
are small positive tolerances, that depends on the problem. A detailed
description of the method is given in Still et al. (2000) and Westerlund et al. (1998). 4. I N C O R P O R A T I O N OF A P S E U D O - C O N V E X O B J E C T I V E F U N C T I O N
Now we consider problem (P) equipped with a pseudo-convex objective. As a first approach we will construct a procedure that computes a sequence of improving upper bounds on the objective. The limit point of this sequence can be proven to the optimum of (P). This procedure can be view as a sequence of MINLP sub-problems (Pr~ ) of the form: min z~ N, oL'~ ,/z<0
{.}
N r "- (z E N I f ( z ) <~ f ( z ~ )} 9 1-
The set
,2 . . . . .
N r contains all non-linear constraints from N plus an additional, so called, reduction
constraint f (z) < f(Zlr ). The linear set L~r contains all original linear constraints plus all linearizations of the objective at the active contour. Thus, the index r counts the number of different contours visited and l the number of solutions on each contour. The procedure is initialized with N o = N and L~ = L . The solution to this "feasibility" problem is a point zl1 e N n L and the first proper upper bound f(z~) is found. The pseudo-convex objective in (P) is then in the first real iteration replaced with a linearization and the pseudo-convex reduction constraint f ( z ) < f(z~ ) is included. If the reduction constraint is active at the next solution a new linearization is added to Lll and L2 will contain two linearizations. On the other hand, if the reduction constraint is redundant a strict improvement in f ( z )
is obtained and all
previous linearizations of f ( z ) will be replaced by a single linearization at the solution. Thus, L~ will contain one linearization and the reduction constraint is strengthened to f ( z ) < f(zl2). Note that problem (Pr") has a linear objective and pseudo-convex inequality constraints and can therefore be solved to global optimality by the o~ECP method. Since the procedure is iterative it can be view at an MILP-level instead of at the MINLP-level. Thus, the solution procedure can fully be integrated within the framework of the ~ECP approach. This means that we can also use previously generated cutting planes for the reduction constraint and the original constraints when solving subsequent sub-problems. Other computational improvements are discussed below. The proof of convergence of the method relies mainly on the definition of a pseudo-convex function. A function f ( z ) is pseudo-convex on a convex set S c R" if f (Zl) < f (z2) ::> V f (z2) r (zl - z2) < 0 for every z~, z2 ~ S. Theorem of optimality. If z* is a solution to sub-problem with/1 = 0 then z* is optimal also in (P).
1 < l.l must be active at Proof. Since we are minimizing /t at least one of the constraints V f ( z , I)T ( z - Zr)
'f(Zrl ) T (Z * -- Z ,)1 = / t = O. From the definition of pseudo-convexity it follows that if there exists a point, ~,e N r n L~, with f(~,) < f(Z~r ) r 1 T (z * - z r )1 = 0 it implies Vf(z~)r(~-z~)
V
conclude that there exists no point ~,~ N r t'~ Lnr with f(~') < f(z~ ) = f(z*). Thus z* is optimal n (P). [] A non-optimal point will, due to the pseudo-convexity, always correspond to a solution with strictly negative /1-value. Thus, the only additional termination criteria required, besides the feasibility and under-estimation criteria previously mentioned in the section of the aECP method, when incorporating a pseudo-convex objective function is the simple check /t = 0 or I/tl < el, in practice. A situation with /1 = 0 will always occur. This is due to fact that the method produces a sequence of sub-problems with feasible regions, N r ~ Znr, of decreasing size (due to the strengthening of the reduction constraint) and in the limit we will h a v e / t = O. 5. C O M P U T A T I O N A L I M P R O V E M E N T S The following improvements have been implemented to increase the computational efficiency and stability of the method. I. Line-search for the contour. Assume that we have obtained points on the same contour r, z~ ..... z n , and that the next MILP solution, ZMILp,does not satisfy the reduction constraint, i.e. f(ZMILp) > f(z~). It is now possible to compute a point on this contour without solving the entire MINLP problem (Pr~) to optimality or feasibility. First compute the "midpoint" ~mid = 1. ~'~=IZtr of the points on the contour. n
Since f('Zmid ) <- f ( z ~ ) (due to convexity of the level-sets off) and f(zM,,~p) > f(z~) (assumption) a point
segment
f ( z ) = f(z~) can be found using a linesearch strategy. The objective can now be linearized at the relaxed point ~r~+l=2*"Zmi d "~-(1--2*)ZMILP
on the line
[ZMILP,"Zmi d ] located exactly on the contour
where 2* is obtained with the line-search strategy. Due to the continuous nature of the line-search the integer restrictions may be violated at ~'r"+~ . Instead of "solving" ( p n ) a s a MINLP problem we have solved it as a single MILP combined with a simple line-search. II. Line-search for the minimum. Suppose that a MILP solution ZMlt.p that violates the reduction constraint has identical integer part to one of the points on the contour, say Zlr ( y~ = YM~LP)" Since the integer parts match we can in this case find the minimum
of
f(z)
on the line segment
"1
1
I instead of locating the contour. In this case the obtained point will be feasible,
i.e. it satisfies all integer restrictions, since the minimization was performed in the continuous space only. This might improve the convergence on the continuous variables. III. Normalization. To avoid non-optimal termination for extreme cases, for example objective functions with almost constant "plateaus" ( V f ( z ) = 0 ) , it is possible to normalize the linearizations in L~. That is, the linearizations are replaced by their normalized counterparts according to (the norm is the Euclidean)
L"r= Z~LIIVI
r
6. N U M E R I C A L E X A M P L E S In this section we apply our method on some examples. Consider the small MINLP problem .B
min,.y{1.1-((2x-1 O)2+(y-5)2) +sin(q.x-1 O)2+(y-5) 2) } 0.7x+y_<7, 2.5x+y<19, O<x,y
rel
6.
The objective is pseudo-convex by convex analysis. The feasible region is drawn with bold lines and the contours of the objective with thin lines. The iterates are marked with dots. The filled dots correspond to MILP solutions and the circles to relaxed points, which are solutions to line-searches. The method converged within 8 iterations, which included the solution of a total of 8 MILPs and 3 one-dimensional line-searches. Line-search strategy 1 was used.
4
2
0
I
. ~,
X
9 ,:~, . . i , , , ~,
F '7 -8
Figure 1: Solution sequence for the example. zm = (0.00,0), z2 = (7.60,0), z3 = (0.00,7),
Test problems. In this section we apply our method on z3r'' = (1.41,5.70),z4 = (4.50,3),z5 = (6.80,2), four different instances of a demanding cyclic scheduling problem recently published in the literature z~" = (5.04,2.77), z6 = (4.29,4), z7 = (7.60,0), (Jain and Grossmann (1998)). These problems are ZT"' = (4.89,3.27), z8 = (4.29,4) optimal MINLP models representing the problem of scheduling multiple feeds on parallel units (furnaces), where the performance of each unit decreases with time. The optimization trade-off is therefore between the performance of the units, the maintenance costs and the loss in production due to shutdowns for cleaning. The problem is linearly constrained with a fractional pseudo-convex objective (originally maximization of a pseudo-concave objective) of the form, (convex function)/(linear function), that includes both continuous and integer variables. The computational results and a comparison with Jain and Grossmanns branch-and- bound method is given in Table 1. The abbreviations bfs and dfs in the B&B (branch-andbound) rows stands for breadth first search and depth first search respectively. The instance-row gives the "dimension" of the problem as (number of feeds)x(number of furnaces). The total number of continuous and binary variables as well as constraints is also given. The abbreviation 1-s corresponds to the number of line-searches applied in the cutting plane approach. The two first instances are taken explicitly from the paper by Jain and Grossmann and we have generated the last two. In the paper by Jain and Grossmann it was also found that the OA method (dicopt++) converged to a non-optimal point in some of the problems.
Instance
3x1
7x4
5x2
cont/bin
11112 931140 37/50
52/60
9
-100
20
140
57
15/9 0.25
144/131 84.0
13/8 0.8
3 0.22
85 49.0
no result
-80
, Im:m.
'~ .,,,~.
Cutting plane
iter (MILPs/1-s) cpu-tirne (sec) B&B (dfs) iter (NLPs) cpu-time (sec) B&B (bfs) iter (NLPs) cpu-time (sec)
I"K]O~ 16013130.
# of constraints
linear
Solution results for inslance 7x4
4x3
# of variables '"
m
85168 6.9 ~" 11~110 no
result
-
1100130-
9
100000,
0
3 0.33
>2000
no result
no
,
,
~
40
,,
,
.
60
80
.
.
100
.
0
1~
140
itera~on mm'i3er
result
Table 1: Problem characteristics and solution results for the scheduling problems
Figure 2: Objective value and CPU-time versus
the iteration number.
p+
The lower bound on the objective (since the original problem is a maximization problem) and the cumulative CPU-time are plotted versus the iteration number in Figure 2 for the instance 7x4. Already after about 40 iterations (-15 CPU seconds) the lower bound is less than 0.2% from optimal. The proposed cutting plane method visited a total of 11 different contours, starting from a value of 105239 at the first. The cutting plane procedure is implemented within the ctECP program package (Westerlund and Lundqvist 2000). The program is written in FORTRAN-90 equipped with a Visual Basic interface. The CPLEX 6.0 package is used to solve the MILP sub-problems. The NLP-code used in branch-and-bound method was MINOS connected to GAMS. The computer used by Jain and Grossmann was a HP 9000/C 110 workstation and we used a 500 MHz Pentium III PC. The efficiency of the branch-and-bound method with dfs was according to the authors due to a very small integer gap. Branch-and-bound with bfs did not converge within 2000 NLPs for the instance 7x4. 7. CONCLUSIONS In this paper we have described an iterative MINLP method for minimizing a pseudo-convex function subject to pseudo-convex inequality constraints. The procedure was first view as a sequence of MINLP sub-problems with linear objective and pseudo-convex constraints. It was thereafter embedded within the framework of an existing cutting plane method, the ~ECP method. An additional termination criterium for the case with a pseudo-convex objective was also derived. The computational efficiency of the proposed method was then improved using two different simple line-search strategies. Finally, the method was used to efficiently solve four instances of a demanding scheduling problem recently published in the literature. ACKNOWLEDGEMENT Financial support from the Nordic Energy Research Programme (NEFP) and from the Academy of Finland is gratefully acknowledged. REFERENCES Duran M.A. and Grossmann I.E. (1986). An Outer-Approximation Algorithm for a Class of Mixed Integer Nonlinear Programs. Mathematical Programming, 36, 307-339. Fletcher R. and Leyffer S. (1994). Solving Mixed Integer Nonlinear Programs by Outer Approximation.
Mathematical Programming, 66, 327-349. Geoffrion A.M.(1972). Generalized Benders Decomposition. Journal of Optimization Theory and
Applications, 10, 237-260. Jain V. and Grossmann I. (1998). Cyclic Scheduling of Continuous Parallel-Process Units with Decaying Performance. AIChE Journal, 44, 1623-1636. Ryoo H.S. and Sahinidis N.V. (1995). Global Optimization of Non-Convex NLPs and MINLPs with Applications in Process Design. Comp. & Chem. Engng., 19, 551-566. Smith E.M.B. and Pantelides C.C. (1999). A Symbolic Reformulation/Spatial Branch-and-Bound Algorithm for Global Optimization of Non-convex MINLPs. Comp. & Chem. Engng., 23, 457-478. Still C. and Westerlund T. (2000). The Extended Cutting Plane Algorithm. Chapter in: Encyclopedia of Optimization, Ed: Floudas C.A. and Pardolos P.M., Kluwer Academic Publishers. Westerlund T. and Lundqvist K. (2000). Alpha-ECP Version 4.0. An Interactive MiNLP-solver Based on the Extended Cutting Plane Method. Report 0-170-A. Process Design Laboratory, ]kbo Akademi University. ISBN 952-12-0606-3. Westerlund T. and Pettersson F. (1995). An Extended Cutting Plane Method for Solving Convex MINLP Problems. Comp. & Chem. Engng., 19, S 131-S 136. Westerlund T. Skrifvars H., Harjunkoski I. and P6rn R. (1998). An Extended Cutting Plane Method for a Class of Non-Convex MINLP Problems. Comp. & Chem. Engng., 22, 357-365.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
A Novel Interactive M I N L P Solver for C A P E Applications Jens Peter Henriksen, Soren F. Stoy, Boris M. Russel & Rafiqul Gani CAPEC, Dept. of Chem. Eng., Technical University of Denmark, 2800 Lyngby, Denmark
1 Abstract This paper presents an interactive MINLP solver that is particularly suitable for solution of process synthesis, design and analysis problems. The interactive MINLP solver is based on the decomposition based MINLP algorithms, where a NLP sub-problem is solved in the innerloop and a MILP master problem is solved in the outer-loop. The interactive MINLP solver allows a purely equation system mode (where all equations are given explicitly) and a simulation mode (where the process model equations are included and solved by the simulation engine of a simulator). These solution modes give flexibility in problem formulation and inclusion of different solution approaches. Applications of the interactive MINLP solver to the solution of typical CAPE problems are illustrated through several test problems.
Keywords: Optimisation, MINLP, NLP, solver, synthesis, design, integrated systems 2 Introduction Current trend of integrated approaches in computer aided process engineering (CAPE) requires the use of mixed integer non-linear programming (MINLP) techniques. In integrated design and synthesis algorithms, it is necessary to determine the optimal flowsheet configuration simultaneously with the optimal process design conditions. This is formulated as optimisation problems involving both integer and real variables with equality/in-equality constraints that are linear as well as non-linear. Solution of MINLP problems requires robust and efficient numerical solvers. In addition, since the process flowsheet needs to be simulated during the solution of these MINLP problems, it is necessary to have available, the process model equations. Most currently available MINLP solvers require the process model equations to be given explicitly as equality constraints. While this approach has the advantage that the solution, if achievable, is obtained very efficiently, it also has some disadvantages. For example, the size of the optimisation problem may become too large to handle and it may not always be possible to supply all the process model equations. As a result, it may be difficult to formulate and solve realistic MINLP problems with such solvers. The objective of this paper is to present a novel new solver that is robust and efficient and at the same time, flexible, easy to use and does not need the process model equations to be explicitly supplied by the user. The new solver integrates a NLP solver, a MILP solver, a computer-aided tool for process model generation, a simulation engine and a user interface, which 'manages' the various tools during the solution of an optimisation problem. With this architecture, it is possible to solve optimisation problems where all equations are supplied
explicitly or solve problems where some equations are supplied explicitly and the rest supplied by the model library in the simulation engine or the modelling test-bed tool (MOT). The MINLP solution strategy is based on the Outer Approximation algorithm, where the problem is decomposed into an inner-loop NLP sub problem and an outer-loop MILP master problem. 3 Proposed Interactive MINLP Solver The outer approximation is described by Duran and Grossmann (1986) and Floudas (1995) for explicit MINLP problems. It is a decomposition technique where the problem is divided into an inner-loop NLP sub problem (giving and upper bound on the optimisation problem) and an outer-loop MILP master problem (giving a lower bound). The MILP master problem is generated by linearisation of the optimisation problem in the optimal point from the NLP sub problem. Convergence is reached, when the upper bound - lower bound _< e. The Outer Approximation is a rigorous method, which under assumptions of convexity guarantees an optimal solution. This paper will focus on how this algorithm can be integrated with a process simulator. The Outer Approximation integrated with the process simulator can be written as follows:
Problem:
NLP sub problem:
min f (d,x, y) d,y
v(y)
= inf f ( d , x , y
k)
MILP master problem: min ~lLiN d , y , [dUN
st.
st.
st.
P(d,x, y) = 0 h(d,x,y)=O g(d,x, y) < 0
P(d,x,
h ( d , x , y k) = 0 g(d,x,y k) <_ 0
d~DcR" xeXcR m
deDc xeXcR
y e Y = {0,1}q
y e Y = {0,1} q
y k) = 0
R" m
Pl~m >---f(dk,xk,Yk)+ Vd,yf(dk'xk,Y k)
yk
0 = h(dk,x k , yk ) + V d,yh(dk,x k , y k () d - d kyk 1 Y O>g(dk'xk'Yk)+Vd,yg(dk'xk'Yk)( d-dk)y yk
i~B k
i~NB k
9d e D c _ R " , x e X c R m , y e Y = { O , 1 } q Here f is the objective function, P constitutes the process models as formulated in the process simulator and which are thus implicit equality constraints in the optimisation problem, h and g are the equality and in-equality constraints given explicitly in the optimisation problem. The binary integer variables are represented by y, the process state variables are represented by x and the problem design variables are represented by d. The proposed integrated algorithm is then: 1. Chose a feasible combination of binary decision variables as starting point, yk=yl 2. Solve NLP sub problem for design variables in the process simulator with fixed yk. Optimal solution jf,.,h op, (d,x,y) and optimal d. 3. Upper Bound: UBD = fopf (d,x, y).
4. Linearise the optimisation problem by perturbing the design variables around the optimal solution from the NLP sub problem jf.,.,b op, (d,x, y) and do feasible step changes in the decision variables y. 5 9 Solve the MILP master problem. Gives fM opt (d,x,y) and optimal yg+l" 6. Lower Bound: LBD
=
fopM (d,x, y).
7. Is UBD-LBD < s a. Yes: Terminate algorithm. Optimal solution found. b. No: yk=yk+l. GO to step 2. The integer cut is introduced in the MILP master problem. This additional constraint can be used in an interactive manner either to filter out 'best solutions' or 'solutions' already tested 9 The linear process model is generated by makin~ perturbations in the design variables around the optimal point from the NLP sub problem d ' in the simulator and by 'measuring' the effect on the state variables x: k
E
x
(d',+,~)-a~'i=l,..,m,j=l,..,n
~Xj
X i -- X i
~~
~d,
Furthermore, the integer part of the MILP problem is modelled through feasible step changes in the y variables:
~'
= x,", Vy, ~ y" ,
Ayj
Ax," =x,-", vy, ~ y" Ay.j
Here 2~ is the numerical value of the state variable in a feasible step change of y . From the above we can now generate a linearised model using the following expressions for the state variables in the objective function and in the constraints:
x,:x:,
+
[~x~d
La
~';"
,
)+~:
y:) _
+...+
~;
y. _,
)1
The MILP master problem is solved with a standard branch-and-bound type MILP solver, for example, GAMS (Brooke et al. (1992)). The optimisation algorithm is integrated in ICAS in the way schematically shown in Figure 1. d, Fobj l g(d,x) ICAS SQP h(d,x) l User Interface Problem definition
[
~
d"ew d,x
I Evaluation uoc,ion
~ - Optimiser ~_Administrator] d, Fobj g(d,x) h(d,x)
T
I
ICASSIM ] Steady State ~ Simulator 1
[ MoT Modelling [Testbed_~
Figure 1" Integration of the optimisation algorithm in ICAS 4
Applications
A number of application examples have been developed using the optimiser in ICAS. As an example of NLP solution the Williams Otto Plant (Edgar & Himmelblau (1988), Biegler,
10 Grossmann & Westerberg (1997)) is tested. Production of cyclo-hexane is used as primary example of the MINLP algorithm. The superstructure proposed by Bek Pedersen et al. (1999) is solved using the interactive MINLP algorithm. Solutions of other examples (NLP and MINLP problems from the literature) can be obtained from the authors.
4.1 NLP example: Williams Otto Plant As an example of the implementation of the NLP solver we will present the traditional Williams-Otto plant (Edgar & Himmelblau (1988), Biegler, Grossmann & Westerberg (1997)). This example will illustrate how a problem, normally formulated as a system of equations, also can be solved in a sequential modular simulator. Flowsheet
Model and solution 100
min Z =
*
600 * 0.00002964 * F R * p 2 2 0 7 . 5 2 * Fp + 5 0 . 0 4 * F D - 1 6 8 "
F A - 252* F~
- 84 * F~ - 60 * 0 . 0 0 0 0 2 9 6 4 * F n * p
)
s.t. 0 = Fproa - 4 7 6 3
o = P(x)
Optimal: Z = 7197.71 Figure 2: Williams Otto Plant
Here A and B are reactants, P is the product, C and E can be sold as fuel, and G is a byproduct which has to be disposed of. The optimisation problem can be formulated in the way shown above, where P(x) denotes the process model, which comprise the mass balance equations. The results of the optimisation using both GAMS, ICAS in equation systems mode and ICAS in simulation mode are identical.
4.2 MINLP example: Cyclo-hexane Production As a primary example of the interactive M1NLP algorithm the determination of the optimal flowsheet for cyclo-hexane production is used. This rigorous process model employing rigorous property models, is highly non-linear. Bek-Pedersen et al. (1999) propose a superstructure containing the following variables: Configuration variables y 9 Reactors o 70% conversion o 80% conversion Separation systems o Distillation o Distillation + membrane o Extractive distillation
Optimisation variables x 9 Hydrogen feed 9 Purge fraction (inner loop) 9 Extractive distillation o Top product, dist. 1 o Boil-up ratio, dist. 1 o Reflux ratio, dist. 2 9 Distillation, Distillation + membrane o Reflux ratio; Reboiler temp.
11 The optimisation problem is formulated as a MINLP problem with the above integer variables and the following free continuous variables d: Note that the number of integer variables is reduced by removing some of the redundant separation systems (see Hostrup et al. 1999). The objective function is set up as: min Z = - Cl*Product-C2*ReactorHeatGeneration (Rx!, Rx2) + C3*FeedCost + C2*(Reboill + Reboil2 + Reboil3 + Reboil4)+ C4*(Condl + Cond2 + Cond3 + Cond4) + Yl*RxlCap + Y2*Rx2Cap + Y3*ExtrDistCap + Y4*DistCap + Y5*DistMemCap st. Reactor feed (Hydrogen/Benzene) e [9,14] ; Product purity > 0.99 YI+Y2=I; Y3+Y4+Y5=l The optimisation results using the interactive MINLP algorithm is shown in Table 1 for two different starting points. This illustrates how the optimal configuration is found, formulating and solving a MILP master problem. Iteration 1. Inner
Y2, Y5
1. Inner
Yl, Y3
1. Outer 2. Inner
Yl, Y4 Yl, Y4
2. Outer
Yl, Y4
Binary variables
NLP-solution Fm = 714.08 PFinnerloop= 0.05 (80% conv, extractive F H 2 - - 714.08 distillation) PFinnerloop-- 0.05 (80% conv, 1 dist column) MILP Evaluation (80% conv, 1 dist column) FH2 = 701.41 PFinner loop"- 0.05 MILP Evaluation (80% conv, 1 dist column) (70% conv, membrane)
FOBJ
380.20 $/hr 327.06 $/hr 741.74 $/hr 705.07 $/hr 732.68 $/hr
Table 1: Optimisation results of the interactive MINLP for the cyclo-hexane plant 4.3 Other Application Examples: MINLP optimisation of Acetone-Chloroform separation using extractive distillation (testing three different solvents (1-hexanal, Benzene, Methyl-n-pentyl ether) and pressure swing distillation has been performed (Hostrup et al. (1999)). The chosen optimisation variables were the reflux ratio in the two columns and the solvent flow rate. The optimal solution recommends the use of Methyl-N-pentyl ether. Another MINLP problem solved is the well known Duran & Grossmann (1986) example, which has been tested using the interactive MINLP employing the branch-and-bound algorithm, the outer approximation algorithm and a special "short-cut" feature. The optimisation problem can be formulated generally as: min Z =- Product + Feed + Compr. + Reactor (cap.) + Reactor (var) + HX, heat + HX, cool s.t. Prod = 3600 kmole/h ; YI+Y2=I ; Y3+Y4=l The optimal solution for two different starting points using the interactive MINLP is given in Table 2. Other starting points also gave the same optimal solution.
12
Iteration 1. Inner
Binary variables yl, Y3
(80% A, 10% conv.)
1. Inner
y2, y4
(80% A, 10% conv.)
1. Outer 2. Inner
yl, y4 yl, y4
(80% A, 20% conv.) (80% A, 20% conv.)
2. Outer
Yl, y4
(80% A, 20% conv.)
NLP-solution F = 10081.8 P=0.1499 F = 11118.8 P = 0.2966 MILP Evaluation F = 8345.9 P = 0.219 MILP Evaluation
FOBJ 1565.85 $/hr -1666.09 $/hr -4063.87 $/hr -2547.2 $/hr -2547.3 $/hr
Table 2" Results using the interactive MINLP for the Duran Grossmann example 5
Conclusions
A new interactive algorithm for solution of typical CAPE related MINLP problems has been developed and successfully tested with several non-trivial examples. The novel feature of the interactive MINLP algorithm is that it is able to accommodate the purely equation system mode of solution as well as the simulator mode of solution. For many CAPE applications, since the process model is not provided explicitly but included as library modules, simulator mode becomes particularly suitable. Also, the interactive MINLP approach is particularly suitable for decomposition based MINLP algorithms since it is able to integrate with any process simulator with a NLP solver. The application examples clearly illustrates that reliable optimal solutions can be obtained efficiently even when rigorous process models are employed. It should be noted that for the interactive MINLP algorithm, a special technique to generate the linear model and a modelling testbed were also developed. Finally, the interactive MINLP algorithm is being further extended in terms of CAPE application areas, size of problems, automatic generation of superstructures (for structural optimisation problems) and in computational aspects related to solution efficiency and robustness. 6
References
Gani, R, Hytoft, G., Jaksland, C., Jensen, A. K., 1997, An integrated computer aided system for integrated design of chemical processes, Computers & Chem Engng, 21, 1135-1146. Bossen, B.S., 1995, Simulation and Optimization of Ammonia Plants, PhD-thesis, DTU, Lyngby, Denmark. Bek_Pedersen, E., Hostrup, M., Gani, R., 1999, Using Driving Force Based Separation Efficiency Curves Within an Integrated System for Syntheis/design, Proceedings Escape-10 (submitted). Duran, M.A. & Grossmann, I.E., 1986, "An outer approximation algorithm for a class of mixed-integer nonlinear programs", Math. Progr., 36:307. Floudas, C.A., 1995, "Nonlinear and Mixed-Integer Optimization", Oxford University Press,
New York. ICAS User's Guide, 1999, CAPEC, DTU, Lyngby, Denmark. Edgar, T.F., Himmelblau, D.M., 1988, "Optimization of Chemical Processes", McGraw Hill, New York. Biegler, L.T., Grossmann, I.E., Westerberg, A.W., 1997, " Systematic Methods of Chemical Process Design", Prentice Hall, Upper Saddle River. Brooke, A., Kendrick, D., Meeraus, A., 1992, "GAMS" A User's Guide", San Francisco, CA, Scientific Press.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
An MILP-Based Scheduling
and
Reordering
Algorithm
13
for Complex
Industrial
Rescheduling
J. RoslSf, I. Harjunkoski, J. Bj5rkqvist, S. Karlsson and T. Westerlund Process Design Laboratory, Department of Chemical Engineering /~bo Akademi University, Biskopsgatan 8, FIN-20500/~bo//Turku, Finland Despite the significant progress made in the fields of operations research and process systems engineering (Puigjaner, 1999; Shah, 1998) the complexity of many industrialsize scheduling problems means that a global optimal solution cannot be reached within a reasonable computational time. In these cases, the production schedule must be generated using e.g. some kind of sophisticated heuristics, which can often lead to suboptimal solutions. In this paper, we introduce a Mixed Integer Linear Programming (MILP) based algorithm, which can be efficiently used to improve an existing feasible, but nonoptimal, production schedule or to reschedule jobs in the case of changed operational parameters. The algorithm has been successfully applied to certain scheduling problems in both the paper-converting and pharmaceutical industry. 1. I N T R O D U C T I O N The combinatorial complexity of many real-life scheduling problems combined with the demand for computational efficiency often requires the use of heuristic rules or aggregated procedures (e.g. Lee et al., 1997; Rosl5f et al., 1999). Although these methodologies can produce very good feasible schedules, the optimality of the solutions cannot be guaranteed. Consequently, in most cases a gap exists between the global optimal solution and the best one obtained. Already a minor tightening of this gap may improve the schedule significantly and thus, have a notable economical impact. The problem has been addressed using different approaches. Hasebe et al. (1991) proposed an algorithm based on two reordering operations, the insertion of a job and the exchange of two jobs. They stated that simultaneous reordering of several jobs would be preferred, but concluded that such a system would be very complicated to design. To simplify the problem, they used a method to aggregate jobs before the reordering phase. Franca et al. (1996) presented a tabu-search heuristic with job insertion and removal procedures including a final adjustment stage in order to improve the obtained schedule. Lee et al. (1997) also used a heuristic rule followed by a post-processing procedure consisting of a series of insertion and swap operations. In the present work, a reordering algorithm based on an MILP framework is introduced. The general idea is to iteratively release a set of jobs in a feasible schedule and reallocate them optimally into the production program. By adjusting the number of simultaneously released jobs, even very large systems can still be addressed, obtaining solutions within a reasonable computational time. The algorithm can be applied to a wide range of scheduling problems both in single and multi-unit environments. It works especially well when applied to certain complex batch process scheduling and rescheduling problems with sequence-dependent setup times. Promising results have been obtained with real-life data from both the paper-converting and pharmaceutical industry.
14 2. P R O B L E M
DEFINITION
Consider a problem which contains a set of production jobs i t o be allocated into processing units. Each job has a predetermined production time pi and possibly also a release date ri and a due date di. In case of a multi-unit environment, there may exist several complicating factors to be dealt with in the scheduling process. For instance, a job can often be produced on alternative processing units each of which has a specific production time and there can be precedence relations between the jobs. In many cases, the problem described is most actual in the chemical processing industry, especially in batch production. In an order-driven industry, e.g. in paperconverting, the batch sizes and thus, even the production times are often predetermined by detailed customer specifications. The challenge lies in handling a very large number of jobs in a dynamic environment, where new jobs continuously emerge and specifications of some of the already received ones can be changed at any time. In mid-term scheduling of pharmaceutical fine-chemical production the batch sizes can as well be considered as fixed ones, mainly due to the authority regulations and extremely high validation costs for modified production sequences. In this case, the ultimate assignment for the scheduler is the handling of the alternative production paths as well as the management of unstable intermediates. 3. S O L U T I O N
METHODOLOGY
The complexity of real-life problems together with the expectations for reasonable solution times often exceed the performance that can be provided by the exact mathematical models using available computational capacity. Consequently, the practical problems are commonly solved using heuristics combined with various degrees of true optimization elements. When the construction of an acceptable feasible solution can already be a difficult and time consuming task, the potential hidden in a possible improvement process often remains unutilized. To be able to improve these schedules towards optimality, we introduce a reordering procedure which is easy to implement and applicable even on large-scale problems. When a feasible schedule has been obtained, a subset of the jobs included is released and then optimally inserted back into the schedule between the fixed ones. The jobs to be released can be chosen e.g. according to a specific given order or they can be picked randomly. By adjusting the number of simultaneously released jobs, the computational complexity can be set to a desired level in relation to the total number of jobs present in the system and the available computer capacity. The reordering algorithm can be executed either by running a certain prespecified number of iterations, or using a convergence criterium, e.g. dealing with the reached improvement of the solution in relation to the number of performed iterations. 3.1 M I L P C o r e F o r m u l a t i o n for a S i n g l e - U n i t Case The core model for representation of an iteration stage in the reordering process can be expressed as an MILP formulation. The most interesting part of the formulation, from the algorithmic point of view, is the constraints needed for the allocation of the jobs based on continuous time representation (e.g. Greenberg, 1968). They can be written for example as follows: t~ - tj <_ - P i - s~j
V
(i, j ) e S I i x e d l i
precedes j
(1)
Equation (1) describes the relative order of the jobs that are not released at the actual iteration (subset Ssixed). It must be written for every pair of jobs preceding each other. The equation ensures that the production of job j cannot be started before the production of job i is completed and the required setup time sij has elapsed.
15
V
k E S vet
(2) (3)
V
k
9S r d
(4) (5)
V
k 9 S~
ti -- tk + ( T + sik -- rk ) . Yikj <_ T -- pi -- rk tk -- tj + ( T + Ski -- r j ) " Yikj <_ T - Pk -- rj tk -- t f + ( T + Skf -- r f ) " y k f <_ T - - p k
V
(i,j) E Sfix~dli
precedes j
--rf
tl -- t k + ( T + s l k -- r k ) " Yik <_ T -- p l -- r k
Y i k j + Y k l + Ylk = 1
(i,j)ess~
(6)
l i precedes j
Constraints (2) and (3) model the possible locations of a released job k (subset S re') in relation to the fixed ones. The binary variable y~kj obtains the value one, if the job k is allocated between jobs i and j. Accordingly, constraints (4) and (5) represent the time slots before and after all the fixed jobs, and the binary variables yks and y~k are set to one if job k is to be produced before the first fixed job f or, respectively, after the last fixed job 1. Equation (6) ensures that the released job k is allocated to one and only one of the locations allowed. The constant T represents the upper bound for the completition times of the jobs.
(7)
ti -- t j A- ( T + s i j -- r j ) " Yij <_ T - p~ - r j t j - ti -
( T + s j i - r i ) " y i j <_ - P j
- 8ji
V
(i,j)
9 Srd,i
# j
(8)
If several jobs are released simultaneously, their internal order is determined by constraints (7) and (8), in which the binary variable y~j is set to one if job i is produced before job j. A complete MILP model must, naturally, also contain a proper objective function and other relevant constraints, dealing e.g. with the release and due dates. A clear strength of this formulation, compared to many other reordering algorithms, is that every iteration contains information of the entire system, i.e. all jobs are present in the optimization procedure. Moreover, the result of the previous iteration can be used as a start condition for the following one, which provides a tight initial bound for the B&B algorithm. On the other hand, it is obvious that the reordering procedure can be driven towards suboptimal solutions as global optimality cannot be guaranteed. This phenomenon can, however, be avoided to some extent by releasing several jobs at the same time.
3.2 M u l t i - U n i t E n v i r o n m e n t The formulation presented above can be further utilized for multi-unit scheduling. The possible additional limitations, e.g. durability requirements of the intermediates, must be considered in the model by introducing appropriate constraints. This concept has recently been applied on a production planning problem dealing with the production of fine-chemicals in the pharmaceutical industry (BjSrkqvist et al., 1999). A very common feature in industrial batch processing is that a specific job can be performed on alternative processing units. Furthermore, the system can be highly coupled, i.e. the allocation decision concerning a job belonging to a production sequence often partly determines even the equipment which can be used for the down-stream jobs. These properties can be dealt with by introducing an additional reordering phase. The job with alternative production paths is first released, whereafter it is inserted into each of the possible processing units at a time. Finally, the processing unit providing the best overall solution is chosen. If there are coupled jobs present, it may be necessary that a part of the sequence must be released and then reallocated into the schedule according to the order given by the precedence constraints. An alternative method is to backtrack the entire sequence and enumerate every possible production path that can be generated by fixing the
16 processing units towards the down-stream jobs. This policy can be beneficial with sequences containing a limited number of alternative scenarios, whereas very complex recipes may be problematic from the computational point of view. 3.3 O p e r a t i o n a l A s p e c t s In a dynamic production environment it is very common that some jobs need to be relocated due to e.g. changed customer requirements, delayed raw-material deliveries or failures occurring on processing machinery. In an order-driven industry, there exists even a continuous need to allocate new orders into the existing schedule. In many cases, it is not desirable to reschedule all the jobs included, instead the required changes should be performed in such a way that the entire system is affected as little as possible. The reordering algorithm provides an efficient method to handle these issues. The modified jobs can be released and thus, rescheduled optimally only with minor changes to the original schedule. If a new job emerges, it can, accordingly, be inserted into the schedule using the same concept. The algorithm can also be used as a background process continuously performing release and insertion operations to find better solutions. Such a feature can be useful for production environments containing a very large number of jobs. If the jobs located in the near future are not preferred to be included into the continuous reordering process, the scanning can be limited to a time-window which is sufficiently separated from the current shop-floor schedule. If the complexity has been adjusted to an appropriate level, one iteration step does not require too high computational resources. Thus, if a sufficiently large number of reordering operations is performed, the possibility of finding improved solutions is considerable. From the operational point of view it is also a notable feature that, regardless of the convergence criterium used, the algorithm can be stopped at any time and is always able to provide the best solution obtained so far. 4. I L L U S T R A T I V E
EXAMPLE
A case study based on data received from a Finnish paper-converting mill producing barrier coated and laminated packaging materials and industrial papers, is presented. The mill is operated on an order-driven basis and the production machinery can be used for producing a very wide range of different product qualities. The task is to allocate the production jobs in such a way that the deliveries are performed in time and the total setup time is minimized. The large amount of jobs produced during a short time-period, continuously changing customer requirements together with strongly sequence-dependent setup times form a fascinating challenge, which is extremely difficult to solve by hand, even after years of shop-floor experience. The scheduling task contains 61 jobs representing 24 different product qualities on a single processing unit. No release dates are present but 11 different due-date groups complicate the system. The setup times required between different qualities can be derived from an expert system based on the data obtained from experienced schedulers. The time period includes two weekends during which the processing unit is not operated. The production sequence at the mill has been determined by a manually scheduled production program. The formulations required for the representation of the weekends do not fall under the scope and extent of this paper. Yet, the reordering iterations were performed using the MILP framework described above. The additional formulations do, however, slightly increase the combinatorial search space, which should be noted when studying the results. If the whole problem was solved in one stage, i.e. all the jobs were released simultaneously, the resulting MILP formulation would contain 1952 binary variables and be impossible to solve within acceptable time with the available computer resources.
17 4.1 T e s t r u n s 250 reordering iterations were performed starting from four different initial schedules: 1) The manually sorted production program (MANUAL) 2) a schedule obtained using a heuristics by first sorting the jobs using the earliest due date first policy and then, within a due date group, applying a priority rule dealing with the qualities (EDD-Q1Q2) 3) a randomized solution (RND) and 4) a solution obtained using a tailored iterative optimizer based on the concept presented by Roslhf et. al (1999) (SUPpre). The objective function is formed by a sum of total weighted tardiness and an overall weighted makespan. It should be noted that the schedule provided by the EDD-Q1Q2 heuristics is not regarded as an acceptable solution despite the fact that it provides a much lower objective function value, with the applied cost factors, than the realized schedule. Although the goal of minimizing the tardiness is very important, the total setup time is even more vital a factor for the efficiency of the mill. The EDD-Q1Q2 rule is not intelligent enough to consider this part to as large extent as desired. Two separate test sequences were performed using a uniformly distributed random selection of the released jobs. The first one releasing only one job at a time and the other releasing two jobs per iteration. The algorithm was implemented using C programming language and the tests were performed on a 300 MHz Pentium II workstation with 128 MB RAM running the LINUX operating system. The CPLEX-6.0 (ILOG, 1999) was used as an MILP solver. Selected results are presented in Figure 1 and Table 1.
T w o r e l e a s e d j o b s at a t ime
O n e r e l e a s e d j o b at a t i m e
300 000
300 000
RND
i RND 250 000
250 000
,~ 200 000
200 000
~ 150000
100 000
L__._,
0
50
100
150
Start value of the Randomized schedule is : 1 925 600
Fig. 1
~" 150 000
.
200
250
100 000
U
0
r
50
100
150
Nurrber of Iterations
200
250
Number of Iterations
The objective function value during the iterations
Table 1. Manually sorted production schedule vs. the best and worst solution after the reordering Indicator
Manual
Best solution
Worst
(SVPpre-2)
(RND-1)
Objective function value Active makespan Total setup time Number of tardy jobs Total tardiness
288,830 158.2 h 24.3 h 4 145.8 h
111,738 150.2 h 16.3 h 1 4h
118,516 157.7 h 23.8 h 1 4h
Total CPU-time used Average CPU-time per iteration Binary variables per iteration
-
3755.1 s 15.02 s 243
205.8 s 0.82 s 183
solution
18 4.2 C o n c l u s i o n s First of all, the results show that the SUPpre was directly able to generate a very good solution to the actual problem. However, the most interesting result is that the reordering process was able to improve even the randomized initial condition near to the very best solution found, with only a minor computational effort. When the obtained solutions are compared with the manually generated schedule, it is notable that even the "worst solution" is considerably more efficient than the one realized at the mill, especially in terms of total tardiness. Moreover, the best solution found leads to a very significant decrease of the total setup time. As expected, the strategy to release two jobs at a time led to a much stronger relative decrease of the objective function value than the option with only one released job. However, the higher complexity required much bigger computational effort. Despite the fact that the both strategies resulted, for this actual case, in schedules very near to each other, it can still be beneficial to release several jobs in order to guarantee high quality solutions. 5. D I S C U S S I O N
In this paper, an MILP-based reordering algorithm for the scheduling and rescheduling of batch processes was presented. It was shown that the algorithm can be successfully used to improve large-scale schedules appearing in the chemical processing industry. Considerable improvements to an existing non-optimal schedule can often be obtained already with low computational capacity. A powerful strength of the proposed methodology is that every phase of iteration contains information of the entire system, which means that each iteration will only improve or maintain the current schedule quality. The method can be applied even to improve already running schedules as a background optimization process. Furthermore, due to the general nature of the concept, the algorithm can, with minor modifications, be applied to various types of scheduling and other optimization problems with a related structure. REFERENCES
BjSrkqvist J., Karlsson S., RoslSf J., RSnnback R. and Westerlund T. (1999). Applying Iterative and Parallel Methods for Production Planning in the Pharmaceutical Industry. Presented at AIChE Annual Meeting, Dallas, Texas, USA. Franca P.M., Gendreau M., Laporte G. and Miiller F.M. (1996). A Tabu Search Heuristic for the Multiprocessor Scheduling Problem with Sequence Dependent Setup Times. International Journal of Production Economics, 43, pp. 79-89. Greenberg H. (1968). A Branch-Bound Solution to the General Scheduling Problem. Operations Research, 16, pp. 353-361. Hasebe S., Hashimoto I. and Ishikawa A. (1991). General Reordering Algorithm for Scheduling of Batch Processes. Journal of Chemical Engineering of Japan, 24, pp. 483-489. Lee Y.H., Bhaskaran K. and Pinedo M. (1997). A Heuristic to Minimize the Total Weighted Tardiness with Sequence-Dependent Setups. IIE Transactions, 29, pp. 4552. Puigjaner L. (1999). Handling the Increasing Complexity of Detailed Batch Process Simulation and Optimization. Computers ~ Chemical Engineering, 23, pp. $929-$943. RoslSf J., Harjunkoski I., Westerlund T. and Isaksson J. (1999). A Short-Term Scheduling Problem in the Paper-Converting Industry. Computers ~4 Chemical Engineering, 23, pp. $871-$874. Shah N. (1998). Single- and Multisite Planning and Scheduling: Current Status and Future Challenges. Proceedings of FOCAPO'98, Snowbird, Utah, USA.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
19
Non-Linear Partial Least Squares through Data Transformations B. Li, E. B. Martin and A. J. Morris* Centre for Process Analytics and Control Technology University of Newcastle, Newcastle upon Tyne, NE 1 7RU, United Kingdom Partial Least Squares (PLS) is one of the most frequently used techniques for process modelling and monitoring with correlated data. In this paper, a Box-Tidwell transformation based PLS algorithm (BTPLS) is proposed to deal with non-linear problems in complex systems. The BTPLS algorithm provides a family of flexible regression models for data fitting. These are shown to out-perform both the linear and quadratic PLS algorithms for non-linear problems in terms of modelling and prediction accuracy. In contrast to neural network based PLS (NNPLS) algorithms, the BTPLS algorithm significantly reduces computational costs and exhibits better performance for many practical situations in terms of Akaike Information Criterion. This is due to the resultant model forms being more parsimonious and the use of a non-iterative search for data transformations and regression coefficients. The BTPLS is a compromise between model simplicity and accuracy, and constitutes a complementary modelling technique alongside linear PLS and NNPLS. 1. INTRODUCTION Partial least squares has been shown to be one of the most frequently used techniques for dealing with highly correlated data in process modelling and monitoring [1]. For systems that exhibit complex non-linear behaviour, linear PLS is inappropriate for modelling the underlying structure. A number of non-linear extensions to linear PLS have been developed. Wold et al. [2] proposed a quadratic extension to linear PLS whilst a number of authors developed variants of neural network based PLS algorithms [3-6]. Most recently, Baffi et al. [7] presented an error based weight updating procedure for quadratic PLS algorithm that was shown to out-perform the quadratic PLS algorithm of Wold et al. [2]. Baffi et al. [8] then combined the error-based weight updating procedure and a neural network approach to obtain a modified neural network based PLS algorithm (NNPLS). In this paper, a compromise between linear PLS and the modified NNPLS algorithm is proposed. A model is a simplified representation of the real world, hence modelling is a trade-off between model simplicity and model accuracy. For example, there are a wealth of regression techniques which differ in terms of complexity. These include linear regression, linear regression through data transformations, non-linear regression, and non-parametric regression. Each technique has its own advantages and disadvantages. Practitioners are then required to select the most appropriate tool according to the modelling objective and the required simplicity and accuracy of the application. *The authors acknowledge the support of the EU ESPRIT PERFECT No. 28870 and SCIENTIA No. GR/L28029
20 In this paper a compromise between the two extremes of the complexity spectrum in terms of PLS, linear PLS and the modified NNPLS algorithm is proposed, the Box-Tidwell Transformation based Partial Least Squares (BTPLS) algorithm. Since the Box-Tidwell transformation family [9] encapsulates both linear and quadratic forms as special cases, it is not surprising that the Box-Tidwell transformation based PLS algorithm can exhibit better performance than either the linear or quadratic PLS algorithms for modelling non-linear behaviour. Compared with NNPLS, the BTPLS has clear advantages in terms of computational effort and model parsimony, especially for industrial data where large amounts are being routinely collected, typically involving hundreds of variables. It is also equally useful for laboratory data where there is only a limited number of samples and there is therefore a tendency for the data to be overfitted when using neural network approaches.
2. PROBLEM DEFINITION Consider a set of data representing the "normal" status of a process. Xuxu represents the process variables and YN• the quality parameters, which are recorded for N time points. The objective of linear PLS is to project the data down onto a number of latent variables (LVs), say tj and uj, where j= 1.... a, and then to develop a regression model between the LVs:
uj = bjtj + e j
(1)
t jp Tj + E and Y =
The matrices X and Y are decomposed as X j =1
fij -l)jtj
fi jq Tj "[- F ,
where
j =1
is the prediction of u j from (1), E and F are the residual matrices for the matrix
decomposition of X and Y, and a is determined by cross validation. The most commonly used PLS algorithm is non-linear iterative partial least squares (NIPALS). This algorithm sequentially extracts each pair of LVs through an iterative procedure. In each iteration, the decompositions of X and Y are related through a linear regression between the latent variables tj and uj. To extend the algorithm from linear to non-linear PLS, Wold et al. [2] proposed a quadratic modification to equation (1), U j -- COj ~- Clj[ j + c2jt 2j -t- ej. As pointed out in [3], the applicability of the quadratic form is limited in terms of the structures it can model. To overcome this problem, several variants of NNPLS have been proposed in which neural networks were used to approximate an arbitrary regressor function f(tj), uj = f(tj)+ej. Clearly, equation (1) plays a key role in the non-linear extension of the PLS algorithm. More importantly, such extensions affect the outer mapping through the weight vectors [7].
3. BOX-TIDWELL TRANSFORMATION BASED PLS ALGORITHM The application of data transformations is one of the most often used techniques in regression analysis. One key advantage of PLS is that it essentially reduces to a simple regression problem with only one-independent and one-response variable, equation (1). Compared with general multiple regression problems, the identification of a transformation for one-independent variable is more straightforward. In addition there exists an important
21 distinction between PLS and ordinary regression; regression between the latent variables is performed automatically in PLS. This means that at each iteration it is not possible to check the structure of the latent variable data and thus decide which transformation should be applied. Therefore, it is necessary to develop a procedure that automatically selects the most appropriate transformation. In this section, the Box-Tidwell transformation family is introduced before describing a modified version. Finally the modified Box-Tidwell transformation family is used to develop the new BTPLS algorithm.
3.1. Box-Tidwell transformation family and its estimation procedure Consider a simple regression problem between an independent variable x and a response variable y. Instead of fitting a linear regression model: (2)
E(y) = g(x, flo, ill) = flo + fll x
Box and Tidwell [9] proposed a transformation family for a positive independent variable x and then investigated the regression problem between t and y:
E(y) = g(t, flo,
1~1)=/~0 + til t
with t = x a for a r 0 and t = In(x) for a = 0
(3)
Clearly equation (3) is more flexible for data fitting than the linear model defined in equation (2). To estimate the unknown parameters fl0, fll and o~, they proposed expanding function g about the initial guess, o~0=1, in a Taylor series and ignoring terms higher than first order:
E ( y ) = g ( t , flo,fll)= flo + f l l x + ( a - a o ) { d g ( t ,
flo,fll)/da}a:ao=flo + f l l x + T z
(4)
Pl
in the linear
where y = (c~ - 1)fl~ and z = x In(x). Denoting the least squares estimate of
model, equation (2), as J31, and the least squares estimate of ),in the linear model, equation (4) as ~, oc is estimated as o~* = ~//31 § 1, and ,80 and fll are estimated by least squares estimates for y and t= x a* in the model equation (3). This procedure can be repeated to ensure convergence. Box and Tidwell [9] noted that this procedure usually converges quite rapidly, and often the first stage result, oc*, is a satisfactory estimate [ 10].
3.2. Modified Box-Tidwell Transformation Family One of the key assumptions of the Box-Tidwell transformation family is positivity of the independent variables. This however is not valid for the inner mapping problems of the PLS algorithm where both the independent and response variables, i.e. the latent variables tj and uj, have zero-means. In order to apply the Box-Tidwell transformation, the transformation in equation (3) is extended from x>0 to any real x by the following piece-wise transformations:
•[sgn(x)] 6 Ix Ia, t = ~[sgn(x)]~ ln(I x I),
o~ r 0 a-0
for 6 = 0 or 1
(5)
22
Based on equation (5), the regression model, equation (3) can be extended as follows to replace equation (2): E ( y ) = g(t, f l o , f l l , 6 , a ) =
flo + ill(sign(x)) 8 Ixla
8=0
or 1
(6)
3.3. Modified estimation procedure of Box-Tidwell transformations It should be noted that the above model is no longer valid when both x = 0 and o~ <0. Therefore, o~ is further constrained as o~ >0 in the above extended Box-Tidwell family, Equation (5). In optimization, a typical implementation of this restriction is to let o: = v 2 for any real 1:. Taking this restriction on o~ into account, the modified estimation procedure can be obtained in a similar way to the Box-Tidwell estimation procedure. The only difference is to expand the function g with respect to v rather than o~. Specifically, for observation data { (xi,Yi), i= 1. . . . . N } , the function g in equation (6) is expanded about the initial guess v 0 = 1 in a Taylor series and the higher than first order terms are ignored: E ( y ) = g(t, flo, fll , 6, v 2 ) = flo + ill(sign(x)) 8 I x l +(V - Vo ){dg(t, flo, fll, 8, vZ ) / dv}v=Vo
= flo + fll (sign(x)) 8 I x l + y z where 7"= 2(v
/~, = a r g m i n
- 1)fl 1
and z= (sign(x)) 8 I x Iln(I x I). The parameters are then estimated as
N
Z [ y i - { f l 0 + fll(sgn(xi)) 8 Ixi I}] 2 flo,fl1,8 i=l N = a r g m i n Z [ y i - { f l o + fll[sgn(xi)] 8 Ixi I+y(sgn(xi)) 8 Ixi Iln(lxi I)}] 2 flo,fll,~,T i=1
o~* = {~/(2/~1) + 1}2
(7a)
(7b) (7c)
N
[flo*,fll*,8*]=arg min Z [ Y i - { f l 0 + fll (sgn(x)) 8 Ixi la*}] 2
(7d)
A~,Pl,8 /=1 The problem in equations (7) is essentially a quadratic optimization which can be solved through least squares by decomposing the problem into two sub-problems for 8 - 0 and ~= 1.
3.4. Box-Tidwell transformation based Partial Least Squares Algorithm The Box-Tidwell Partial Least Squares algorithm is based on the quadratic PLS algorithm of [2] and the error-based weight updating procedure of [7]. The estimates of the first iteration are used as approximations of the Box-Tidwell transformations. This avoids additional iterative steps in the search for the optimal Box-Tidwell transformations during the iterative identification of the latent PLS structures. Specifically, the BTPLS algorithm is the same as the error-based weight updating procedure of [7] (Table 3) except for step 5 and step 14, of Table 3 in [7], which is modified to estimate the parameters [fl0*,fll*,~*,ot*] from equation (7).
23 It is noted that in BTPLS, appropriate transformations are automatically chosen through the modified Box-Tidwell estimation procedure. No intervention is needed to identify which transformation should be applied. Moreover, in contrast to a non-iterative procedure, the modified NNPLS algorithm [8] requires a neural network to be trained at each iteration to identify the latent structures. This clearly demands increased CPU time. 4. E X A M P L E The cosmetic data of Wold et al [2], comprising N=17 different formulations of face creams forms the basis of the study. This particular data set has been chosen as it has formed the basis of a number of other comparative studies on PLS [3]. It is also well known that it is problematic in terms of its tendency to be overfitted. Each formulation is composed of 8 chemical constituents including glycerine, water, emulsifier, vaseline, and 11 quality variables such as ease of application, greasiness, skin smoothness, skin shininess and overall appeal. To illustrate the flexibility of the Box-Tidwell transformation family for modelling the LV structure within the BTPLS algorithm, scatter plots of LVs tj and uj (j=1,2,3,5) and the resulting fits from the BTPLS algorithm are given in Figure 1. It can be seen from Fig. 1 that the Box-Tidwell transformation family is quite flexible for fitting data that exhibits non-linear features. For instance, for the latent variables given, latent variables tj and uj, it is clear that neither linear nor quadratic regression can capture the underlying trend, while from Fig. 1, fitting tj and uj through a Box-Tidwell transformation, the result is acceptable. Secondly, in comparison with the modified NNPLS where a neural network is trained to approximate the relationship between tj and uj, BTPLS ensures a parsimonious model. For instance, for LVs tl=[tM and Ul=[Ulk], the resulting regression model fitting is given by Ulk=O.25+4.94tlk4"42+Ekfor tlk_>0 and Ulk=O.25--4.941tlk14"n2+Ekfor t~k <0. Moreover, when the latent LVs exhibit a linear relationship, the BTPLS reduces to linear PLS. For instance, for t3 and u3, the BTPLS gives an estimate of o~as c~ = 0.98. Thus, for the third pair of latent variables, the BTPLS gives almost the same result as linear PLS.
"~ o
Y -0.5
ooj o
oo S
ou o
o
0 tl
0.5
-0.5
0 t2
0.5
0 t5
1
4
f
~oO
o
4'
-2
-i
0 t3
i
2
--2
-1
Fig. 1. Scatter plots of latent variables and regression fitting ( - )
2
for the BTPLS
24 The balance of model simplicity and accuracy can be further evaluated in a quantitative way by using Akaike Information Criterion (AIC). Table 1 gives a comparison of AIC for BTPLS and modified NNPLS (with a feedforward network consisting of one hidden layer and h hidden neurons) for the first eight pairs of LVs. It can be seen that many of the fits through neural network approximations are seriously overfitted. This means that too high a price on model parsimony is paid for in terms of accuracy of the modified NNPLS algorithm. Table 1. Comparison of AIC in LV Pair 1 BTPLS 20.14 NNPLS (h=2) 59.93 NNPLS (h=3) 43.58
latent variable regressions 2 3 4 39.84 26.40 31.52 60.08 66.45 55.09 32.05 48.40 40.88
for BTPLS and NNPLS 5 6 7 6.61 28.36 27.70 31.87 53.33 33.68 33.11 30.09 25.28
36.08 40.60 30.69
Finally from a simulation study (not reported) it is shown that the modified NNPLS approach has much higher computational costs than the BTPLS (the BTPLS takes only 0.42% of the CPU time needed by the modified NNPLS used in the simulation study). This will also be true for all sigmoidal neural network based PLS algorithms. This is a consequence of the fact that at each iteration in the loop for defining the latent structure, a neural network is trained, whilst for BTPLS, this step is superfluous. Computational costs will further increase if good initial values for the training of the neural networks are to be obtained, and the appropriate number of neurons in the hidden layer is also to be identified. 5. CONCLUSIONS Box-Tidwell transformation based PLS is proposed to deal with problems exhibiting nonlinear features. Advantages of the BTPLS algorithm include: (i) enhanced flexibility in fitting data with non-linear features compared to the linear and quadratic PLS algorithms; (ii) lower computational costs than neural network based PLS algorithms; (iii) better performance in terms of AIC than neural network based PLS algorithms. Based upon these features, the BTPLS is a comprise between model simplicity and model accuracy. This is especially appropriate when modelling large amounts of industrial data in terms of savings in computational time, as well as for laboratory data where there is only a limited number of samples, in this case application of BTPLS algorithm can avoid overfitting. REFERENCES
1. T. Kourti, and J.F. MacGregor, J. of Quality Tech., 28 (1996) 409. 2. S. Wold, N. Kettaneh-Wold and B. Skagerberg, Chemo. Intell. Lab. Syst., 7 (1989) 53. 3. S.J. Qin and T.J. McAvoy, Computers Chem. Engng., 16 (1992) 379. 4. T.R. Holcomb and M. Morari, Computers Chem. Engng., 16 (1992) 393. 5. E.C. Malthouse, A.C. Tamhane and R. S. Mah, Computers Chem. Engng., 21 (1997) 875. 6. D.J.H. Wilson, G.W. Irwin and G. Lightbody, ACC, Albuquerque, New Mexico, 1997. 7. G. Baffi, E.B. Martin and A.J. Morris, Computers Chem. Engng., 23 (1999) 395. 8. G. Baffi, E.B. Martin and A.J. Morris, Computers Chem. Engng., 23 (1999) 1293 9. G.E.P. Box and P.W. Tidwell, Technometrics, 4 (1962) 531. 10. D.C. Montgomery and E.A. Peck, Introduction to Linear Regression Analysis. New York: Wiley, 1982.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
25
"Optimisation of an Industrial Cogeneration System by means of a MultiObjective Genetic Algorithm" G. A. Effhimeros", D. I. Photeinos", I. G. Katsipou a, Z. G. Diamantis a, D. T. Tsahalis a a Laboratory of Fluid Mechanics and Energy, Department of Chemical Engineering, University of Patras. P.O. Box 1400, GR26500, Patras, Greece E-mail: [email protected]~:...C.~.H..~M.E,.N....G..,_~,P.AT, R.A.S_:_.G_R_ URL: http ://LFME. CHEMENG.UPATRAS. GR 1. INTRODUCTION The key word in the process industry is energy. Energy is required in multiple forms (electricity, steam, mechanical energy, fuels, etc.) and large amounts for the proper operation of an industrial process plant. Very often, multiple sources of energy are incorporated in large process plants. These sources can be utilities such as steam turbines, gas turbines, fuel-burning boilers, exhaust gas boilers, or even the electricity grid. It is obvious that the operating cost of a plant strongly depends on the amount of energy the plant requires, especially in cases where the energy cannot be generated internally in the plant (as by-product of separate processes in the plant) and has to be purchased in the form of fuels and/or electricity from the local grid. The need for intensification and self-efficiency, in terms of energy, of the process that consequently results in reduction of the operating cost is satisfied to a certain extent by the cogeneration systems. Such systems incorporate various utilities (usually gas or steam turbines, and exhaust-gas boilers with supplementary firing of fuel) to produce needed steam of various grades and electricity, which can be exported to the electricity grid in case of excess. Due to the variety of the utilities incorporated and their interdependencies, as well as the variation of technical and economic conditions with time, the task of optimizing the operation of a cogeneration system is complex. In order to emphasize this fact, it is not rare for the operation of such systems to be empirically determined. In order to solve these types of problems, various methods are still being developed. This is mainly due to the nonlinearity of such problems, which have multimodal objective functions that may invoke both discrete and continuous variables. Up to now, the authors are not aware of a single method that has solved such optimization problems successfully. In this paper, the problem of optimizing the operation of an industrial cogeneration system is addressed with an advanced-level optimization technique that is based on a MultiObjective Genetic Algorithm (GA). This optimization method, which is usually referred to as the Pareto technique, is widely used in optimization problems where multiple parameters must be optimally configured simultaneously. While maintaining the robustness of Gas, this method utilizes multiple objective functions to better describe the problem. For the calculation of the fitness functions, the Pareto GA
26 utilizes both the simulation model of the system and specified constraints, in order to solve the optimization problem considered.
2. THE OPTIMIZATION P R O B L E M
The optimisation problem at hand consisted of a cogeneration system with a recovery boiler. This system comprised a "superstructure" that in general could include the following technologies: combustion, air preheating, oxygen enrichment of air, gas turbine, advanced gasturbine, partial oxidation gas-turbine, gas or diesel engine. During the EXSYS project, a model was developed for this system, that took into account only generic technologies, i.e., technologies and utilities available in the market, along with the necessary information about them [2]. In this system, the role of the gas turbine is proven to be pivotal. The size of the gas turbine influences not only the cost of the system, but also its efficiency. According to the generic model developed for this system, the following equations [2] governed the selection of the appropriate gas turbine: ]]loss - - 0.1576 + 0.0129 log(WinCe),
(1)
Cost =
(2)
1.466
(We)0"845,kECU
Maintenance costs = 0.0851 (We)"0"3081,in kECU/kWh
(3)
Where: ]]loss is the loss factor for the gas turbine, expressed as 1 - ]]heat- ]]mec ]]heat is the thermal efficiency factor
]]mccis the mechanical efficiency factor, corresponding to the electricity produced by the gas turbine with respect to the fuel consumption Wmcc is the mechanical power of the gas turbine (in kW) Wo is the electrical power of the gas turbine (in kWo) Information about the mechanical and electrical power of commercial gas turbines existed for 41 commercial gas turbines in the databases developed during the EXSYS project [2]. It is obvious that the optimal gas turbine for the cogeneration system under consideration should be the one with the minimum loses, cost and maintenance costs. Thus, the optimization problem consisted in finding the optimal gas turbine that satisfied equations (1)(3) simultaneously, or in other words, to find the mechanical and electrical power that minimized equations ( 1)-(3). At this point it should be noted that the optimization problem is not of extreme complexity. However, it was selected as a test case for the multiobjective optimization algorithm that was developed. Both the algorithm and the results are preliminary, while it is envisaged that in the near future this algorithm will be further improved.
27 3. THE MULTIOBJECTIVE GA As mentioned in the Introduction, the optimization method that was selected for this problem, is the Genetic Algorithms (GAs) method. This method is part of a class of algorithms, called Evolutionary Algorithms. The characteristics that distinguish the GAs from the classic optimisation techniques are the following: 1. GAs do not need any type of information other than a measure of how good each point in the optimisation area is. Consequently, they may be applied in cases where only a black-box model of the system-to-be-optimised exists 2. GAs work with a coding of the parameter set and not with the parameters themselves 3. GAs search for the optimum from a set (population) of points, and not from a single point 4. GAs use probabilistic transition rules and not deterministic ones 5. GAs are a robust optimisation method, i.e. they perform regardless of the optimisation problem. 6. GAs provide more than one equivalently good near-global-optimum solutions. This allows the selection of the most appropriate solution for the problem at hand. From the above it seems that for the application of a GA to a problem, a) a good knowledge of the problem and b) an experience using GAs are not necessary. Even though this may be generally true, both (a) and (b) are necessary for the successful application of the GA. The GA developed for the optimization problem at hand, utilizes the classic GA operators of reproduction, crossover and mutation. For an introduction to these operators and the classic structure of a GA, the reader is referred to Goldberg [ 1]. The fitness, which is a measure of the "goodness" (optimality) of each possible configuration (string) for Wo and Wmoc in the optimization space, should consider all three objective functions (eq. (1) - ( 3 ) ) simultaneously. This was achieved by assigning a "dummy" fitness to each string in the population, according to the values of the three objective functions. Moreover, a weighting factor was used for each of the three objective functions, which corresponded to the importance of each objective function, i.e. the weighting factors should have the same value for all objective functions when the objective functions are considered to be of the same importance. Thus, the "dummy" fitness assigned to each string in the population and in each generation, was based on the following formula:
F= Ewi" fi
(4)
i
where: wi is the weighting factor of the objective function i t] is the value of the objective function i for a string Based on the value of the "dummy" fitness of each string, the population in a generation was classified in ranks. Each rank consists of strings (configurations) of equal fitness, i.e. of equivalently optimal configurations, and the optimal configurations were stored in Rank 1. Finally, a comparison was performed between each pair of "parent strings" and the corresponding pair of "offsprings (4 strings in total) and the best two strings were included in the initial population of the next generation.
28 4. RESULTS AND CONCLUSIONS The GA multiobjective optimization algorithm was developed in the FORTRAN 90 programming language, and the code executions were performed on a Pentium II 450 MHz with 320 MB of RAM. Depending on the size of the population and the limit to the number of generations, the code takes a few seconds to execute. Various runs were performed, and the results indicate that the optimal gas turbine(s) should produce a mechanical power between 5960 and 24630 kWmee,electrical power between 5841 and 24150 kWo. Concerning the three objective functions for the optimal range of gas turbines, the corresponding values are: rllossbetween 0.34 and 0.358, cost between 15 and 20 Million EURO and maintenance costs between 0.0027 and 0.0037 EURO/kWh. The results obtained with the present multi-objective optimization method, although preliminary, are satisfactory. This conclusion stems from the comparison of the results obtained with those obtained from the single objective optimisation method, which was applied under the framework of the EXSYS project. With the new optimisation technique a remarkable decrease in the maintenance cost is achieved for a specific range of the generator output, 584124150 kWh. The range of the maintenance cost that corresponds to the referenced generator output in the case of the single optimisation method is 0.0031-0.0062 EURO/kWh. The coding of the parameter space and certain aspects of the code can be further optimized. However, the optimal values that the GA found correspond to at least 4 commercial gas turbines that satisfy the needs of the case study, while remaining the most economic solution available. From the work performed, the following conclusions can be derived: 1. GAs are a robust optimization method that can successfully address any optimization problem in the process industry. This conclusion has already been pointed out in similar work performed by LFME on this field. [3], [4], [5] 2. The introduction of multiple objectives in the robust method of GAs takes optimization a step further. This is specifically true for the optimization problems of the process industry, where the cost has always been the only parameter for optimization, regardless of any other improvements. 3. Due to their fast execution, multiobjective optimization algorithms based on GAs is a tool that can be used by the process engineer, not only during the development phase of a process, but also on-line, for control purposes. 6. ACKNOWLEDGEMENT Part of the work presented in this paper was supported by the European Commission in the JOULE3 project EXSYS (Contract No. JOE3-CT95-0020). REFERENCES
1. Goldberg D. E., "Genetic Algorithms in Search, Optimisation, and Machine Learning", Addison-Wesley Publishing Company Inc., 1989. 2. EXSYS Final Report, submitted to EC in December 1997. 3. Manolas D.A., Gialamas T.P., Tsahalis D.T., "Development of a special Genetic Algorithm for the Selection of the Energy Best Available Technologies and their Optimal Operating Conditions", presented at the First European Congress on Chemical Engineering (ECCE-1), Florence, Italy, 7 May 1977.
29 4. Manolas D.A., Effhimeros G.A., Tsahalis D.T., "Development of an expert System Shell based on genetic Algorithms for the Selection of the Energy Best Available Technologies and their Optimal Operating Conditions for the Process industry", Proceedings of IFAC Minisymposium on Optimisation using Evolutionary Methods, 15-17 July, 1998, Patras, Greece. 5. Manolas D.A., Gialamas T.P., Fragopoulos C.A., Tsahalis D.T., " A Genetic Algorithm for Operation Optimisation of an Industrial Cogeneration System", Journal of Computers & Chemical Engineering, Vol. 20, Suppl. B, p.p. S1107-Sl 112, 1996
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European Symposiumon ComputerAided ProcessEngineering- 10
S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
31
Grid refinement in multiscale dynamic optimization* T. Binder 1, L. Blank 2, W. Dahmen 2, W. Marquardt 1 1 Lehrstuhl fiir Prozesstechnik, 2 Institut fiir Geometrie und Praktische Mathematik, RWTH Aachen, Germany. Abstract
In the present work we explore an adaptive discretization scheme for dynamic optimization problems applied to input and state estimation. The proposed method is embedded into a solution methodology where the dynamic optimization problem is approximated by a hierarchy of successively refined finite dimensional problems. Information on the solution of the coarser approximations is used to construct a fully adaptive, problem dependent discretization where the finite dimensional spaces are spanned by biorthogonal wavelets arising from B-splines. We demonstrate exemplarily that the proposed strategy is capable to identify accurate discretization meshes which are more economical than uniform meshes with respect to the ratio of approximation quality vs. number of used trial functions. dynamic optimization, large scale systems, direct method, adaptive mesh refinement, input estimation, state estimation, simultaneous approach, wavelets Keywords:
1
Introduction
The numerical solution of dynamic optimization problems is typically quite challenging for large-scale applications. The challenge becomes even more severe when real-time applications are envisaged, since the response time where a valid solution has to be prompted is fixed. In on-line applications usually the numerical cost and the associated solution time is conservatively estimated and the optimization problem is solved with well developed numerical methods (e.g. Tanartkit and Biegler, 1997, and Bock et al., 1999). The numerical effort is determined to a large extent by the complexity of the finite dimensional approximation. In fact, the computational effort grows at least superlinearly with an increasing number of decision variables. Problem adapted discretization of the control vector might be considered if one is interested in lowering the computational burden associated to a specified accuracy while at the same time the robustness of the solution algorithm is improved. In particular a discretization which is too coarse on a uniform grid or which locally resolves the profiles in the wrong location on a nonuniform grid cannot meet prespecified accuracy requirements. On the other hand, a discretization which is too fine often leads to inappropriately high computational cost and might cause robustness problems in addition. However, it is not trivial to generate a problem adapted mesh a *This work has been supported by the "Deutsche Forschungsgemeinschaft" under grant no. MAl188/6
32 priori. There have been only few attempts to incorporate adaptivity into the discretization scheme with the objective of automatically determining an appropriate mesh (e.g. Waldraff, 1996; Tanartkit and Biegler, 1997; Betts and Huffman, 1998; and Binder et a/.,1998). In this contribution, we want to address adaptive discretization strategies for input and state estimation problems which are formulated as dynamic optimization problems. The optimization problem is approximated by a nonlinear program (NLP) where states and inputs are discretized simultaneously. We suggest a systematic framework which automatically generates sequences of nonuniform grids with an increasing degree of resolution. The method is started from a coarse grid (t~ = 0) with few decision variables which can be solved efficiently and robustly. In every following refinement cycle (t~ := t~ + 1) the discretization mesh is improved in order to meet local accuracy requirements. In particular the optimization problem is formulated in a weak sense and discretized using a Wavelet-Galerkin approach. Biorthogonal wavelets arising from B-splines are employed to span the test and trial functions. The aim of this paper is to explore potential reuse of the current solution for an adaptive refinement. Improvements on the approximation of the measurements are established employing thresholding techniques. Error analysis techniques based on one-step methods are used to monitor the discretization error of the model equations whereas unknown input functions in the estimation problem are refined based on gradient information of potential trial functions. The suggested refinement procedure leads to problem adapted approximations with local resolution to guarantee prespecified local error bounds. 2
Problem formulation
We consider estimation problems formulated as a dynamic otpimization problem given by min fo(x Xo,W J(to*~f
y, w, z) dT
(P1)
under the restrictions
y
:
f(X, U, w,t),
t e [to, tf]
:
C(X, U, W, t),
t e [to, tf]
o >_ o
-
te[to, tf] z(to)-xo
.
,
(1) (2) (3) (4)
x(t), x0 C ]RTM denote the state and initial condition, y(t) e IR'~ are the model outputs and u(t) E ]Rn~ are the inputs which are assumed to be measureable and therefore known. w(t) E R n~ is an unknown time variant input function which might reflect disturbances in the process or uncertainty in the process model, z(t) c ]RTM is a reference or measurement function and known, u and z are interpolated functions based upon possibly denoised measurement samples z(ti), u(ti), l a n d c denote the process model of appropiate dimensions. Mixed constraints are denoted by g. f0 specifies the cost functional where to, tf denote the initial and final times.
33 A multiscale formulation of the process model which permits model representation of various resolutions is obtained by a weak formulation of equations (1)-(4): = 0 V ~ e (L2(I)) n~ (y - c(x, u, w, t), ~y) - 0 V C y e (L2(I)) ~ (g(x,w,t),v) << 0 V u e (L+(I)) ng
(& - f (x, u, w, t), ~ )
-
to))
=
o
(5) (6) (7) (s)
Here 5 denotes the dirac function and I := [to, tf]. For convenience we gather v := ( x T yT, U T , w T zT)T e T and r "- (~T ~T ~T) e M where T and ~ refer to appropiately chosen functions spaces. We choose appropriate finite dimensional wavelet spaces T A and 2~4A as subsets of T and ~4. Wavelets Cj,k, are specialized functions of local support which enable an efficient approximation of an arbitrary function f, e.g. J f ~ ~keZ~ o djo-l,k~jo, k -+-~-~j=jo ~keJj dj,kCj,k where Zjo , ffj denote appropiate index sets 9 Wavelets are constructed such that the Euclidean norm of the wavelet coefficients dj,k is always proportional to the L2 norm of f (see e.g Dahmen, 1997 for a survey). Thus discarding small coefficients dj,k will cause only small changes in approximate representations of f. Moreover, in regions where f is smooth the values Idj,kl will be quite small. Therefore functions that exhibit strong variations only locally, can be approximated very accurately by linear combinations of only relatively few wavelets corresponding to the significant coefficients dj,k. Denoting the set of the corresponding indices (j, k) by A, the sparse basis ~h : = {~)j,k : (j, k) C A} is expected to lead to a particularly adapted approximation to f. In particular we discretize (P1) using hat wavelets r to approximate x, y, z and Haar wavelets r for u, w , e. g. x ..~ x A "= d~ @A~, Y ~ Y A "- d ~A~, z .~ z h "= dT~A~, u ~ uA " - d uTm aH~ w ~ w h := d T~ A H . The state and output equations (5),(6) are tested by appropriately chosen dual wavelets (see Binder et al., 1998 for further details on the discretization). Since the trial functions employed are piecewise constant (r or piecewise linear (r it is sufficient to evaluate the inequalities (7) pointwise on a possibly nonuniform mesh nh:= {ti E I } which captures the finite number of knots in the approximation. The optimization problem in discretized form is now given by min OA(dx, dy, d~, dz)
(P2)
under the restrictions f A(dx, d ~ , d w , t ) cA(dx, dy, du, dw, t) gh(dx, d~, ti) eh(dx, xo)
---
u 0
,
<_ 0, =
0
V t i E/kA,
,
where f A , CA,gA a r e the finite dimensional model equations resulting from (5), (6) and (7) after projection, e i 9= d T x ~ i (to) approximates (4) and Oh "-- f f0(Xh, YA, Zh)d7 is the discretized cost functional of problem P1.
3
Adaptive Algorithm
R e f i n e m e n t of k n o w n q u a n t i t i e s : z, u are measurement functions, which have been obtained by interpolation of the possibly non-uniform discrete measurements. We limit
34 ourselves to the treatment of z since refinement of u is obtained in a similar way in every refinement step g. We are interested in compressed approximations z~A~satisfying II
-
z
!
<
,
(9)
with given tolerance c~. Let dzJ be the coefficient vector of the expansion zi - dJziT~A~ where ~ h [ is sufficiently rich to capture all measurement samples. On account of the norm equivalences (Dahmen, 1997) we simply have to neglect the elements of dz~ which are smaller than a threshold c which uniquely depends on c~. The remaining entries are the s i g n i f i c a n t coefficients and their indices form the index set Aezi. Typically the number of significant wavelet coefficients is far less than the number of discrete measurement samples. During the refinement sequence the approximation quality is increased Q+I' -< Q.' Thus the approximation from step g to t~+ 1 is refined adding trial functions with indices F ez i "- ~Ae+l A A~ " Hence, P~z~is called refinement set of step g. R e f i n e m e n t of s t a t e s x a n d o u t p u t s y: Although the discretization structure allows individual non-uniform meshes for the components of x and y we restrict ourselves to the simpler case where a common mesh is used to discretize all xi, Yi. The refinement is based on an error analysis of x~ only since errors in YA are directly linked to errors in xa. Error analysis either involves local error estimation or residual evaluation. The latter is computationally inexpensive and gives usually a qualitatively good grasp on the error behavior. However, sharp error estimates are difficult to obtain. Therefore the refinement is based on local error estimation where x~[ is compared to locally refined solutions keeping x~ e, U~A and W*A'e at their current optimal values. In particular error tests are performed at the midpoints of the current mesh associated to A~. The error tests require the solution of equation systems of size nx for each midpoint. At first glance this seems to be a substantial numerical effort. But one has to realize that the local solutions are computed by solving a sequence of decoupled equation systems of relatively small size. This is obviously much cheaper than resolving the complete refined optimization problem. The local error estimates are then checked if they satisfy error bounds which depend uniquely on the approximation quality of z~, u~, w~. Regions where the error bounds are violated are locally refined with appropriate trial functions whose indices are collected in the refinement set F eX" R e f i n e m e n t of u n k n o w n i n p u t s w: The central idea for an improved approximation of w is to evaluate gradients of the Lagrange functional s "= OA + ,kTfa + ttTcA + V~g A+~/~eAof problem P2, where AA, tt h, Vh, ~Th are the associated Lagrange multipliers, with respect to p o t e n t i a l l y n e w trial functions collected in the set II~, and encoded by e their expansion coefficients dw~,iii. Large gradients are taken as an indication to identify parameterization functions of large impact. In particular, the current W~A~is expanded by zero, i. e. w a~ e - d,~ e T ~ h ~ + dw~,ni~n~i eT e i -- 0 Local neighbors of the trial with d~,n functions in A~w~are excellent candidates for for II~ where we require II~w~A A~ = 0. Once the gradients e r i :=
Os
~
Odw,,n,
]~=~e, i - - 1 , . . . , n w
35 101
I~
-e-
.................
,
........
,
.......
scalewise I 10o
10 0 o o ._1
..J
10 -~
i
- B - scalewise I adaptive - e - adaptive accumulated
10 -1
200
400 600 800 number of trial functions
1000
10;20_2 .................................. 10-1
10~ computation time [s]
101
102
Figure 1" Comparison of scale wise vs. adaptive refinement
are computed at the current optimal solution ~ "- {x~~, y~, A~, u~l, v~t, ~7~e} we collect the trial functions associated to the larger absolute gradients in the F~ while the smaller ones are neglected. The directional gradients can be inexpensively evaluated using the adjoint formula (Fletcher, 1987) taking the already computed Lagrange multipliers A~l, tt~e v~ associated to the optimal solution ~ into account. In areas where inequality restrictions are active large gradients might be obtained. Here, refinement might not increase the approximation quality since the active inequalities already bound the solution. However, simple heuristics can be applied which exclude refinement in these areas by analyzing the active inequalities. E l i m i n a t i n g trial functions: The discretization structure of the approximations for x, y, w might change during the cycles of refinement such that we are interested in the elimination of currently unnecessary trial functions in x~, y~, w~. Similar to the previous section, the compression is based on the norm equivalences and the vectors dex, dy, d,~ and are searched for their significant coefficients which are collected in A~ c_ A~, A~ c_ A~, Final algorithm: The refinement algorithm proceeds as follows: (1) Solve problem P2 at A~. (2) Refine z, u by thresholding -+ F~, F~. (3) Estimate discretization error of x~, Y~h and locally refine areas with large errors (4) Perform gradient tests of potential refinement functions for w -+ rw. Check error and gradient tolerances and terminate refinement if satisfied. (5) Compress current approximations x~h, y~, w~ by thresholding --+ ~.~, A~,/~ and define A~z "- A~z, A~ "- A~. (6) Construct a new approximation A~+1 :-- /~etA F t taking interdependencies into account. (7) t~ := t~+ 1, go to 1. Potential savings are illustrated in Figure 1 where a comparison of the outlined adaptive
36 algorithm with respect to uniform (scalewise) discretization is shown. The figure is based on a linear estimation problem generated from nonlinear model equations of an ethylene glycol process. In the adaptive case we start with a non-uniform mesh which has been computed by analysis of the measurement functions as outlined. Adaptive refinement uses less trial functions to obtain similar approximation quality compared to the scalewise approach. The CPU times of the QP solver needed to execute the refinement steps are shown as well. Since the computation time is strongly coupled to the number of trial functions adaptive discretization is more economical, even if one considers the accumulated times. Note, that here, although in principle possible, the computation does not employ a warm start using the previous solution ~ - 1 . 4
Conclusions
We presented a framework which steadily refines the discretization of a dynamic optimization problem starting from an initial coarse grid and suggested a methodology which reuses the current solutions to generate improved discretization meshes. The suggested procedure leads to economic and nonuniform grids on which the optimization problem might be evaluated at less computational expense then on a uniform mesh of comparable accuracy. The approach explores differences in the time-frequency behavior of the solution. Local areas containing low frequencies are parameterized with large step sizes while on the high frequency areas the step size is chosen to be rather fine. Reductions in numerical expense should not be expected if the optimal solution does not reveal local differences in frequency content. The framework is not restricted to estimation problems, it can be applied to other more general dynamic optimization problems including nonlinear model predictive control (MPC) or nonlinear moving horizon estimation (RHE). References Betts, J.T. and W.P. Huffman (1998). Mesh refinement in direct transcription methods for optimal control. Optim. Control Appl. Meth. 19, 1-21. Binder, T., L. Blank, W. Dahmen and W. Marquardt (1998). Towards multiscale dynamic data reconciliation. In: Nonlinear Model Based Process Control (R. Berber and C. Kravaris, Eds.). N ATO-ASI Series. Kluwer. Dordrecht. pp. 623-665. Bock, H.G., M.M. Diehl, D.B. Leineweber and J.P. SchlSder (1999). A direct multiple shooting method for real-time optimization of nonlinear DAE processes. In: Nonlinear Model Predictive Control (F. Allg5wer and A. Zheng, Eds.). Birkh~iuser Verlag. Basel. Dahmen, W. (1997). Wavelet and multiscale methods for operator equations. Acta Numerica pp. 55-228. Fletcher, R. (1987). Practical Methods of Optimization. Wiley-Interscience. Tanartkit, P. and L.T. Biegler (1997). A nested, simultanous approach for dynamic optimization problems- II: The outer problem. Comput. Chem. Eng. 21(12), 1365-1388. Waldraff, W. (1996). Modellgestiitzte Uberwachung und Fiihrung von Fed-Batch-Bioreaktoren. VDI-Fortschrittsbericht, Reihe 8, Nr.592. VDI-Verlag. Diisseldorf.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
37
Numerical Strategies for Optimal Experimental Design for Parameter Identification of Non-Linear Dynamic (Bio-)Chemical Processes Julio R. Banga 1, Karina J. Versyck 2 and Jan F. Van Impe 2 (1) Chemical Engineering Lab, Instituto de Investigaciones Marinas (C.S.I.C.). C/Eduardo Cabello 6, 36208 Vigo (Spain). Fax: +34-986-292.762, e-mail: [email protected] (2) BioTeC - Bioprocess Technology and Control, Department of Food and Microbial Technology. Katholieke Universiteit Leuven, Kardinaal Mercierlaan 92, B-3001 Leuven (Belgium). Fax: +32-1632.19.60, e-mail: [email protected], [email protected]
1. A B S T R A C T The problem of optimal experimental design (OED) for parameter estimation of non-linear dynamic systems is considered. It is shown how this problem can be formulated as a dynamic optimization (optimal control) problem where the performance index is usually a scalar function of the Fisher information matrix. Numerical solutions can be obtained using direct methods, which transform the original problem into a non-linear programming (NLP) problem via discretizations. However, due to the frequent non-smoothness of the cost functions, the use of gradient-based methods to solve this NLP might lead to local solutions. Stochastic methods of global optimization are suggested as robust alternatives. A case study considering the OED for parameter estimation in a fed-batch bioreactor is used to illustrate the performance and advantages of two selected stochastic algorithms.
2. I N T R O D U C T I O N Sound mathematical models are the fundamental elements of modern methods in process systems engineering. The development of a model is usually an iterative process which involves two basic steps: designing the model structure, and estimating the model parameters for a given structure from experimental data. Here we will consider this latter task of parameter estimation considering dynamic models of chemical and biochemical processes. These processes are generally described by systems of differential and algebraic equations (DAEs), with the particularity that these models are frequently highly non-linear and stiff. The parameter estimation problem can be formulated as finding the parameter vector p to minimize some norm of the weighted differences between a vector of measured outputs ym(t) and the model predictions y(p,t). Performing experiments to obtain a rich enough set of ym(t) is a costly and time-consuming activity, very especially when considering real industrial units. The purpose of optimal experimental design (OED) is to devise the necessary dynamic experiments in such a way that
38 the parameters are estimated from the resulting experimental data with the best possible statistical quality. The latter is usually a measure of the accuracy and/or decorrelation of the estimated parameters. Although the OED applied to linear steady state models is a well known subject, in the more challenging case of non-linear dynamic models, as those arising from (bio)-chemical processes, fewer references are available (e.g. Munack and Posten, 1989; Lohmann et al, 1992; Baltes et al, 1994; Versyck et al, 1997; Bauer et al, 1999). Mathematically, this OED problem can be formulated as a dynamic optimization (optimal control) problem. The objective is to find a set of time-varying input variables (controls) for the experiments so as the quality of the estimated parameters is optimal in some statistical sense. This optimal control problem is subject to a set of equality constraints (the DAEs describing the system dynamics) and a set of inequality path and/or point constraints on the state variables. These inequality constraints are usually associated with restrictions concerning issues like practical implementation, safety and/or model validity. The cost function reflecting the statistical quality of the parameters is usually constructed as a certain function of the corresponding Fisher information matrix F:
(1)
with between the brackets the sensitivity matrix of the model outputs (in the vector y) with respect to the parameters (in the vector p) and Q the inverse of the measurement error covariance matrix under the assumption that the outputs are corrupted with zero-mean gaussian noise. Thus, the mathematical formulation of the OED problem can be written as: Find u(t) to minimize J = ~(F) subject to f[x,x,p,u,t]=O X(to)- Xo
(2)
h[x,y,p,u,t]= 0
(3),(4)
x L_<x_<x v
g[x, y, p, u, t] _<0
u L
u
(5),(6)
where J is the performance index (cost function), x is the vector of state variables, u the vector of control (input) variables, Eqn. (2) is the system of ordinary-differential equality constraints with its initial conditions, Eqns. (3) and (4) are the equality and inequality algebraic constraints, and Eqns. (5) and (6) are the upper and lower bounds for the state and control variables. The performance index is a scalar function ~(F) of the Fisher information matrix. Examples of common functions are the so-called D-criterion (maximizing the determinant of F), where J =-~V I , which measures the global accuracy of the estimated parameters, and the modified E-criterion (minimizing the condition number of F, i.e. the ratio between the maximum and minimum eigenvalues), where J = A ( F ) = Am.x(F)/Ami.(F ) , which measures the parameter decorrelation. Thus, OED of these processes implies the solution of a class of very challenging non-linear, highly constrained dynamic optimization problems. Besides, additional difficulties may arise due to the non-smooth nature of the abovementioned cost functions.
39 3. S O L U T I O N M E T H O D S Dynamic optimization methods can be classified in two major groups: indirect approaches, based on the minimum principle of Pontryagin, and direct approaches, which transform the original problem into a non-linear programming (NLP) problem via discretization of the control and/or the state variables. In the case of the problem stated above, indirect approaches are not applicable due to the non-differentiable nature of the costs. Thus, one must resort to direct approaches, reducing the problem to the solution of an NLP, either following the control vector parameterization (CVP) strategy (Vassiliadis et al, 1994) or the complete (controls and states, via collocation) parameterization strategy (Cuthrell and Biegler, 1989). In any case, standard gradient-based NLP-solvers may fail to converge, or converge to a local solution, due to the non-smooth and highly non-linear nature of the problem (Barton et al, 1998). In this study, we have followed the CVP approach, where the control variables are approximated by some type of discretization, usually based on low order polynomials (e.g. piecewise constant or piecewise linear). This discretization results in an NLP main problem, where the decision variables are the coefficients used to build the approximation. This main NLP problem requires the solution of the initial value problem formed by Eqns. (2) and (3) for each function evaluation. The most popular methods for solving the NLP are gradientbased local optimization techniques, usually variants of the powerful sequential quadratic programming (SQP) method. But, as already mentioned, since this NLP may be non-convex due to the non-smooth nature of the performance indexes, SQP solvers may converge to local solutions, or even fail to converge. Global optimization techniques are needed in order to surmount these difficulties. The use of stochastic or hybrid stochastic-deterministic methods have been suggested as efficient and robust alternatives for similar difficult optimal control problems (Banga and Seider, 1996). Here, we will apply two stochastic techniques, namely Integrated Controlled Random Search (ICRS) and Differential Evolution (DE), which have been found particularly suitable for this type of problems (Balsa-Canto et al, 1998). ICRS is an adaptive stochastic method developed by Banga and Casares (1987), while DE is a population based method recently presented by Storn and Price (1997). Both methods are very simple to implement and use, which is an important characteristic for many potential users.
4. CASE STUDIES We illustrate the advantages of these strategies considering an important problem in biotechnology: the estimation of kinetic parameters of unstructured microbial growth models. The dynamics of a fed-batch bioreactor where one biomass is growing on one limiting substrate are described by the following model: dCs - - o ( C S )C x -[- Fin (Cs,in -- C S ) dt
dC~ dt
-
g. = ~t(Cs)% --:-:-c~ v
(7) (8)
40
dV dt
--=
Fi,,
(9)
with Cs [g/L] the (volumetric) concentration of limiting substrate, Cx [g DW/L] the biomass concentration, V [L] the volume of the liquid phase, Cs,i,, [g/L] the substrate concentration in the influent with flow rate Fi,, [L/h], cy [g/g DW h] the specific substrate consumption rate and ~t [ l/h] the specific growth rate. These rates are functions of Cs: 1
cr(Cs) -yx~s ~(Cs)+ m
(10)
CS
~z(c~)-lz,,, X,, +C~ +C~/K,
(ll)
with Yx/s [g DW/g] the biomass on substrate yield coefficient and m [g/g DW h] the (overall) specific maintenance demand. Eqn. (11) is known as the non-monotonic Haldane growth kinetics, with parameter Kp [g/L] indicating how fast the optimum for the specific growth rate is reached, and K; [g/L] the inhibition parameter. In this case study, our objective is to estimate parameters Kp and Ki based on measurements of the substrate concentration Cs and the biomass concentration Cx in a fed-batch experiment. The information matrix F (1) can be computed from the sensitivities obtained by solving an extended initial value problem. Values for initial conditions, operating conditions and other model parameters can be found in Versyck et al (1997). The OED problem is formulated as finding the optimal substrate feed rate so as maximum global accuracy and/or parameter decorrelation is obtained, subject to the differential equality constraints (model dynamics, Eqns. 7-11) plus bounds for the control and the states. The following two sub-problems are considered: a) Optimal control for parameter decorrelation" the cost function is the modified E-cost (ratio of the maximum and minimum eigenvalues of matrix F). b) Optimal control for parameter accuracy: the cost function is the D-cost (determinant of matrix F). Preliminary runs indicated that path inequality constraints on Cs must be imposed in order to guarantee model validity. More specifically, wild jumps in the substrate concentration are not acceptable because the condition of balanced growth (which is at the basis of the unstructured growth kinetics considered) would not hold (Roels, 1983). Considering sub-problem (a), it should be noted that the modified E-cost should be as close as possible to 1, which corresponds to a situation of complete decorrelation. Versyck et al (1997) followed a heuristic procedure, based on theoretical analysis of the optimal process performance feed rate profile, to design optimal control inputs which provide the desired complete decorrelation. Here, we started from no a priori assumptions and solved the problem in a purely numerical way. Apart from the stochastic techniques mentioned in the previous section, we also tried to solve the problem by using a deterministic (SQP-based) solver. Many
41 difficulties were encountered by the SQP solver, and in fact convergence failures or convergence to local solutions after excessive computation times were found, in agreement to the reasoning presented above about the non-smoothness of the cost function. In contrast, the stochastic solvers arrived to very good solutions (cost functions close to 1 within a tolerance of 10-5 in most cases) in very reasonable computation times (order of minutes in a PC Pentium-II). These solutions, which in this case we know are globally optimal, agree quite well with those of Versyck et al (1997). For example, Figure 1 shows the optimal feed rate found by these authors, while those obtained with the stochastic methods are presented in Figure 2 (for the particular case of fixed tf = 86.3 h and Cs(O)= 38.4 g/L).
,
........
,
0.1
0.25
,
,
....
~
~
0.08
El
0.2
2" "1 ~.0.15
~
0.06
u.
0.04
,...., t---
._ LL
0.1
ICRS 0.02
0.05
0
..... 0
I
........
20
40
time [h]
6'o
80
0
1oo
o
2'0
4'o
time [h]
60
8'0
,oo
Figure 2.- Optimal feed rates obtained with ICRS and DE for subproblem (a)
Figure 1.- Optimal feed rate by Versyck et al (1997) for subproblem (a)
Considering sub-problem (b), the determinant of F must be maximized. There is no a priori knowledge about the globally optimal value for this cost function, so we have to limit ourselves to compare the solutions obtained by the different stochastic solvers and the heuristic solution obtained by Versyck et al (1997). These authors obtained a D-cost of IF1= 2.2143-10 , with Cs(O) = 68.5 g/L and tf 218 h. Here, the problem was solved for TM
-
-
several time horizons with the different solvers. In this way, a compromise between statistical quality and process time can be reached by using the results as a pareto-optimal response curve. In agreement with the previous results, the SQP-based deterministic solver failed to arrive to good solutions. In contrast, the ICRS stochastic method arrived to a better value of IF[ = 1.9770.1016 for a shorter process time of 86.3 h and Cs(O) = 38.1 g/L. The DE method clearly outperformed the ICRS method in this sub-problem, arriving to a value of 2.7155.10 , with Cs(O) = 31.7 g/L and the same process time.
IFI
TM
When considering the optimization results obtained for sub-problem (b) with respect to the modified-E-cost in (a), and vice versa, it should be noted that the minimization of either cost function is at the expense of the alternative cost. For example, the best solution (obtained by DE) for subproblem (b) yields a modified E-cost of 5.1580.1011. This confirms the results reported for heuristic solution procedures in Versyck and Van Impe (1998). Current research focuses on the stochastic optimization of a novel cost combining both criteria in a weighted sum. As such, a trade-off between the two parameter estimation quality measures is aimed at.
42
5. C O N C L U S I O N S In this contribution, we compared several numerical strategies for solving two optimal experimental design problems in the context of parameter estimation for (bio-)chemical process models. The stochastic global optimization methods show the best performance in terms of the criterion values and the computational efficiency for these non-linear and highly constrained dynamic optimization problems.
Acknowledgements Author Karina Versyck is a research assistant with the Fund for Scientific Research - Flanders (FWO). The scientific responsibility is assumed by its authors.
REFERENCES Balsa-Canto, E., A. A. Alonso and J. R. Banga (1998). Dynamic optimization of bioprocesses: deterministic and stochastic strategies. In Proceedings of ACoFoP IV, 21-23 Sept., G6teborg. Baltes, M., R. Schneider, C. Sturm and M. Reuss (1994). Optimal experimental design for parameter estimation in unstructured growth models. Biotechnol. Progr. 10:480-488. Banga, J. R. and W. D. Seider (1996). Global optimization of chemical processes using stochastic algorithms. In "State of the Art in Global Optimization", CA Floudas and PM Pardalos (Eds.), p. 563-583. Kluwer Academic Pub., Dordrecht. Banga, J. R. and J . J. Casares (1987). Integrated Controlled Random Search: application to a wastewater treatment plant model. IChemE Symp. Ser. 100,183-192. Barton, P. I., R. J. Allgor, W. F. Feehery and S. Galan (1998). Dynamic optimization in a discontinuous world. Ind. Eng. Chem. Res., 37:966-981. Bauer, I., H. G. Bock, S. K6rkel, and J. P. Schl6der (1999). Numerical methods for initial value problems and derivative generation for DAE models with application to optimum experimental design of chemical processes. In Proc. of Int. Workshop on Scientific Computing in Chemical Engineering, May 26-28, vol. II, TU Hamburg, Germany. Cuthrell, J. E. and L. T. Biegler (1989). Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng., 13:49. Lohmann, Th.W., H. G. Bock and J. P. Schloder (1992). Numerical methods for parameter estimation and optimal experiment design in chemical reaction systems. Ind. Eng. Chem. Res., 31:54-57. Munack, A. and C. Posten (1989). Design of optimal dynamical experiments for parameter estimation. In Proceedings of the American Control Conference, ACC89, Pittsburgh, USA, pp. 2011-2016. Roels, J. A. (1983). "Energetics and kinetics in biotechnology". Elsevier, New York. Storn, R. and Price, K. (1997). Differential Evolution - a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11:341-359. Vassiliadis, V. S., R. W. H. Sargent and C. C. Pantelides (1994). Solution of a class of multistage dynamic optimization problems. Part I. Ind. Eng. Chem. Res., 33:2111-2122. Versyck, K. J., J. Claes and J. Van Impe (1997). Practical identification of unstructured growth kinetics by application of optimal experimental design. B iotechnol. Progr., 13:524-531. Versyck, K. J. and J. F. Van Impe (1998). Trade-offs in design of fed-batch experiments for optimal estimation of biokinetic parameters.In Proceedings of the 1998 IEEE Conference on Control Applications, Control Applications in Biological Systems, 51-55, IEEE Catalog Number: 98CH36104 (also on CD-Rom)
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
43
Solution of population balance equations for prediction of particle size distribution in emulsion polymerization: comparison and evaluation of different numerical methods. A.H. Alexopoulos and C. Kiparissides Department of Chemical Engineering and Chemical Process Engineering Research Institute, Aristotle University of Thessaloniki, P.O. Box 472, Thessaloniki, Greece. 1. INTRODUCTION The quantitative description of the development of latex particle size distribution (PSD) in emulsion polymerization reactors is very complex. Important process variables, such as type and concentration of initiator and surfactants, polymerization temperature, and variation of the physical properties of the latex during the polymerization, have been included in a recent mathematical model for predicting the evolution of particle size distribution in emulsion polymerization reactors (Kiparissides et al. 2000). The formation of panicle nuclei can occur by both homogeneous and micellar nucleation mechanisms (Gilbert, 1995). The nucleated particles can subsequently increase in size either by growth due to polymerization or/and coagulation with other particles. The combined action of nucleation, growth, and aggregation determine the evolution of the latex panicle size distribution PSD (Feeney et al. 1984). The evolution of the latex PSD is further complicated due to the strong coupling between reaction kinetics and panicle size during emulsion polymerization. The use of population balance equations (PBE) to describe the development of panicle size distribution in a paniculate process is a well-known method and has been utilized in a diverse range of problems including crystallization, liquid-liquid dispersions, polymerization (Ramkrishna, 1985). The continuous distribution of panicles in an emulsion polymerization reactor is usually described by a number density function n(x,t) which represents the number of panicles within a differential volume size range per unit volume of latex. The rate of change of the panicle number density function is described by a nonlinear integro-differential population balance equation. o~n(x, t) alv (x)n(x, t) c3t + ax = 8 ( x - Xmin)So(t) y=X_Xmin
+-2
y=Xmax
(1)
I I 3 ( y ' x - y ) n ( y , t ) n ( x - y , t ) d y - n(x,t) II3(Y'x)n(y't)dy y=xmin
y=Xmin
where Iv is the particle growth rate, So is the nucleation rate, and 13(x,y) is the aggregation rate kernel between particles of volumes x and y. The numerical solution of the PBE generally requires the discretization of the particle volume domain into a number of elements leading to a usually stiff set of nonlinear differential equations. The application and subsequent solution
44 of the PBE in reacting, and, especially, nucleating systems, is difficult due to the complexity of particle nucleation mechanisms that are often not well-understood or/and the numerical difficulties involved in the solution of PBEs. In this work two different numerical methods, namely, the discretized PBE method of Hounslow et al. (1988) and the orthogonal collocation on finite elements (OCFE) method, are applied for the solution of the particle population balance model (1). A detailed study is carried out to assess the accuracy of the numerical methods and to establish the conditions for the convergence of the numerical solution to the correct PSD. This is accomplished by comparing the numerically calculated PSDs with the corresponding analytical solution for the case of a constant aggregation kernel. 2. NUMERICAL METHODS FOR THE SOLUTION OF PBE 2.1. Discretized Population Balance. The particle size distribution can be determined using a discretized PBE which assumes a constant number density function ni within the size interval "i" such that a particle number density Ni can be defined as:
gi+l (2)
N i = ~ nidW = nii(Wi_,- Vi) =ni--Vi
vi Accordingly, following the developments of Hounslow et al. (1988), one can derive an equivalent to eq. (1) discretized PBE including particle aggregation, growth, and nucleation mechanisms.
I'Ir
'
dN i dt -8(i-1)B(t)+(l+r)V i r2-1Ni-1 + N i - r 2 rNi+l
1 (3)
"~'2"~i_l,i_lNi_ 1 - N i Lj=
2J-i~i,j Nj )"]-~(~i,j Nj j=l
The terms on the right hand side of eq. (3) represent particle nucleation (B=VoS0), growth, generation due to aggregation, and depletion due to aggregation, respectively. Unfortunately, for many problems the geometric discretization of eq. (3) (based on the rule: Vi+l=2Vi) is often inadequate. As a result Lister et al. (1995) proposed a more general discretization method based on the following fractional geometric discretization rule:
Vi+l - V i * 21/q
(4)
where q is an integer greater than one. This approach has been further generalized by Kumar and Ramkrishna (1996) for any type of discretization which guarantees the correct determination of any two moments of the PSD (e.g., zero moment and first moment which correspond to the total number of particles and total particle volume, respectively).
45 2.2. Collocation on Finite Elements The PBE can also be solved in the continuous form (eq. 1) using orthogonal collocation on finite element methods (OCFE) (Gelbard and Seinfeld, 1978; Nickmanis and Hounslow, 1998). The collocation on finite elements was used by Gelbard and Seinfeld (1978) with cubic Hermite basis functions and equally spaced (non-orthogonal) nodes to simplify the "bookkeeping". The method was very successful to the solution of pure aggregation problems and also in the case of a combined growth-aggregation problem. In the collocation on FE method nc collocation points are defined within ne elements and the solution is obtained by an expansion of the unknown number density function in terms of appropriate basis functions over each element. nc+2
(5)
n(x,t) = Y'~n](t) qoj(x) j=l e
,,
where nj indicates the value of n(x,t) at node "j of the element "e" and (pj are Lagrange basis functions. Based on the above discretization procedure eq. (1) can be recast into the following ne(nc+ 1)+2 residual equations: 9e n i
(t)= -
1 +--2
nc~.+2(dq) j "~ j=~l' ~--'~-)
(div 1
i
ns ns
e e e e f f nJ (t)ljlelv(x)- \ dx Ji n i (t) - n i (t) f=l j=l w(j)lJI e 13(xi ,xj )nj (t) nc+2
w(J)lJlgl3(x~ -- xgj'xjg)n~
-- g=l
1 (pfjn
\
+~-
]
ng
(6)
[3(xe - x , xng)n(x)n(x e - x)dx Xlng
where f = map(x~, xjg) is a mapping function indicating the element that the difference xie --xjg belongs to. The last term of eq. (6) corresponds to the "fractional" birth element, 1
ng = map(x~,xl), which is integrated separately. The integrals are determined using GaussLegendre formulas at the node positions xj with weights of wO). The mapping of element e into the integration domain ~ ~[-1,1 ] is described by the Jacobian of the mapping function [Jle The system of equations (6) is closed with two boundary conditions and ne-1 continuity conditions and integrated with respect to time using standard DAE solvers. 2.3. Full-Discrete Simulation Many particulate polymerization processes (e.g., emulsion, dispersion, etc.) at the early stages of reaction can be approximated by a discrete population balance model assuming zero particle growth rate and the simultaneous presence of particle nucleation and aggregation mechanisms. For example, in emulsion polymerization, the precursor-particle model of Feeney et al. (1984), which assumes a homogeneous coagulative nucleation mechanism, can take the following PBE discrete form: 1 i-1
dN~ _ B o S ( i - 1 ) +
ne
~13(i_j,j)Ni_jNj _~[3(i,j)NiN j
(7)
46 The above full-discrete model (Hidy, 1965) assumes that particles of a specific size consist of i-multiples of a unit size corresponding to the particle nucleus. This approach allows the use of realistic aggregation kernels and nucleation rates although, due to the computational burden, its application is limited to small domain sizes. Furthermore, it can not account for particle growth. 3. R E S U L T S In Figure 1 the accuracy of the collocation and discretized PBE methods is shown. Both numerical solutions are compared to the analytical solution n(x)=N0/V0e 2v/v~247 where "c is a dimensionless integration time (x=130N0t) and 130 and No denote the characteristic aggregation kernel and particle number, respectively. As can be seen the OCFE method produces accurate PSDs even for extended particle aggregation times provided that sufficient collocation points are used. The OCFE method requires more computation time than the discretized PBE method but it is more accurate than the latter, especially, at larger particle diameters (see Fig. 1b "c=10). The solution of the combined nucleation and aggregation problem did not present any difficulties for the discretized PBE method except for the case of kernels promoting small to small particle aggregation. The numerical solutions of the PBEs were assessed by a direct comparison with the solution obtained by the full-discrete model. In Figure 2 the full-discrete method is compared with the discretized PBE solution for a Brownian kernel with a dimensionless nucleation rate of B0/130N0=10-3. The PSD peak positions were produced accurately despite the observed numerical dispersion of the PSD. In Figure 3 it is shown that in the presence of an initial seed (Fig. 3a) the solution obtained with the discretized PBE method is smoother and presents fewer numerical difficulties. Figure 4 shows that PSDs which include nucleation, growth, and aggregation can develop a "secondary nucleation" at a time that depends on the dimensionless nucleation rate (Fig. 4b). After the "secondary nucleation" the small diameter distribution converges to a steady state as with the Brownian aggregation case shows in Fig. 3a. This behavior is a result of a dynamic balance between nucleation and aggregation of small with the larger particles in the PSD (Sathyagal and McCormick, 1998). |
'~
x =0
1
Analytical
_,
c)
x=0
z z
0.1!
-- " "
",
Numerical
z o" .,..,
x=l
x~
E
0.01-
1E-3
~= 1 E - 3 -
IE-4
"~
'
'
''""1
'
'
''""l
1.0 10.0 Volume Ratio, V/Vo
'
'
"
.
.
100.0
.
.
Numerical - ~-
%
/ I :1oo
1E-4 -!
z 1E-5
, x=l
0.01
z
Analytical
0.l-
\
-.
1E-5 . .
.
0.
'"l
'
'
''""l
1.0 10.0 Volume Ratio,V/Vo
'
'
'
..... 100.0
Fig. 1. Simulation of a PSD undergoing only aggregation. Comparison between analytical and numerical solutions" (a) Collocation on FE (ne=10,nc=4); (b) Discretized PBE (q=2).
47
0.1 o
0.01 -
y
~ 1
x=100
[
~
o
x=200 x=300
1E-3 -
I
:
0.01 1
Z=100
X
~=20o x:300
1E-3 !
.,-,
.,..,
~
1E-4 -
~-
IE-5 -
>
1E-6 -
x=400
= o
x=400]
1E-4
x=500
x=500 1E-5 o
1E-7
> '
' '''"'1
'
' ' ....
1E-6 1E-7
'1
'
'
'''"'1
10 100 V o l u m e Ratio V / V 0
'
' '''"'1
'
.....
'"
10 100 V o l u m e Ratio V / V o
Fig. 2. Simulation of a PSD undergoing aggregation and nucleation. Comparison between full discrete (a) and discretized (b) PBE solutions. (a) N= 1000 points" (b) ne=20points (q= 1) 1
1
9
o
"
Z
0.1-1
.~
0.01 -~
~:o
~
~=2
Z d
0.01
~
1E-3
1E-3~
~
IE-41
z=100
~
1E-4
~
1E-5
x~ 1 E - 5 ~
.~
1E-6
~ Z
~ Z
1E-7
1E-6 -!
~// \ ~
1E-8
1E-7
. . . . . . . .
I
. . . . . . . .
I
1.0
10.0
Diameter
Ratio, D / D o
~--0.2 x=2.4
"""
x=34 ~
1~~"\
'
X
'
' ' ''"1
~
'
"r=4.6
' ' ''"
. . . .
1
10
100
D i a m e t e r Ratio, D / D o
Fig. 3. Simulation of a PSD undergoing Brownian aggregation and nucleation kinetics. Comparison between (a) seeded and (b) unseeded case. Nucleation rate Bo/13oNo=10 .2 l
o
100.0
--
~=0
--
Z
T=2
Z ._ .,_., r
_
z
0.1~
x=6 T=I0
0.01~
10.0 --= [.,
.,_,
1.0
1E-3~ E
.,-,
Z
IE-4
!
10 Diameter
100 Ratio, D/Do
!
0.1 1E-6
1E-5
IE-4
Dimensionless
1E-3 Nucleation
Fig. 4. PSD evolution with constant aggregation, nucleation, and growth rates. (a) Dimensionless nucleation and growth rates: Bo/13oNo2=10-3, Iv/13oNoVo=l (b) Dimensionless time required for the development of secondary nucleation.
0.01
0.1
Rate So/rioN o2
48 10
><(
6-
N o seed
"F..
t=100 min
8-
E
t=30 min
--i--
~=
t=170
8-
7>
4-
o
"=
t=30 min
~
o
o
t=0 min
,~L
6-
,....
;:>
[ ~(
(.r.,
0
. . . . . 9",,,r,, - ,r",,,r,, v , , r - r , ~
10 100 Diameter Ratio, D/Do
1000
A /
/ ! /
l,
42-
2-
t=100min t=17Omin
eel [.r.,
Seed: 60nm
......
2 ?j
r--'-""'-"-'" r-'" " 10 100 Diameter Ratio, D/Do
1000
Fig. 5. Simulation of the PSD for an emulsion polymerization process. Nuclei size Do=2nm. The discretized PBE was subsequently used to study the behavior of an emulsion polymerization process. The nucleation, growth, and aggregation functions were obtained from an emulsion polymerization model (Kiparissides et al. 2000). The dynamic nature of the polymerization prohibits the formation of the small-diameter "steady-state" discussed above. The mean diameter of the PSD depends on the growth rate and the total amount of nucleated particles while the shape of the PSD is determined by the aggregation kernel (Fig. 5). Different nucleation transients including secondary nucleation were examined and were found to affect mostly the small-diameter portion of the PSD. REFERENCES Kiparissides, C., Achilias, D.S., Fratzikinakis, C., and Samaras, S. 2000. The effect of oxygen on the homogeneous particle nucleation in emulsion polymerization. I & EC. (to be submitted). Feeney, P.J., Napper, D.H., and Gilbert, R.G. 1984. Coagulative Nucleation and Particle Size Distributions in Emulsion Polymerization. Macromolecules, 17, 2520-2529. Gelbard, F. and Seinfeld, J.H. 1978. Numerical solution of the dynamical equation for particulate systems. J. Comp. Phys., 28, 357-375. Gilbert, R.G. 1995. Emulsion Polymerization. Academic Press, London,. Hidy, G.M. 1965. On the theory of the coagulation of noninteracting particles in Brownian Motion. J. Colloid Sci., 20, 123-144. Hounslow, M.J., Ryall, R.L., and Marshall, V.R. 1988. A Discretized Population Balance for Nucleation, Growth, and Aggregation. AIChE J., 34, 1821-1832. Kumar, S. and Ramkrishna, D. 1996. On the Solution of Population Balance Equations by Discretization-I. A Fixed Pivot Technique. Chem. Eng. Sci., 51, 1311-1332. Lister, J.D., Smit, D.J., and Hounslow, M.J. 1995. Adjustable Discretized Population Balance for Growth and Aggregation. AIChE J., 41, 591-603. Nickmanis, M. and Hounslow, M.J. 1998. Finite-Element Methods for Steady-State Population Balance Equations. AIChE J., 44, 2258-2272. Ramkrishna, D. 1985. The Status of Population Balances. Rev. Chem. Eng., 3, 49-95. Sathyagal, A.N. and McCormick, A.V. 1998. Effect of Nucleation Profile on Particle-Size Distribution. AIChE J., 44, 2312-2323.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
49
Solution of the hyperbolic model for heat and mass transfer in packed bed reactors A. A. Iordanidi, A. E. Kronberg, J. A. M. Kuipers and K. R. Westerterp Department of Chemical Engineering, Twente University of Technology, P.O. Box 217, 7500 AE Enschede, The Netherlands 1. INTRODUCTION The importance of packed bed reactors in chemical industry has prompted extensive research on mathematical modeling of such systems. One of the key problems involved is the formulation of conservation equations for the smoothed concentration and temperature profiles in the flowing fluid. The mass and energy balance equations for axi-symmetrical problem and at steady state can be written as: OC 1c3(rjmr ) OT lO(rjhr) ajh ~ u--+-~+ - + - ~ r- J - c3r =Q(C,T) (1) C3Jmx = R t,C, T ), "OCpU -ox & r Or & Ox In order to use these equations to determine concentration and temperature distributions additional relations for the mass and energy dispersion fluxes jmx, jmr, jhx and jhr are needed. The only approach used for practical calculations is to represent the mixing in terms of modified Fick's law of diffusion and Fourier's law of heat conduction:
j.~ =-D~ OC ~ '
J~r =-Dr aC -&r'
Jhx = - K x OT ~x'
OT Jhr = - K r Or
(2a)
where effective dispersion coefficients Dx, Dr, Kx and Kr are used instead of molecular transport parameters. Eqs. (1), (2a), are of parabolic (diffusion) type and together with appropriate boundary conditions constitute standard packed bed reactor model. Its numerous modifications are described in the usual textbooks [1,2]. The capabilities of the standard model are recognized. However, experiments at high reaction rates and in the reactors with low tube to particle diameter ratio demonstrate that the model breaks down [3,4]. This is not unexpected. More surprising is that the model can be made to yield good agreement with experiments in many cases because nearly all packed bed reactor problems violate the requirements for the applicability of Eqs. (2a). The same problems with the gradient transport models are recognized in many other fields of transport phenomena. In particular to overcome drawbacks of the Fourier's law the concept of heat transmission by waves has been introduced [5,6]. Recently the wave (hyperbolic) model for longitudinal dispersion has been derived [7]. Kronberg and Westerterp [8] derived two-dimensional wave model for packed bed reactors similar to that proposed by Stewart [9]. An essential feature of the wave model is that mass and heat dispersion fluxes are taken to be additional to C and T state variables. According to the packed bed wave model the relations between the dispersion fluxes and T and C to be used instead ofEqs. (2a) are [8]: U aim r c3C 0 - rmP)Jmr + "Cm- - ~ = -Dr--, g Ox Or
U Oj,,~ OC 0 - rmP)J~x + rm - ~ = -Dxg Ox Ox'
(2)
50
aT
U ajh r
where
u ajhx
K r --,
0 -- "chB)Jhr + I'h . . . . cox
aT
(1- rhB)Jhx + r h . . . .
Or
P(C,T)=aR(C,T)/aC,
(3)
K x --
e ax
ax'
rm "rh are the model
B(C,T)=(pCp)-laQ(C,T)/aC;
parameters of time dimension. The boundary conditions of the wave model are: X=0"
C=Co,
r=0"
Jmx=0,
Jmr=O,
Jmr=O,
Jhr=O;
T=T
o,
jhx=0
(4)
Jhr=hw(T-Tw)
(5)
Jhr=O,
r=F t " Jmr=O,
Here we assume that pCprhU 2 > Kx and rmU2 >_Dx, otherwise upstream mass and heat transfer are allowed and the boundary conditions over axial coordinate will be different. Note that in this case the wave model - in contrast to the standard model - doesn't require the boundary conditions at the reactor outlet. Fundamental advantages of the new model have been demonstrated [8]. For the calculations of packed bed reactors the new model has never been applied. Since Eqs. (1)-(3) are of the hyperbolic type, their solution requires methods different from those well developed for diffusion-type models. The purpose of this paper is to develop a suitable finite difference method to numerically solve the wave model and to identify some conditions at which predictions of the standard and wave models diverge. 2. N U M E R I C A L A P P R O X I M A T I O N For the numerical approximation of hyperbolic equations method of characteristics is extensively used. The method maximally exploits the properties of the differential equations, converting them to ordinary differential equations [10]. It is particularly convenient if the slopes of characteristics are constant (as in the case of Eqs. (1)-(3)). The main shortcoming of this approach is that to integrate system (1)-(3) along its characteristic curves one must fix the ratio of the radial and the axial integration steps. As a result excessively tiny partitioning determined by the characteristic slopes - may appear in one of the directions. In this section we shall show that the Wendroff's [ 11 ] algorithm adopted to the problem (1)(5) is unconditionally stable, does not depart much from the characteristic method, requires less computer memory and is faster. To derive the scheme we introduce new dependent variables
II1
-
T
-
+
mh T 0
~
Jhx '
Y2 - -
lOC p U T 0
C
Jinx
2.Vh
Jhr
2
pC p U[ o
k2 C
--'-I- - -
uC o /, I"
2Vm C O
where
= [8Kr/[t7~7 p
Vh
r i pCpU2~.h
2
,
1/2,
Vm --
r
~2 T
Y3 -
'
~ I +
2
/ u r2 m 1/2,
o~Dx Wh
2V m C O
_
2
k2 C Z 3 =
uC o
k
2.Vh
i
2 Vh T O
k V m Jmr
Z 2 = - -
ZI=Wmc o
=
A,2 T 2 Vh TO
_
Jhr
[~7,p U [ 0
k V m Jmr "]
2
uC o'
,
2 =VhKx/Kr
Wh =
/K
In terms of the new variables Eqs. (1)-(3) become: aY~
&
=
5(Y~,Zi,r)
,
aZ, __ G , ( y . , Z i , r ) , az
OY2 /l aY2 = F~(Yi , Z,,r) az
az= az
as
,
k aZ2 =G2(Y,.,Zg,r ) , as
aY3 az
k/],
ar, as
= F 3(Yi,Zi,r)
aZ 3 + k aZ 3 = G3 ( y i , Z , , r ) az
as
(6)
(7)
51 where i=1, 2, 3; Fi, a n d Gi are the known functions determined by Eqs. (1)-(3); z = x / r t , s=r/rt; 2 and k determine the directions of heat and concentration wave propagation. Boundary conditions (4), (5) shall be rewritten as
,~2 z = O"
YI = Wh ,
s=0:
Y3=Y2,
k2
Yz = }"3 - Z V h
Z3=Z2;
Z I = Wm
s=l:
Z2=Z 3
Yz=a, Y3-fllTw,
2Vm Zz=Z 3
(8) (9)
where a 1, fll are the known constants. The new variables allow us to treat the first Eqs. of (6) and (7) as ordinary differential equations. They are solved by the second order explicit Runge-Kutta method. A combination of the Wendroff's and the second order Runge-Kutta methods is exploited to solve the second and the third Eqs. of (6) and (7). To derive the finite difference analog of the second equation of (6) we approximate the axial derivative at the n + 89level as the average of the derivative at the n and n + 1 levels and the radial derivative at thej + 89level as the average of the derivative at thej and j + 1 levels:
k C~Z J j+l/2
i /n lJ2 \ Os Y j+I/2
n
.,
= -'2
]j+l _(r
L1
Az
-
n1
+
-2
As
(11)
As
Using Eqs. (10) and (11) and auxiliary functions Y~, i = 1, 2, 3; we construct difference scheme: ( r"~ z ) J = ( Y z ) J +nl
. 2Az . l1+- -pp ((Y2)j+I- (Y2)j) + l + p F2(Yi,Zi,I")j+I (12)
(Y2)~ +1 = (Y2)j+I
11 +- pp
Y2,j+,
)j + 1 + p
,Z,,r
+Fz(Y~,Z,,r)j+,
wherej = 0 ... M-l, n = 0 ... N-1 refer to the radial and the axial grid points respectively; As = 1/M, Az = 1/N and p = 3.Az/As. Similar approximation is applied to the second equation of (7). For the third equations of (6) and (7) the derivatives are approximated at the levels n + 89andj 89 The first of Eqs. (12) represents the Euler's method for the corresponding differential equation, therefore we set for the grid functions Yi the same boundary conditions as for Y,: z =0:Y2 = Y3 = A//ZVh ,
s=0"
~ = I72 ,
s = 1" ~ = a l ~ - f l , r w
(13)
The scheme has the local truncation error O(Az 2 +As2). The error is determined by the approximation of the differential operators O/Oz +_ A.O/Os and can not be improved using more accurate integration of the source terms F2,3 and G2,3. Because of coupled boundary conditions (13) Eqs. (12) must be solved simultaneously with the corresponding difference equations for Y3. To do so we shall rewrite them in the form:
=a(Yz)j+l +bj, j E [ 0 , M - 1 ] ; where bj and dj depend on (g)", (Z)"; (Yz)j
(Y3)j = a(Y3)j-1 + d j , j E [ 1 , M ]
(14)
a = - (l-p)/(1 +p), la[ < 1. Running index j from 0 to M - 1 in the first equation of (14) and from M to 1 in the second one we get:
52 ~(a
(Y2)o :aM(y2)M +
M-1
Ybj),
(Y3)M:aM(Y3)o + Z(aJdM-s) 9
j=0
j=0
Together with boundary conditions (13) they give a system of four linear, independent equations for four unknowns. Solution of this system determines the boundary values of Y2.3. Knowing (Y2)Mand (Y3)o we apply Eq. (14)to calculate (Y2,3)j. The same procedure is used to calculate (Y2,3)"+1.Similarly we introduce the grid functions 22,3 and calculate (Z2,3)n+1. Note that Eq. (14) gives the
estimation [(Y2)~+l< 1(I,,2]j+l] +)n+l Klll(Y2)n[]+ K2I(F2)n ]l '
where K1, K2 are positive constants. Similar estimations are valid for all other unknown grid functions. These conditions ensure unconditional stability of the algorithm. Very important properties of the difference scheme as dissipation and phase shifting are discussed in Appendix A. 3. RESULTS For the calculations we use data from [ 12]: dp = 0 . 0 0 9 5 m de = 0.042 m
p=
Cp - 1046 J/(kg K)
To=Tw= 7 4 0 K
L = 0.5 m u = 0.8 rn/s
-AH=278854 J/mole K0=84000 s l
Co = 1.8 m o l e / m 3 c = 0.4
1.12 k g / m 3
E = 1 0 0 k J / ( m o l e K)
Pem, r = 5, Peh, r = 7 Bi = 2.5
R( C, T)=(1-e)Ko C exp(_E/R T) Q(C, 7) : (-AH)R(C, 7) O.l
For simplicity we assume that homogeneous model is valid and put jmx = jhx =0. The latter simplification does not influence much the numerical algorithm whereas it is helpful for the analysis of the two models. We expect larger difference between the diffusion type model and the wave model without these assumptions. The detailed comparison is a subject of the future publications. Comparison of Eqs. (2) and (2a) shows that the difference between the models is determined by the reaction dependent terms rhB(C, 7) and "cmP(C,7/) and the axial derivatives of the radial fluxes. For our calculations we set Az = As = 0.005. For the method of characteristics Az = 0.005, whereas As is determined by the values of'chUMp and rmu/dp. Analysis shows that there is little difference in accuracy between the method of characteristics and the developed method whereas the new method is certainly advantageous from a computational perspective. For example, in the case c, Fig.l, the method of characteristics requires 3-4 times more radial grid points, moreover different meshes are required to solve Eqs (6) and (7) and then interpolation is needed to prepare data for the next step. The developed method does not have these shortcomings and as a result is faster. For the case of rh = rm (interpolation is not needed), the new method is 5 times faster. If rh ;e rm the difference increases. Taking into account that there are 3 independent functions for each balance equation, available computer memory can also be a limiting factor for the characteristic method in case of many components. Figure 1 also shows that the difference between the diffusion and the wave models becomes significant as the relaxation times approach u/dp. This difference becomes crucial for fast reactions when influence of the reaction on transport processes is taken into account, see Fig.2. For the test problem the value of the product rhB(C, T) reaches 0.5 - 0.8 near the hot spot. According to Eqs. (2) this can notably influence the radial fluxes in this region. Finally note that the difference between two models for the problem at hand is determined mainly by the energy conservation equation. The mass distribution for this case is well described by diffusion type model, see Fig. 3.
53
1170
Fig.1. Temperature profiles in the center o f the tube. Influence of the reaction on transport processes is not taken into account i.e. rhB(C, T) = rmP(C, T) = O. a - the diffusion type model; b - the wave model solved by the new method and the method of characteristics ' r,u/d = 0.06 ' r,m u/dp = 0.08"' n p c - the wave model solved by the new method and the method of characteristics, rhu/dp = 0.6, rmu/dp = 0.7; d - the wave model solved by the new method rhu/dp 0 . 8 , Tmu/dp -- 0.6.
1120
i
1070 ~lg
1020 970
/
920 0,,
E
1--
870 820 770
f
720 0
0.2
0.4
0.6
length
dimensionless ~ a
--~--b
0
c
0.8
- ,El-,d
1070
1070 ~"
1020
1020 970
"~
/
/,.,
920 870
E
1--
~
970
~
920
~
870
820
~"
820
770
~.
770
~f
./
i
720
720 0
0.2
0.4
dimensionless A
a
-s
0.6
0.8
length - {3-.c
1
0
0.2
0.4
0.6
0.8
dimensionless length ~ a
~ - -b
Fig.2. Temperature profiles in the center o f the tube. a - the diffusion model; b - rhu/dp = 0.5, rmu/dp = 0 . 5 ; c - Z'hU/dp = 0.06, TmU/dp -- 0 . 0 8 ; .
Fig.3. Temperature profiles in the center o f the tube. a - ThU/dp = 0.06, -Cmu/dp = 0 . 0 8 ; b - rhu/dp = 0.06, TmU/dp = 0 . 5
NOTATION Biot number (:hw dt /(2Kr) Bi concentration in a fluid phase C fluid specific heat per unit mass Cp pellet diameter ap dimensionless axial position Z tube diameter d, effective mass dispersion oefficient D dispersion flux per unit bed area J wall heat transfer coefficient hw effective heat dispersion coefficient K reactor length L heat production Q tube radius rt temperature o f a fluid phase T reaction rate per unit reactor volume R radial position r
Pe Peclet number (--p Cp u dp/Kr or udp/D r s dimensionless radial position u superficial velocity x axial position Greek letters c porosity of the bed p fluid density ~relaxation time Subscripts 0 inlet, at x = 0 h heat m mass r radial w wall x axial
54
REFERENCES 1. G. F. Froment and K. B. Bischoff, Chemical Reactor Analysis and Design, Wiley, New York, 1979. 2. K. R. Westerterp, W. P. M. van Swaaij and A. A. C. M. Beenackers, Chemical Reactor Design and Operation, Wiley, Chichester, 1987. 3. H. Hofmann, Ger. Chem. Eng. 2 (1979) 258-267. 4. M. J. Schwedock, L. C. Windes and W. H. Ray, Chem. Eng. Comm. 78 (1989) 45-71. 5. D. D. Joseph and L. Preziosi, Rev. Modern Phys. 61 (1989) 41-73. 6. I. Mfiller and T. Riggeri, Extended Thermodynamics, (pp. 1-16), Springer, New York, 1993. 7. K. R. Westerterp, V. V. Dil'man and A. E. Kronberg, A.E. Ch.E.J. 41 (1995) 2013-2028. 8. A. E. Kronberg and K. R. Westerterp, Chem. Eng. Sci., 54 (1999) 3977-3993. 9. W. E. Stewart, AIChE Symp. Ser. 61(58) (1965) 61-65. 10. D. C. Wiggert, J. Heat Transfer, February, 35-40 (1977). 11. B. Wendroff, SIAMRev. 3 (1961) 237-242. 12. G. Eigenberger, Chem. Eng. Sci. 27 (1972) 1909-1915. Acknowledgements - The investigation is supported by Dutch Foundation for Chemical Research (STW) with the financial aid from the Dutch Foundation for Science Research (SON). Apendix A. Let us consider some qualitative properties of the exact and numerical solution of the homogeneous equation OF OY --+A--=O ~z ~s The analytical solution exhibits neither damping or amplification of waves nor alteration in the velocity of propagation. Indeed, using the complex Fourier analysis, assume an elementary solution in the form
Y(z,s)= ~~F, exp(Ifl,(s-Az)),
I2=-1
k =-oo
The absolute value of the amplification factor represents the ratio of the amplitude of the kth Fourier component after an axial step Az to its amplitude at the beginning of the step. Yk(z + Az, s) = F k exp(Iflk (s - A(z + Az))) = Yk(z, s) exp(- IAflkAz)= Yk(z,s)~ k . Since [~'kl= [exp(-I2flkaz ~ = 1 no damping or amplification occurs. The Von Newmann analysis of Eqs. (12) shows that for the component (Yk)~ = Fk exp(IffkJAs + Iflknaz) the amplification factor is ~k
(1 + P) + (1 -- p)exp(//~kSsl = (1 - p) + (1 + p)exp(I/3kAs)
(15)
and I~kl= 1, This it means that the numerical method does not introduce any dissipation or growth. The phase error can be expressed in terms of real and imaginary parts of amplification factor 9tg(q~k) = Im(~k)/Re(~k) 9For the analytical solution it gives q~ = flkAAz = flkpAs. Whereas the numerical amplification factor (15) gives 2p sin(/~kAs)
tg(~k)= O_ pz)+ O + pZ)cos(BkAs) " Obviously the difference approximation has a phase error A~bk = ~bk - ~bk unless p ~ 1. The error vanishes when As ---y0. If p = 1 the method is reduced to the method of characteristics.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
55
Moving finite difference method for tracking a shock or steep moving front Y. I. Lira, S. H. Jeong*, J. M. Le Lann, and X. Joulia ~ Laboratoire de G6nie Chimique (LGC, UMR-CNRS 5503), INPT-ENSIGC 18 Chemin de la loge, F-31078 Toulouse, France *International Cooperation Laboratory (ARC & INPT-ENSIGC) For tracking a shock or steep moving front in the convective Partial Differential Algebraic Equations (PDAEs), some spatial discretization methods such as central, upwinding, Essentially Non-Oscillatory (ENO) and Weighted Essentially Non-Oscillatory (WENO) schemes, and the moving grid techniques are examined respectively and simultaneously with regards to accuracy, temporal performance and stability. The uniform fixed-grid method is efficient owing to relatively short CPU time. However, the solution of the Fixed Stencil (FS) approach such as the central and upwinding spatial discretization methods is unstable. Many mesh points are needed for all of the fixed grid methods to obtain enough solution accuracy. In the moving grid techniques, stability of the solution is enhanced even at small mesh numbers but it is prohibitive because of the addition of a complex and nonlinear mesh equation to solved PDAEs. The combination of the ENO/WENO schemes (based on an adaptive stencil idea) with moving grid techniques gives a stable and accurate solution even at small mesh numbers and is efficient to track a moving shock front.
1. Introduction The numerical study of Partial Differential Equations (PDEs) whose solutions display steep moving fronts has demonstrated the need for numerical solution procedures with space adaptation. For solutions possessing sharp spatial transitions that move (e.g. traveling wave fronts, emerging boundary and interior layer), a grid held fixed for the entire calculation is computationally inefficient since this grid often contains a very large number of nodes (in order to yield a good solution). In such cases, methods which attempt to automatically adjust the size of the space step are likely to be more successful in solving critical regions of high spatial activity. Solving PDEs for heat and mass transfers in chemical engineering problems, a shock or steep moving wave front can appear for the case of convection-dominated packed-bed adsorption and gas-solid catalytic/non-catalytic reactions in a porous solid. It is well-known that moving grid methods and upwinding ENO (Essentially Non-Oscillatory) schemes are two kinds of techniques for tracking the shock or steep wave front in the solution of PDEs. It is expected that their combination should produce more robust methods (Li and Petzold, 1997). Many works have been devoted to the construction of shock capturing schemes on uniform grids, such as TVD (Total Variation Diminishing), MUSCL (Monotonic Upstream centered Schemes for Conservation Laws), ENO (Shu and Osher, 1989) and WENO (Jiang and Shu, 1996). These schemes need enough points to capture the shock or steep wave front, and consequently often put too many points in the area where the solution is very flat and smooth. zTo whom correspondence should be addressed, e-mail: [email protected], phone: +33 5 6225 2355
56 Moving grid methods (Dorfi and Drury, 1987; Huang et al, 1994), on the other hand, can redistribute the nodes to the area where the solution varies rapidly at each time step in order to gain computational efficiency. During discretization using MOL (Method Of Lines), the most troublesome terms are convection terms that cause instability. The principle of MOL is to separate the space and the time discretizations by approximating all partial derivatives in space. Thus a system of DAEs (Differential and Algebraic Equations) is obtained from PDAEs and their boundary conditions. Although the moving grid method can place enough nodes in the wave front and works very well for a convex flux function and sufficiently smooth initial conditions, there are still some oscillations as well as inaccuracies appearing in the solution of nonconvex flux functions or piecewise initial conditions. Therefore, reliable spatial discretization methods are desirable in moving grid techniques in order to take information about the 1st order derivatives. Furthermore, for multidimensional large PDAEs containing mass, momentum, energy balance and equilibrium equations etc, it is necessary to discretize spatially PDAEs with methods which are stable and accurate even at small numbers of grid points. In this study, moving finite difference methods based on Equidistribution Principle (EP) and accurate shock capturing methods for discretization are tested and analyzed. As an illustration, dynamic simulation of a convection-controlled chromatography column is presented. 2. Spatial discretization of the convection term For the convective conservation laws, the one-dimensional hyperbolic PDE is expressed,
au
af
at
~x
or
ut=-fx
(1)
In the frame of MOL, the convective term, fx, can be discretized by using central, upwinding, ENO or WENO schemes. Here, the upwinding concept is used in the sense that a present value (e.g. ui) is updated by values to the left (e.g. ui-1) when the fluid flows from left to right (namely, the fluid speed, v>0). The central and upwinding schemes are classified as the fixed stencil (FS) approach because the number and the position of used spatial points are fixed at the given accuracy order over the discretization procedure. The first-order upwind scheme (named, FS-upwind-1) well-known as the 2-point backward method gives very stable solutions but poor accuracy because of many truncation errors: fxm(fi-fi-1)/(xi-xi-1) (2) The second-order central scheme (named, FS-central-2) is used commonly for the discretization in fixed and moving grid methods: fx--(fi+ 1-fi - 1)/(Xi+ 1-Xi-1) (3) In general, the FS approach may not be adequate near discontinuities or steep fronts because of oscillations (termed the Gibbs Phenomena), which motivates the idea of the adaptive stencil (AS). In order to avoid numerical dissipation in a discretized cell i, the convective term, fx, is newly defined in conservative form, e.g. fx= 2(f/+,/2- Z-,/2)
(4)
Xi+ 1 - - X i _ 1
where fj+l~2is called the numerical flux. The ENO and WENO schemes, which use the AS and the upwind ideas, are uniformly high-order accurate right up to the discontinuity. They achieve this effect by adaptively choosing the stencil based on the absolute values of divided differences (DDi). For the lSt-order divided differences, DDr is defined as follows:
57
--(I),
DDi+,, 2 =(fi+1-fi)/(Xi+ 1-Xi)
(5)
A smaller DDi implies the function is smoother in that stencil. Thus, the one with a smaller absolute value is picked by comparing the two relevant divided differences. In the 2nd-order ENO scheme (named, ENO-Roe-2), if ID/)i+)/21>]D1)j_,/2,1 '
~(1)
'
~(l)
DD~)I/2 is picked and then the
numerical flux is for the positive flow speed (v>0)"
~(2) =fi+ DD[~),,2 "(Xi-Xi-1)/2
(6)
i+l/2
Otherwise, ~(2) =fi+ D/~+),2 .(xi+)-xi)12 ,+,~2
-,-- ( 1 )
,
(7)
In this study, the upwind idea used in the AS approach is achieved by the Roe scheme (Roe, 198 I) where the numerical flux is differently defined according to the flow speed signs (v>0 or v<0). One of the efficient ENO schemes, PHM-REF (Piecewise Hyperbolic Method based on Roe Entropy-Fix correction named by ENO-PHM-3; Marquina, 1994) and two WENO schemes (named, WENO-Roe-3 and 5; Jiang and Shu, 1996) are also tested with the convective adsorption column containing a moving shock front. 3. Moving mesh PDE (MMPDE) Under the moving grid techniques, an additional convective term (Ux) from the Lagrangian description (Dorfi & Drury, 1987) appears in the equation (1). 0u =u- Ux /~=-fx (8) at where, u and /~denote ordinary derivatives of u and x with respect to time (t). The stable and accurate discretization method is thus needed for u,, and f,, that may cause instability to the numerical solution. Moving grid techniques based on EP (Equidistribution Principle) move the grid continuously in the space-time domain, using a monitor function, M(x,t), which means that for 0.0<x(~,t)
,,,(~,t)
i Lo
~0.0 M(x,t)d~=~- ~)0M(x,t)d~=~0(t), i=l, N
(9)
where N denotes the number of grids. The computational coordinate (~=i/N) transforms the nonuniform physical mesh (xi) into a uniform computational grid (~i). From equation (9), one can obtain various mesh equations involving node speeds, so-called MMPDEs (Moving Mesh PDEs; Huang et al, 1994) that are employed to move a mesh having a fixed number of nodes in such a way that the nodes remain concentrated in regions of rapid variation of the solution. The mesh equation and the original PDEs are generally solved simultaneously for the physical solution and the grid. The MMPDE4 proposed by Huang et al (1994) is described:
(M-0--~)-- - ~-0-~-(M-~)
(10)
The parameter "c is representative of a timescale in order to force the mesh toward equidistribution. Note that this mesh equation is shown to satisfy the no node-crossing 0x condition ( ~ >0). The arclength monitor function (1 St-order derivative monitor function) is described as follows" M(x,t)= 41 + (f~) 2
(11)
where, f~ is the convective term defined above. For most discretization methods, abrupt variations in the mesh will cause a deterioration in the convergence rate and an increase in the error (Huang and Russell, 1997). In this study, the arclength monitor function is locally smoothed, employing some spatial discretization
58 methods (central, upwinding, ENO and WENO schemes) to the convection term discretization. 4. N u m e r i c a l results
As an illustration, numerical experimentation for hyperbolic conservation PDAEs exhibiting the shock moving front from a initial condition is carried out. The DAE system is solved on a Digital Alpha server 4000, using the double precision in the DAE solver DISCo (Sargousse and Le Lann, 1998) at the absolute and relative tolerances < 104. To check the accuracy, the Loo error is measured at each time level: N
s
.... ,-u X com te l x = lU Xi ex c,--U Xi com ute I i=l
The liquid-solid (L-S) adsorption column without diffusion effects is described for onecomponent models as follows:
+
1-e ,6'
L
Cf +V.Cx = o
~Cf -k.(C I -cS)=o
(13)
I
CI
-
K
. C L = 0
where the concentration in fluid phase (CL), the concentration in solid phase (cS), the L-S interface concentration (d), the void fraction (e=0.4), the liquid velocity (v=0.1m/s), the mass transfer coefficient (k=0.0129/s) and the equilibrium constant (K=0.85) are denoted. Indices t and x are used for temporal and spatial derivatives. The piecewise initial step-input concentration (CL(x,0)) is set at 2.2 mol/1 and the column length (L) is equal to 1.5m. The equation (13) and the MMPDE4 are discretized on a fixed or moving grid by means of some discretization methods such as FS-upwind-1/3, FS-central-2, ENO-Roe-2, ENO-PHM-3 and WENO-Roe-3/5. The temporal smoothing parameter (x) is set at 0.2. In Fig. 1, numerical solutions of the liquid concentration (CL(x,t)) solved on the uniform fixed 200-grid (Ax=l.5/200) are compared along the axial direction (x) at t=10s. As mentioned above, the FS-upwind-1 is stable but inaccurate, while the FS-central-2 and FS-upwind-3 are unstable near the shock. In fact, the FS approach is efficient with regard to temporal performance, but is not reliable on the solution containing a shock or steep moving front. The ENO-Roe-2, which can overcome the drawback of instability, is very stable and relatively accurate. Note that the other AS approaches (ENO-PHM-3 and WENO-Roe-3/5) also show the accurate and stable solution. However, the AS approach is more time consuming than the FS approach. In order to examine relations between accuracy and temporal performance, we introduce the multiobjective concepts (Lim et al, 1999) to simultaneously minimize the L~ error and the CPU time. In Fig. 2, accuracy and temporal performance of the fixed 200-grid is compared with those of the moving 20-grid according to stable spatial discretization methods (FSupwind-1, ENO-Roe-2, ENO-PHM-3 and WENO-Roe-3/5). It is clear that accuracy conflicts with temporal performance and one can choose a compromise discretization method, considering these two criteria. It is worth noting that the ENO-Roe-2 and the WENO-Roe-3 are reasonable in the multiobjective point of view among the considered moving grid methods. The most accurate discretization method under consideration, WENO-Roe-5, and the method consuming the least CPU time, FS-upwind-1, are shown in Fig. 3 on the fixed 200-grid and
59
the moving 20-grid. More accurate spatial discretization methods on the fixed 200-grid also produce more accurate results on the moving 20-grid. 2.5
~-- 2.0 0
g = 1.5
0 .,,-~
, Real Solution = 1.0
FS-upwind- 1
o
-.
0
0.5
~ 9
o.o
-
-*
FS-central-2 FS-upwind-3 ENO-Roe-2
'
I~ 1.0
I
0
0.5
1.5
Axial direction (m)
-0.5
Fig. 1. Comparison of liquid concentration profiles (C L) in the uniform fixed-grid method at t= 10s and N-200 along the spatial direction (x). 60 Fixed 200-grid
50
9
rid
, ~ 4O O
E
30
.,..~
~
20
r..) 10 0
r
0
5
"
r
i
i
10
15
20
25
Fxror* 100 ( m * l ~ l / 1 )
Fig. 2. Comparison of accuracy (L= error at t=10s) and temporal performance (CPU time during 30s integration) in the multiobjective point of view (the order of points from the right hand side: FS-upwind- 1--)ENO-Roe-2---~WENO-Roe-3---)ENO-PHM-3---~WENO-Roe-5).
5. Conclusion Even though the mathematical modeling equation and its parameters are well constructed and carefully obtained by experiments, one has to also carefully choose the numerical methods in order to more exactly simulate chemical engineering problems since there could be some undesirable errors on the numerical simulation procedure. Furthermore, temporal performance and stability of numerical methods are considered, in particular, for delicate PDAE systems exhibiting a shock, discontinuity or steep moving front. For tracking the shock or steep moving front, some spatial discretization methods and moving grid techniques were taken into account, respectively and simultaneously. According to the numerical experiments, ENO schemes based on the AS approach give accurate and stable solutions. The moving grid techniques also increase accuracy, stability, as well as the calculation time at even small grid numbers. The moving grid with the AS approach can track
60 more accurately the shock or steep front in a stable manner than the commonly used FS fixed grid methods (e.g. FS-upwind-1 and FS-central-2). However, since accuracy conflicts with temporal performance, it is necessary to select a compromise discretization method between the two conflicting criteria. There are many user-selectable elements (rather than tuning parameters) in moving grid techniques. Hence it is desirable to select suitable elements, depending on the problem. This work is a part of the project relevant to PDAE solution techniques, which should be in the nearest future included in the object-oriented numerical kernel DISCo for dynamic simulation. Real solution 2.5
~
Fixed 200-grid with FS-upwind- 1 Fixed 200-grid with WENO-Roe-5
~ 2.0
Moving 20-grid with FS-upwind- 1 Moving 20-grid with WENO-Roe-5
g 1.5
~
1.0
8 .~0.5
0.0
,
0.0
t
0.5
,
-
1.0
1.5
Axial direction (m)
Fig. 3. Numerical solution comparison of the convective adsorption problem.
Reference
Dorfi, E. A. and L. O'C. Drury, Simple adaptive grids for 1-D initial value problems, J. Comput. Phys., 69, p175-195(1987). Huang, W. and R. D. Russell, Analysis of moving mesh partial differential equations with spatial smoothing, SIAM J. Num. Anal., 34, p1106-1126(1997). Huang, W., Y. Ren and R. D. Russell, Moving mesh methods based on moving mesh partial differential equations, J. Comp. Phys., 113, p279-290(1994). Jiang, G. and C. W. Shu, Efficient implementation of weighted ENO schemes, J. Comp. Phy., 126, p202-228(1996). Li, S. and L. Petzold, Moving mesh methods with upwinding schemes for time-dependent PDEs, J. Comput. Phys., 131, p368-377(1997). Lim, Y. I., P. Floquet, X. Joulia & S. D. Kim, Multiobjective optimization in terms of economics and potential environment impact for process design and analysis in chemical process simulator, Ind. Eng. Chem. Res., 38, p4729-4741(1999). Marquina, A., Local piecewise hyperbolic resolution of numerical fluxes for nonlinear scalar conservation laws, SIAM J. Sci. Comput. 15, p892-915(1994). Roe, P. L., Approximation Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys., 43,357-372(1981). Sargousse, A. and J. M. Le Lann, The stiff DAE solver DISCo, ENSIGC, INP de Toulouse, 1998. Shu, C. W. and S. Osher, Efficient implementation of essentially non-oscillatory shockcapturing schemes II, J. Comp. Phy., 83, p32-78(1989).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
61
Neural Network in Physical Equilibria Prediction S. Oregki a, J. Zupanb and P. Glavi6 a aFaculty of Chemistry and Chemical Engineering, Smetanova 17, P.O. Box 219, SI-2000 Maribor, Slovenia bNational Institute of Chemistry, Hajdrihova 19, P.O. Box 34-30, SI- 1000 Ljubljana, Slovenia
Abstract In the paper Kohonen neural network as an alternative tool for fast selection of suitable physical property estimation method that is very important for efficient chemical process design and simulation is described. Neural networks should advice appropriate methods of phase equilibrium estimation on the basis of known physical properties. In other words, they should classify objects into none, one or more possible classes (possible methods of phase equilibrium) and estimate the reliability of the proposed classes (adequacy of different methods of phase equilibrium). From among several different artificial neural networks, Kohonen neural networks were chosen as the most appropriate for the specific problem. Probability maps for each specific phase equilibrium method were obtained as a result. The analysis of the results confirmed that the hypothesis to use Kohonen networks for separation of the classes was correct. Keywords: Physical properties; Phase equilibrium; Neural networks; Classification. 1. I N T R O D U C T I O N During the past years, neural networks have become widely used in chemical engineering. First attempts appeared about a decade ago. The application of neural networks in various areas of chemical process engineering, such as fault detection, diagnosis, process control, process design, and process simulation (Hoskins et al., 1988) has been discussed. Later on, a whole book was edited on neural networks for chemical engineers where principal fields of application within chemical engineering like modeling and simulation of complicated processes, process identification and control were represented (Bulsari, 1995). Also some very useful books which make readers acquainted with the basic concepts of networks and possible applications of them have been published (Schuster, 1991; Schuster, 1992; Zupan et al., 1999). Another promising field of application is to use a neural network as a tool for optimization, where neural networks substitute a mathematical solution of a set of complicated partial differential equations (Dong et al., 1996; Calderon et al, 1999; Acufia et al., 1999). In the field of phase equilibria, we can mention the use of neural networks as a part of or complete predictive tool for vapor-liquid-equilibrium using different single phase equilibrium methods (Petersen et al., 1993; Kan et al., 1996; Alvarez et al., 1999). In all the above mentioned fields of application, neural networks have been used for the simulation of complicated systems where the available information was experimental. In this
62 study, we use neural networks as an alternative tool to help an engineer to choose a suitable phase equilibrium method for further process calculation. The physical property estimation method is crucial for good design and simulation of a chemical process. Neural networks are trained with inputs describing several combinations of physical properties associated with the corresponding methods of phase equilibrium. Although many methods of phase equilibrium exist or are appearing day by day, a whole domain of all possible combinations of chemical components, their concentrations, temperatures, and pressures are not covered by them up to now. Our domain consists of a large number of data but, nevertheless, it still does not describe all the possibilities. 2. P R O B L E M C O N S I D E R A T I O N AND NEURAL N E T W O R K M O D E L S E L E C T I O N Artificial neural networks are a set of several different models featuring a wide variety of different architectures, learning strategies and applications. The nature of the problem we are trying to solve determines which neural network will be employed. In our application, three main characteristics could be exposed: a large number of data, the data not describing all the possibilities, and the classification by the neural network. A large number of data transformed into objects for training neural networks demands larger neural networks with more neurons and longer time for training. An incomplete domain requires an "unsupervised" approach because all the responses are not available. A neural network should also be able to classify objects into none, one or more classes and not only into one out of several pre-defined (known in advance), existing classes. According to the problem described, a Kohonen neural network was chosen among several different neural networks as one with the most appropriate architecture and learning strategy. Kohonen neural network is a "self-organising" system which is capable to solve unsupervised rather than supervised problems where it is not necessary to know in advance to which cluster or group the training object belongs. It has only one layer of neurons; therefore, the specific input variable, let us say the i-th variable, xi, is always received in all neurons of the neural network by the weights placed at the i-th position of each neuron. If the neurons are presented as columns of weights then all the i-th weights in all the neurons can be regarded as the weights of the i-th level (Zupan et al., 1999). Because neurons in our Kohonen neural networks are ordered in a two-dimensional formation, the layout of neurons is especially important to us as it will be seen later. 3. DATA P R E P R O C E S S I N G From the numerous phase equilibrium methods, we have chosen fifteen of them most often used in practice. They are divided into methods of phase equilibrium which use only equations of state and methods which employ activity coefficients. Among semi-empirical equations of state two cubic ones were chosen, Soave-Redlich-Kwong and Peng-Robinson method. Out of multiparametric equations Benedict-Webb-Rubin, Starling and Lee-Kesler methods were included. The last of the equations of state was the theoretical Virial equation. From activity coefficient methods Margules-1 and Margules-2 methods, slightly more complex Van Laar and complex Wilson, NRTL, UNIQUAC, ASOG and UNIFAC methods were chosen. In the same group there was the Regular Solution Theory, too. The phase equilibrium methods mentioned are known and well represented in the corresponding literature, so they will not be described here again. The data about physical properties
63 represent information describing chemical bonds, structure of the components, working conditions, further calculations desired, accuracy of methods, simplicity and speed of calculations, and data availability. The data collected from the literature and assigned by experts were expressed with objects of the form X = ( y , x l . . . . . X9). The variables Xi are carrying information about physical properties like chemical bonds, working pressure, and temperature of the mixture, etc. For instance, Xl represents chemical bond (xl=l indicates nonpolar mixture of components, x1=2 slightly polar mixture, x1=3 polar, etc). The variable y represents the corresponding method of phase equilibrium out of the fifteen possible. For example, y with the value 1 represents S o a v e R e d l i c h - K w o n g equation of state, y with the value 10 W i l s o n activity coefficient method, etc. To exclude multiple objects and to include possibilities overlooked, several preprocessing steps like sorting, detection and elimination of identical objects, simple object classification were executed. In the improved database each object was represented as a multidimensional vector X=(x~ .... ,x9,yl .... yls) where Xi represented the values of separate physical properties, and the fifteen-target variables yi which also constituted a target vector Y=(yl ..... y15), indicated the methods of phase equilibrium with the values "1" or "0" assigned to the target variable yi when the particular method i seemed to be appropriate or not appropriate for the phase equilibrium, respectively. 4. T R A I N I N G OF N E U R A L N E T W O R K S AND RESULTS After scaling the variables of the vector X, the target vector Y having values 1 to 15 was substituted by "binary" (1/0) variables assigning "1" to a particular method. Similarly, each of the 9 xi variables was substituted by as many "binary" (1/0) variables as it had different values. Altogether the final representation consisting of 41 binary variables (26 and 15 representing 9 different xi variables and a target vector Y, respectively) was obtained. Finally, there were 3780 objects Xs arranged in 41-dimensional vectors in the bank of objects. According to the number of objects, it was estimated that neural networks of size from 50x50 to 60x60 were needed where the place for 2500 to 3600 neurons was available. For the present, several square Kohonen networks of the dimension 50x50 with 41 weights in each neuron were trained at different learning steps (epochs) using competitive learning (the "winner takes it all method") according to the criteria that the winner c is the neuron having the weight vector Wj(Wjl,Wj2 ..... Wjm) most similar to the input signal Xs(xsl,Xs2 .... ,Xs,,): (Xsi -- Wji ) 2
c 6-- m i n
,
j -- 1, 2, ..., n
I. i=1
The index j refers to a particular neuron, n is the number of neurons, m is the number of weights per neuron, and s identifies a particular input. During the training, the correction of all the m (i=1 ..... m) weights on the j-th neuron was carried out according to the following equation: W ji(new) _. W ji(old) Jvq(t)a(dc - d j ~ x s i -w(Old) ) ji The learning rate term q(t) was decreasing linearly between 0.5 at the beginning of the training and 0.01 at the end of it. The triangular neighborhood function a(dc-dj) was used for scaling corrections on neighbor-weights (Zupan, Gasteiger, 1999).
64 When training, we could notice that with increasing number of epochs the number of activated neurons also increased, but errors of learning and the number of conflict situations diminished. Regardless of the prolongation of the training the number of conflicts in any neural network could be diminished below 10 %. When analyzing the conflict situations it was found that almost all of them were associated with activity coefficient methods. In certain regions Kohonen learning proposed a liquid phase when a two-phase vapor/liquid region was expected. Because activity coefficient methods were applicable to the liquid part of the twophase region (the vapor phase was ideal or it was simulated with appropriate equation of state), these conflict situations were not considered as very bad ones. When inspecting several other conflict neurons it was found out that they where activated by the objects Xs which differed mutually only in some values of the basic "no-binary" variable x4 representing the physical property temperature. The difference in only one dimension of the basic ninedimensional objects X was obviously not informative enough to enable the Kohonen neural networks more precise decisions. In Figure 1 the Kohonen map for 50x50 neural network with 41 weights per each neuron obtained after 900 epochs is represented. By request the pictures are uncolored, although colored pictures are much more informative. + 50 49 48 47 46 45 44 43 42 41 40 39
3 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 1 11
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ILLL L LLLS V VVVVVVVV VVVVV V V VVVV VVVV VVV V VVI IL LLLL L V VV V V VV VVV V V VV VV V VI IL LLL L S V VV VV V VV V VV V V V VV VV V I ILLL L S SVV V V V V VVV VVV V V VV V VVVVVVV VI IL L LL SS VV V V V VV V V V VVV V VV V I IL LLL S V VVV VV VV VV V VVV V V V V VVV VVVVVVI ILL L LL SV V V V V V V V VV VV V V V V VVVI IL LLL S V VVVVVVV VV VV V V V V VVVVVV I IL L L SS V V V VVV VVV VVVV V V VV V VVI ILL LLLS V V V V VV VVV V VV VVV VV V V I ILL L S S S VVV VVVVVV VVVV VVVVV S V V VVVV SSI ILLLLSS SS S VVVV V V VVVV VV VV S SSS V S SI IL S SVV V V VV VV V V V VVSS SS SS SI ILLL S SSSSSS VVV VVV V V V V VVVV S SSS S S SI IL L L S S V VVVVVVV V V VVV SS S I IL L LSL SS V V V V V V V V V V VV SSV VV VSSSS SI IL LLLL S L VV SV VV VS S V VVVV VS S SI IL L LL LSLLL S S SSSSS VVV VSS V V V SVSSI IL L LL L LS LS S S V V VVV V V V VS LI IL LL LLL SL L L L L SSS S V VV VV LI I L L L L L LLS S VV VVVV VVVV S LLLI ILL L LLLL L LL L LL S VV VV VVSS LI I L L L L S L L LLLL LL S SSVVVV VVSSS L I I L LL LS L LL L LLL L S S V VV SS SLL I IL L L LL L L L L L LLL LL SSS VS S L LI ILL L L L L L L LL L L L LLL SSSSSSSS L LI I L L LLL L L L L L LL SSS S SSL LI IL L LL L L L L L L LLLL L SS S LLI IL L L L L L L LL L L L L LLLLL LLL S SS LI ILL L LL L L L L L SL LL SS SSLLI I LL LL L LL L L L L L L SLLLL SS SS L I ILLLL L L LL L L L LL S L L S LL LLI I LL L L L L L L LL LLL SLL LLSI ILSS L L LLLLL LL L LLL L SS L LL I I L S L L L L L LLL LLLSS SS S L SI I LL S L L L L L L LL L LL S LL S SSSS S LI L L L LL L LL S SS SS i ILS L L IL L L LL L L L L L LL L LL LL S S S S L LLI I L L LL L L L L L L SS S S L SI IL L L L L LLLL L L LL L L L S SS LL S SI ILL L L L LL L LL L L L L L L L SSI IL L L L L LL L L L L L LL LL LI ILL L L L L L L SL L L LL LLLL LI IL L L L L L L L L LLL I ILL LL L L L L L L L L L S L LL LI ILL L L L LL L L L L L LL L L LI IL LL LL L LL L L L L L LLLL LL LI I L L L L L L L L L L LLL L LLI I L L L L L L LL L I ILLL L L L LLL L LLLLLLL LL LL L L L L L L L L S L L LLL L LLI + + 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
Figure i"
Kohonen 50x50x41 map trained with 900 epochs.
65 Although learning was "unsupervised", almost clearly separated clusters can be seen representing homogeneous vapor, heterogeneous vapor/liquid and homogeneous liquid phase regions ('labels' V, S or L respectively). Because of the limitation of the space only one separate target map representing one phase equilibrium method is shown in Figure 2.
50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 ii i0 9 8 7 6 5 4 3 2 1
I 8469 I 8 I I I 18 I 9 I 9 I 199 9 I 9 9 39 198 9999 I99 9 9 9 I889999 76 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I + 1 2 3 4 5 6 7 8 9 0
6 9
9
9 98
225 98 9 87 95 5 7 5 571 5 9 8
16 4
96 96
4
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4 291 581 594 7899951 999688 29 99 99997 2
8 7
7997 7 76 9 6 7 7 8 6 9 99 9 7 6 99 3 91 9 9 8 3 199 99 93 3 3 9 9 9 9 9 999 9 9 9 99 3 19 999 95 8 9 9 9 77 1 996 9399 469 9 999 999 99 7 999 59 97 79 9 5 788 769 7 9 9 969 998 2 9 1 9 77 7 7
I I I I I I I I I I I I I I I I I I I
9191 9I 9 I ii I I I I I 1 I 89 I 71 I 8 8 I 919 I 99 I 81 92 I 1 992 1 I 76 88 I 7 I I I 9 I 88 I 7952 I 99 I 5 52 91 1 8 9 9I 8999 99 9I 69 9 95I 69 9 I 6 999 7 7I + 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0
Figure 2" Target map yl represents Soave-Redlich-Kwong method. In each target map, the positions of the activated neurons are exactly the same as the positions of the activated neurons in the Kohonen map. If the separate target map is overlapped with the Kohonen map it can be seen in which areas (vapor, vapor/liquid or liquid) the neurons are activated for the phase equilibrium method defined. The values of weights of activated neurons are scaled between 1 and 9. The neurons with the highest values are the neurons that satisfy the chosen criteria of competitive learning best. With the distance from those central neurons the values of neighbor neurons diminish. As expected the equations of state except the Virial method appear in all the three regions V, S, L (see for example the Soave-RedlichKwong method in Figure 2). The activity coefficient methods appear only in L and S regions. In the different target maps the same neurons can be activated but with different strengths. Such areas represent which phase equilibrium methods are appropriate for certain
66 combinations of physical properties and with what levels of certainty. If in the certain target map the same area is not activated the corresponding method is not appropriate at all. When for separate neurons all fifteen target maps are inspected, an information about adequacy of phase equilibrium methods for particular combinations of physical properties is obtained. 5. CONCLUSIONS AND FUTURE RESEARCH Although Kohonen learning is an "unsupervised procedure" it was able to cluster vapor, liquid and vapor/liquid regions. It can self learn the characteristics and the applicability of the phase equilibrium methods. The trained neural network can estimate the reliability of appropriate phase equilibrium methods. Despite the less precise neural networks that have been trained one can make a conclusion that the hypothesis to use Kohonen network was correct. In future studies all conflict situations should be resolved with an intervention into the bank of objects to improve their informativity. According to the number of objects larger neural networks will be interesting to be trained in order to prevent too many objects to activate the same neuron. With more precise neural networks further sub-clusters inside the clustered homogeneous vapor, homogeneous liquid and heterogeneous vapor/liquid regions are expected. Active neurons are distributed rather evenly through the whole Kohonen map. All the neurons (not only the activated ones) carry the weights trained. Hence, we expect to perceive important data about the missing knowledge in our bank of objects in return. This is an advantage over classical expert systems which, in the best case, can only warn the user against unsolvable situations. REFERENCES Acufia, G., Cubillos, F., Thibault, J., Latrille, E. (1999). Comparison of Methods for Training Grey-Box Neural Network Models. Comput. Chem. Eng. Supp., $561-$564. Alvarez, E., Riverol, C., Correa, J.M., Navaza, J.M. (1999). Design of a Combined Mixing Rule for the Prediction of Vapor-Liquid Equilibria Using Neural Networks. Ind. Eng. Chem. Res., 38, 1706-1711. Bulsari, A.B., (Ed.) (1995). Neural Networks for Chemical Engineers, Amsterdam: Elsevier. Calderon, Z., Espufia, A., Puigjaner, L. (1999). Minimising Waste Generation Using Neural Networks Based Dynamic Optimisation. Comput. Chem. Eng. Supp., $463-$466. Dong, D., McAvoy, J., Zafiriou, E. (1996). Batch-to-Batch Optimization Using Neural Network Models. Ind. Eng. Chem. Res., 35 (7). Hoskins, J.C., Himmelblau, D.M. (1988). Artificial Neural Network Models of Knowledge Representation in Chemical Engineering. Comput. Chem. Eng., 12, 881-890. Kan, P., Lee, C.H. (1996). A Neural Network Model for Prediction of Phase Equilibria in Aqueous Two-Phase Extraction. Ind. Eng. Chem. Res., 35, 2015-2023. Petersen, R., Fredenslund, A., Rasmussen, P. (1994). Artificial Neural Networks as a Predictive Tool for Vapor-Liquid Equilibrium. Computers Chem. engng., 18, $63-$67. Schuster, H. G., (Ed.) (1991). Nonlinear Dynamics and NeuralNetworks. Weinheim: VCH Verlagsgesellschaft, New York, NY: VCH Publishers. Schuster, H. G., (Ed.) (1992). Applications of Neural Networks. Weinheim: VCH Verlagsgesellschaft, New York, NY: VCH Publishers. Zupan, J., Gasteiger, J. (1999). Neural Networks in Chemistry and Drug Design. Weinheim: Wiley- VCH.
European Symposiumon ComputerAidedProcess Engineering- 10 S. Pierucci (Editor) 9 2000ElsevierScienceB.V. All rights reserved.
67
Novel Methods for the Efficient Evaluation of Stored Mathematical Expressions on Vector Computers B.R. Keeping and C.C. Pantelides Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BY, United Kingdom
1. Introduction Equation-based process simulation depends on the evaluation of the residuals and Jacobians of large systems of nonlinear equations. These highly repetitive tasks can represent a major component of the CPU time requirement of the overall solution, both in the steady state case (where a single non-linear system is solved by some form of Newton iteration) and the dynamic case (where a similar system is solved at least once per integration step). It was once common practice to achieve rapid evaluation of the residuals and the analytic entries of the Jacobian by generation of code in a procedural language (usually FORTRAN) for these calculations, which is then compiled and linked with the simulator code to produce a custom executable program specific to the given problem. Thus, the task of achieving high efficiel~cy is transferred to the FORTRAN compiler. However, there are several disadvantages associated with this, including the excessive cost of compilation and linking; the loss of symbolic information that becomes inaccessible to the mathematical solution methods and cannot, therefore, be used for providing full diagnostic information to the user; and the difficulty of handling changes to the form of equations taking place at discontinuities during dynamic simulation (see, for instance, [1]). For all the above reasons, more modern equation-oriented process modelling tools such as OMOLA [2], ASCEND [3] and gPROMS [1], [4] have adopted a different approach, in which the model equations are held in memory using an appropriate data structure such as a binary tree. The latter is then used during the solution phase to evaluate the equation residuals as required. The above approach has several advantages. Model development times are reduced dramatically because of the removal of the compilation and linking steps; the numerical lnethods have access to the full symbolic information available; and dynamic changes to the equation structure become much easier. Moreover, arithmetic errors (such as divisions by zero, square roots of negative numbers etc.) can be captured and reported in a manner meaningful to the user rather than relying on the compiler's exception handling mechanisms. On the other hand, it should be recognised that storage requirements are increased because of the memory required to hold the binary trees. Fortunately, this is not of major significance given the ever increasing availability of memory on modern workstations. In any case, it is counterbalanced by the fact that there is no longer the need for a large residual evaluation code. A more serious drawback is that the time required for the evaluation of residuals is increased because of the overheads associated with traversing the binary trees. This effect is
68 important, especially when one makes the comparison with the highly optimised code generated by most modern FORTRAN compilers. This paper presents an approach to improving the speed of evaluation of expressions stored as binary trees. Section 2 briefly reviews the binary tree representation of algebraic expressions. Section 3 describes an entirely different representation mechanism suitable for vector computers. Finally, Section 4 presents results that demonstrate the substantial efficiency gains that can be achieved by using this alternative representation.
2. Binary tree representation of algebraic expressions The binary tree representation of algebraic expressions is effective because the basic arithmetic operators are binary themselves: thus, our trees will naturally include the multiplication, division, addition, subtraction and exponentiation (raising to a power) operators. Other naturally arising binary operators include max(.,.) and min(.,.). The simplest unary operator, the minus, can be dealt with by introducing a zero left-hand node and using the binary subtraction operator instead. We will also need to handle some more complex unary operators, more often thought of as functions. These include the usual transcendental functions In(.), exp(.), cos(.), sin(.) etc. In the interests of simplicity, we introduce a special 'binary operator', /F, whose left branch contains an integer identifying the function contained while the right branch is the function's argument. Probably the most natural way to carry out the evaluation of an expression represented as a bi~lary tree is a simple recursive algorithm. However, albeit elegant in form, such algorithms require considerable condition testing as well as the function call overheads implied by the use of recursion. An obvious alternative strategy is to carry out the same process as the recursive code, but avoid the actual recursive calls. However, an implementation of this approach (based on that of [5] with simple modifications to handle the function nodes) has shown that this results in no significant acceleration compared with the recursive code. This merely indicates that modern compilers produce near-optimal code for recursive procedure invocations. In the next section we will present an approach to carrying out evaluation of the expressions represented by such trees, which requires a complete alternative representation of the problem, btll [t/l'llS o u t [O be much more suited to vector parallel computation.
3. Operation storage methods In [6] it was shown that significant savings to linear algebra operations in dynamic simulation can sometimes be achieved by creating, storing and executing explicit lists of all arithmetic operations involved in matrix factorisations. In a similar vein, here we investigate the question of whether greater CPU time savings can be made by generating fiat storage structures to represent the operations required for the evaluation of the equation expressions.
Basic concepts We consider a set of equations as a 'forest' of binary trees. We define a leqfloperation to mean one whose operands are both leaf nodes. We can then collect all the leaf operations in the forest into separate lists, one for each type of arithmetic operation (e.g. addition, subtraction, multiplication, division, exponentiation and transcendental functional transformation).
69
In implementing the algorithm, it proves useful to have an additional memory location associated with each node of the tree, which is used to hold the value of the subtree rooted at that node: it is thus referred to as the v a l u e field of the node. When evaluating the residuals of the set of equations, the leaf operations are those that can immediately be executed since their operands are known constants and/or variables. Of course, once a leaf operation is executed, we can view it as a leaf operand itself since its value is now known. We can therefore start considering a second level of 'leaf operations' coral)rising all nodes where e i t h e r both operands were leaf operations in the original forest or one operand was a leaf operation and the other a leaf. All of these second-level leaf operations may be collected into lists analogous to those for the first level. The above procedure may be repeated until all the operator nodes in the original forest have been collected into lists of identical arithmetic operations. The maximum number of lists for any particular type of arithmetic operation is equal to the maximum depth of any binary tree in lhe forest. Once all the lists have been constructed, the set of equation residuals can be computed whenever necessary by evaluating all the elements of the first-level lists, then those in the second-level lists and so on. The representation of the set of algebraic expressions as lists of identical arithmetic operations immediately raises the possibility of vectorised execution of the equation residual evaluations. In particular, we note that each list essentially consists of 3 arrays a, b and c of pointers to real numbers. The latter are simply the V a l u e fields in the nodes of our binary trees. The evaluation of a list of, say, multiplications translates to a loop of the form shown in Figure 3. Here, b { i l and c [ i ] point to the v a l u e fields of the operands of the ith multiplication operation in the list while a[ i ] points to the Value field of its result. FOR
i:=l
a[i] ~
END
to
NumberO :=
fElement
( b [ i ] ~)
*
s InLis t
(c[i]~)
DO ;
Fi~oure 1" Execution of a list of multiplication operations Modern vectorising compilers have no difficulty with vectorising loops involving indirect addressing of the type appearing in Figure 1, provided they are informed explicitly that it is safe to do so.
Generation of operation lists Having described the basic ideas of our approach, we now turn to consider more formally an algorithm for the generation of the operation lists from a forest of equation expressions stored :~s binary trees. This is done by applying a recursive routine to each expression in turn in order to distribute each arithmetic operation in this expression to the appropriate list. The latter will depend (a) on the type of operator involved, and (b) on the height from the bottom of the tree at which the operation occurs I. Alihough, in principle, this algorithm is fairly straightforward, there is a possibility for some optimisation that may result in substantial benefits.
In this context, the height of all leaves in the tree is taken to be zero.
70
This is best explained with an example. Consider the arithmetic expression (x, + x 2 )• (x 3 + x 4) + xj • x s . A direct implementation of the operation lists idea would result in the following lists being generated: 9 l~evel l l i s t s - A d d i t i o n s : z ~ = x ~ + x 2 ; z 2 = x 3 + x 4 M u l t i p l i c a t i o n s : z 3 = x ] •
5
9 Level 2 l i s t s - Additions : NIL
a
9 Level 3 l i s t s - Additions : z,s =
Multiplications :
z 4 = z~ •
Multiplications : NIL
z4 + zs
Here.v are the original problem variables while z indicate intermediate calculation results. In I:he interests of clarity, we only show the additions and multiplications lists; all others are empty (NIL). Now suppose that the above lists are executed on a single vector processor, and that, at each level, we execute first the additions list and then the multiplications one. In this case, we can see that the single multiplication operation that appears in the level 2 list could, in fact, have been carried out at level 1 since the values of both its operands will have already been delermined by the execution of the level 1 additions. Thus, we could envisage a different slrucl:ure which involves two levels only: 9 Level 1 l i s t s - Additions : z~ = x~ +x2; z 2 = x 3 + x 4 m u l t i p l i c a t i o n s : z 3 = x~ xx5; z 4 = z] x z 2 9 Level 2 l i s t s - A d d i t i o n s :
zs
= z4 + zs
Multiplications: NIL
Of course, the above is based on the assumption that additions will always be performed before multiplications. There is really no fundamental reason why this should be so: we could ,just as well choose to execute the multiplications list at each level b e f o r e the corresponding additions list, and this will lead to a different list structure. Our experience, based on 'typical' process engineering problems, indicates that a good evaluation order is F , ^, x , / , +, -. The above considerations are important as the efficiency of the vectorised evaluation code will be higher if we have fewer levels of longer lists rather than more levels of shorter ones. iNUNCTION MinParentLevel(T:BinaryTree; 2 ParentOp:OperatorType) 3 LOCAL Level : INTEGER ; 4 IF T ~ . T y p e = L e a f THEN 5 MinParentLevel := 1 ; 6 7
8 9 i0
ELSE Level
:=
(T~.Left
MAX
( MinParentLevel
, T^.Operator), MinParentLevel (TA.Right, T~.Operator)) ;
ii 12 13 14 15
16 17
AddToList IF
(T ^ , T A . O p e r a t o r , Level) ; EvalOrder(TA.Operator) < EvalOrder(ParentOp) THEN MinParentLevel := L e v e l ;
ELSE MinParentLevel
18 END 19 END 20END
:=
Level
+
1
;
Figure 2 : Algorithm for allocation of binary tree nodes to operation lists With the above in mind, Figure 2 presents an algorithm that takes a given binary tree T and allocates its operations to lists of the appropriate type. The algorithm is i m p l e m e n t e d as a function which returns the m i n i m u m level for the parent node of T. The algorithm of Figure 2 does not actually do anything with leaf nodes, but insists that their parents be allocated to lists of level 1 or higher (line 5). For non-leaf nodes, it seeks to establish the level of list (Level) to which the current node should be added. This is done by referring to its left and right branches (lines 8 and 10), and taking the larger of the m i n i m u m level values dictated by them. Once the correct level for the current node is established, the node's operation is added (via an invocation of procedure AddToList) to the list that corresponds to its type of operator and level (line 11).
71 The rest of the algorithm of Figure 2 is concerned with establishing a minimum level for the parent node of tree T. It does this by comparing the evaluation order Eva-1 Order of the parent's operator to that of its own (see Table 1). If the parent operator has a higher evaluation order than T, then it can be allocated to a list at the same level as T itself (see line 15); otherwise, it must ~o to a list at the next level (line 17)
Vectorised residual evaluation Once the operations lists are constructed, they can be used to evaluate the residuals of all the equations incorporated in them. This can be achieved by a simple loop that processes all the lists for each level in turn, following the choice of evaluation order given earlier. The structure of each of these list processing operations is identical to that of the simple loop presented in Figure 1. We note that the process of evaluating function lists (corresponding to the /F operator) is unlikely to vectorise well given the fact that it may involve transcendental functions of different types; however, this is not a major concern as typical process engineering applications do not involve excessive numbers of such functions. On the other hand the vectorisation of the other evaluation procedures is straightforward and effective. The execution of this evaluation procedure will leave the v a l u e field of the root node of each binary tree containing the correct residual of the corresponding equation. The speed of execution of this algorithm will be influenced by the vector length of the machine for large problems, but also by the maximum height of any tree in the forest, which will determine the number of times the outer loop of needs to be repeated. It is interesting to note that reformulation of the problem (with intermediate quantities being assigned to additional unknowns) could affect the latter.
4. Numerical experiments This section presents some results obtained using two dynamic simulation examples. Problem I involved 84 equations, with the residuals evaluated 14008 times and the Jacobian 3401, while problem 2 was considerably larger with 1809 equations, 23694 residual evaluations and 2639 Jacobian evaluations. l~rob I 1 2 2
Algorithm Resid Jacobian Other CPU CPU CPU Recursive 4.3 1.0 6.0 List 1.2 0.6 6.2 Recursive 335.7 64.0 424.7 List 144.7 23.1 430.7
Table 1" CPU timings for test problems on a SUN UltraSparc 1
Prob 1 1 2 2
Algorithm Resid Jacobian Other CPU CPU CPU Recursive 43.0 9.4 39.6 List 4.6 3.3 39.5 Recursive 1955.7 467.7 1894.4 List 152.1 137.6 1920.4
Table 2: Table 2: CPU timings for test problems on a CRAY J90
Table 1 compares the performance of the recursive and operation list algorithms on a single 1)roces.s'or SUN UltraSparc 1 workstation. It can be seen that the operation list approach is at least twice as fast as the recursive method for both problems. However, an alternative approach presented in [7] shows that for a serial machine, a similar improvement is achievable with greater melnory efficiency. The speed up for Jacobian evaluations is generally similar to that for residuals. On the first problem, it is less noticeable, probably because the equations were largely linear, resulting in very simple Jacobian expressions.
72
Table 2 presents comparative results obtained on a low-end J90 CRAY machine. Comparing lhe acceleration obtained here with that on the Sun machine, we can clearly see that vectorisation is indeed a benefit. Our test problems involve a number of discontinuities which force changes to the form of a subset of the system equations. In the vectorised case, adding or removing a single equation will generally necessitate recalculation of the entire set of operation lists. For example, in problem 2 this occurs no (ewer than 216 times. Nonetheless, because each such recalculation is essentially no more complex than a single original recursive evaluation algorithm, its impact on the overall solution time is not significant. Overall, then, it is reassuring to note that the analysis stages recluired for the new methods have not had a significant effect on total execution time.
5. Concluding remarks This paper has demonstrated that alternative approaches to evaluation of expressions represented as binary trees are well worth considering. These methods can reduce the CPU demands of such evaluation to the point that they are relatively small compared with other costs involved in process simulation (e.g. linear algebra computations). The Jnethod presented for the vectorisation of equation residuals is of particular interest. It has been generally accepted until now that the equations arising from general process engineering models are too diverse to offer significant scope for vectorisation - as opposed, for instance, Io those arising from the discretisation of partial differential equations in computational fluid dynamics applications. Therefore, efforts for the exploitation of novel computer architectures in process simulation have mostly focussed on the use of multiple-instruction multiple-data (MIMD) type machines (see, for instance, [8]). However, as demonstrated in section 3 of this paper, vectorisation is possible provided one is willing to go to a sufficiently fine granularity. Given the extensive symbolic information held by modern equation-based modelling systems, l lais is a relatively straightforward task.
References 1. P.I. Barlon and C. C. Pamelides, Modeling o.fComhined Discrete/Continuous Processes, AICHE Journal, 2. 3. 4. 5. 6. 7.
g.
411. pp. 966-979 (1994) M. Andersson, Discrete Event Modelling and Simulation in Omola, IEEE Symposium on Computer-Aided
Cr162 System Design, Napa, California (1992) P. Picla, ASCEND - An Ol?/ect Oriented Environment /br the Development of Quantitative Models, Ph.D. Thesis. Carnegie Mellon University, Pittsburgh (1989) M. Oh and C. C. Pantelides, Modelling and Simulationfor Combined Lumped and Distributed Parameter 5,)',stems, Comput. chem. Engng., 20, pp. 611-633 (1996) I). J. Hatter, A generalised non-recursive binaw tree traversal algorithm, Computer Journal, 27, pp. 178-184 (1984) F;. R. Keeping, Efficient methods for tlle solution of large systems otdifferential-algebraic equations, Ph.D. Thesis. Universily o1"London (1996) B.R. Keeping and C. C. Pantelides, Novel Methods ti)r the Efficient Evaluation of Stored Mathematical l~:xprcssions on Scalar and Vector Computers, Paper presented at AIChE Annual Meeting, Los Angeles
(1997) .I.R. Paloschi, Steady state process simulation on MIMD machines: solving nonlinear equations, Comput. chem. Engng., 19S, pp. $721 -$728 (1995)
European Symposium on Computer AidedProcessEngineering- 10
S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
73
Global Optimization of Nonconvex Problems with Differential-Algebraic Constraints W i l l i a m R. E s p o s i t o and C h r i s t o d o u l o s A. F l o u d a s Department of Chemical Engineering, Princeton University, Princeton, N.J. 08544-5263,USA Differential-algebraic systems of constraints, in particular, initial value ordinary differential equations, appear in numerous optimization problems in the chemical engineering field. A difficulty in the solution of this formulation which has not been throughly addressed, is the problem of multiple local minima. In this paper, a novel deterministic global optimization method using a sequential approach will be presented.
1. Introduction The solution of optimization problems involving differential-algebraic constraints arises often in the chemical engineering field. Examples include the optimal control of batch and semi-batch processes as well as the determination of kinetic constants from time series data. Difticulties arrise from both numerical and optimization points of view. Two different approaches exist for the solution of the DAE system within the optimization problem. The first approach, referred to as a simultaneous method, involves the complete discretization of the dynamic system. The resulting formulation is algebraic in nature and can be solved using known nonlinear programming methods. A second approach, referred to as a sequential method, involves the solution of the DAE system through an integration routine at each iteration of the local solver. Control parameterization is also used when necessary. In each case, do to the nonconvex nature of formulation, multiple local minima arise. In this paper a deterministic global optimization method using the sequential approach will be presented for the solution of dynamic optimization problems. The method is based on the (~:BB (Adjiman et al., 1998b,a), a branch and bound algorithm for the determination of the global minimuln for twice continuously differentiable NLPs. 1.1. F o r m u l a t i o n
(1)
rain f ( x , v) xv
s.t. z j = g(z,v,~)
j<,]
0 = h(z,v,t)
~(to)
= ~o
t, c
[to, ts]
where z are the state variables, J are the set of states whose derivatives appear explicitly in the system, and 11I is the set of states which appear in the set of point constants P. v C Iv L, v U] are time invariant parameters which appear in the dynamic system, a: C Ix L, x U] are a set algebraic variables which do not appeal in the dynamic system, f (x, v) and c(x) are twice continuously ditTerentiable functions, g(z, v,/.) and h(z, v, t) are twice continuously differential functions with respect to the states, z and the parameters, v. Within (1), the control (if it exists) has already been parameterized by, u = /d(v, t) .
74
2. Global Optimization Approach The proposed approach is based on the oeBB (Adjiman et al., 1998b,a), for twice continuously differentiable problems. Within this branch and bound framework, a sequence of upper bounds on the global solution is obtained by solving the full nonconvex problem to local optimality from multiple starting points. A lower bound is determined by solving a valid convex underestimation of original formulation, e-convergence is obtained by successive subdivision of the original region at each level of the branch and bound tree. A through treatment of deterministic global optimization methods and applications can be found in the recent book by Floudas (2000).
2.1. Convex underestimation The key to the success of the approach is in the ability to generate a valid convex underestimation. The algebraic functions within the formulation (c,~,t,,(x) and f ( x , v)) are underestimated using techniques shown by Adjiman et al. (1998b,a). To underestimate the dynamic part, consider the differential-algebraic system of equations as a simple input-output map.
.+j = g(z, v, t~) j e J v
0 = h(z, v, t)
>
> z(t)
(2)
zj(to) = zo ~ e [to, t~] Pontryagin (1962) showed that under the assumptions given previously, this map is continuous and twice differentiable with respect to the parameters, v. Therefore, the states, z, at given time points, t/,, can be written as twice continuously differentiable functions: z(t,,)
=
a:(t,,v)
-
a:,~(v)
.
(3)
Substituting (3) into (1) results in: rain f ( x , v)
(4)
X~V
~t
~,.(~)+
7~,~(v)
= 0 ,~ c M ~ c p
The underestimator of this term is generated by adding a quadratic function in v
iCI
where I is the set of v variables, and s represents the underestimator of the function 2t-. A simplification of (5) has/3,,.,iL,i equal Vi C I. The value of these/3 parameters needs to be large enough to ensure convexity, but not too large as to overly underestimate the original function. The full convex underestimator takes the form:
rain ~.f(x, v)
(6)
X~V
9
t~. c,,\, ( ~ ) + c , p,
+
..~-rr~,
p,
(v) < o --
~
c~,(~)+c;: m,
t
,
(v) < o --
where 12(:+,,.,, is the underestimator of the function c,,.,l,., 12~.,,,.,L is the underestimator of the function -c:,,,,/, and 12+~m,., 12~-.~,,,, and 12.f are all similarly defined. It is necessary to split the equality point constraints into a positively and a negatively signed inequality, each being underestimated separately.
75
2.2. Determination of fi Parameters The/~ parameters are positive quantities calculated fiom the Hessian matrix of the function This matrix is generated using the second order sensitivities of the state z,,,,, with respect to the parameters v,
S,,,,/,(v).
7-L~,l~
0 2 Z,m
-
O v 2 (tlL)
(7)
.
In the case when the/3 parameters are equal for each variable, their values can be calculated by: /~,,,, t, > - 1/2 min~, ~,,~in '~,i,, (v) (Maranas and Floudas, 1994), where A''~i'~ is the minimum eigenvalue of the Hessian matrix 7-L,,~,z,. The difficulty arises from the fact that 7-L,~,z,,can not be written as an analytical function of v. The elements of the matrix, however, can be determined through an integration of the augmented system at given values of v. As a result of this, three different methods for the determination of/3 values have been developed. ,
,,
,
__
j
" m ~ I z
Constant or Semi-Constant Values: The values used are preselected and can either be a constant or a function of tree level. Sampling Approach: The values of the elements of 7-L are determinable at given values of the parameters v. In each region, a number of random points are selected to evaluate the Hessian matrix and calculate the eigenvalues of these matrices. The minimum of these eigenvalues is then used to calculate the value of/3. Sampling with Interval Calculations: In this approach, the values of each element of ?-L are determined at given values of the parameters, but the eigenvalues of the matrices are not directly determined. Instead, an interval Hessian matrix is generated by determining the minimum and maximum of every element over the sampled set. A valid lower bound on the minimum eigenvalue of this matrix can then by determined using methods presented by Adjiman and Floudas (1996); Adjiman et al. (1998b).
2.3. Illustrative Example In order to illustrate the above concepts, consider the system: =
',, -
z 3,
Z(to)
=
9
tE
[0,1]
(8)
where the control is bounded by ~t C [-5, 5], Consider the value of the state, z, at t = 1, which is an implicit function of the variable u, 2r(u). The first order sensitivity of the state with respect to the parameter, o~ is determined by integrating an additional equation with the one given above (Vassiliadis et al 1994) The second order sensitivity, 02~ is calculated using a finite difference approximation. The minimum of the second order sensitivity is found to be -0.2214. Therefore a/3 value of 0.1107 is needed to generate a convex relaxation of this t\~nction. The original function and the underestimator are plotted in Figure 1. Notice the continuity of the original function, and the convexity of the underestimator.
3. Algorithmic Outline S t e p 1." Initialize the problem:
Set the relative, e~d, or the absolute, e TM convergence tolerance. If a sampling method is being used to calculate the/3 values, set the number of points to use at the first iteration, ])i,,.i~,,z, and the minimum number at each subsequent iteration, pever~j, i t e r = O. S t e p 2." Calculate initial/3 values:
76
1.5 1 0.5 0
A(,
-0.5 -1 -1.5
...~..~'~7~........
......... ~ .....................
-2
. . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . .
-2.5 -3
-4
-2
0 u
2
4
Figure 1 Function and Underestimator for the Illustrative Example.
A3"~k, ~ -~
A7
k3",7 ~ k4
kl Aj
/k k
\As k(,lk5 A4
Figure 2: Bifunctional Catalyst Reaction Scheme.
Integrate, with second order sensitivity evaluations, the system at f,
4. Example problems The solution of an optimal control problem and a parameter estimation problem are presented to illustration the performance of the algorithm.
4.1. Bifimetional Catalyst Problem This example concerns the optimization of a bifunctional catalyst in converting methylcyclopentane to benzene. The catalyst contains a hydrogenation component and an isomerization component. The objective is to determine the mixture of the two along the length of the reactor
77 which maximizes the concentration of the desired product, At, in the reaction scheme given in Figure 2. The formulation for this problem using a piecewise constant control profile on 10 equally spaced intervals can be found in the in Floudas et al. (1999) and Esposito and Floudas (2000a). The problem exhibits over 300 distinct local solutions. Using 1000 random starting points, only one resulted in the determination of the global solution to the problem. Using 1000 sampling points, the/~ value need for convexity was determined to be 0.0698. The problem was solved using a ,/~ value which started at 0.0698 and was reduced at each level of the tree. A relative convergence tolerance of 0.1% was used. The global solution determined had an objective value of 10.095 • 10 .3 with a control profile of: v = [0.66595, 0.67352, 0.67500, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9]. This example illustrates an interesting characteristic of the approach. The algorithm acts as a very effective search to determine the global solution. Initial points for the solution of the original problem are determined by solving a convex relaxation. This relaxation acts to smooth the nonconvex nature of the original formulation, thus supplying starting points which are relatively close to the global solution. Consider in this problem that using 1000 randomly chosen starting points only results in the global solution being identified once. In the proposed approach the global solution was identified in at most 169 iterations and more often less than 100 iterations. Each iteration produces two different starting points for the solution of the upper problem. Therefore, it took no more that 200 points (on average) to find the global solution. This is 5 times better than simply choosing random points. 4.2. Lotka-Voltera Problem This problem involves the estimation of parameters in the predator-prey model used in ecology. The model is described by two differential equations:
dzi
d~, = 0 1 z ~ ( 1 -
z2),
dz2
d~, - 0 2 z 2 ( z ~
-
1),
z 0 - [ 1 . 2 , 1.1],
t C [0, 10]
(9)
where z~ represents the population of the prey, and z2 the population of the predator. The solutions to these equations are cyclic in nature and out of phase with each other. The data used in the study was generated with parameter values of 0 = [3, 1] at 10 equally spaced time points with a small amount of normally distributed random error with o- - 0.01 and zero mean added. The problem is formulated as an error-in-variables estimation problem, and the full formulation appears in Floudas et al. (1999) and Esposito and Floudas (2000b). This problem has been shown to have at least 20 local solutions, with the most prevalent solution being the third best (Esposito and Floudas, 2000a). This problem was solved to global optimality using both of the sampling based approaches. The average and standard deviation of three runs with different numbers of sampled points are shown in Table 1 Notice how increasing the number of sampling points results in less variation within the algorithm. Also, the use of interval methods have been shown to result in looser underestimators. This gives slower convergence to global solution, which is clearly evident in the results. In every case, in spite of the large number of local minima, the global solution was obtained.
5. Conclusions In this paper a deterministic global optimization approach has been presented to address nonconvex problems with differential-algebraic constraints. The proposed method is based on
78
No. of Points
Sampled CPU sec.
l/'"iti('~llS"'<'J
Inter.
25/10 50/25 100/50
123-F 14 145 + 6 152 + 3
331.04+ 56.54 540.04 + 41.50 884.14 + 49.50
Sampled/Interval Inter. CPU sec. 168-F 4 195-+- 3 199-F 2
396.25-+- 20.00 699.73 + 13.24 1119.38-F 10.16
Table 1: Results for the Lotka-Voltera problem. a branch-and-bound fi-amework in which the solution of a convex relaxation is used to generate a valid lower bound on the global solution. Two examples, a parameter estimation problem and an optimal control problem, were presented to illustrate the computation and theoretical aspects.
6. Acknowledgments Financial support from the National Science Foundation is gratefully acknowledged.
References Adjiman C.S., Androulakis I.R, and Floudas C.A., 1998a, A global optimization method, (~BB, for general twice-differentiable NLPs - II. Implementation and computational results. Coral). Chem.. Engng. 22, 1159-1178. Adjiman C.S., Androulakis I.R, Floudas C.A., and Neumaier A., 1998b, A global optimization method, (~(BB, for general twice-differentiable N L P s - I. Theoretical advances. Comp. Chem. Engng. 22, l137-1158. Ac[jiman C.S. and Floudas C.A., 1996, Rigorous convex underestimators for general twicedifferentiable problems. Journal of Global Optimization 9, 23-40. Esposito W.R. and Floudas C.A., 2000a, Deterministic global optimization in nonlinear optimal control problems. Journal of Global Optimiz.ation, Accepted for Publication. Esposito W.R. and Floudas C.A., 2000b, Global optimization for the parameter estimation of differential-algebraic systems. I&EC Res., Accepted for Publication. Floudas C.A., 2000, Deterministic Global Optimiz.ation: Theory, Methods and Applications. Nonconvex Optimization and its Applications, Kluwer Academic Publishers. Floudas C.A., Pardalos RM., Adjiman C.S., Esposito W.R., Gtimti~ Z., Harding S., Klepeis J., Meyer C., and Schweiger C., 1999, Handbook of Test Problems in Local and Global Optimization. Kluwer Academic Publishers. Maranas C.D. and Floudas C., 1994, Global minimum potential energy conformations of small molecules. Journal of Global Optimization 4, 135. Pontryagin L.S., 1962, Ordinal. Differential Equations. Addison-Wesley Publishing Co., Translated from Russian by L. Kacinskas and W. B. Counts. Vassitiadis V.S., Sargent R.W.H., and Pantelides C.C., 1994, Solution of a class of multistage dynamic optimization problems 1. Problems without path constraints. I&EC Res. 33, 21112122.
European Symposiumon Computer Aided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
79
Scheduling to Minimize Expected Completion Time in Flowshop Plants with Uncertain Processing Times Jayanth Balasubramanian and Ignacio E. Grossmann* Department of Chemical Engineering, Carnegie Mellon University Pittsburgh, PA 15213, USA Email: [email protected], [email protected] Abstract We present a Mixed Integer Linear Programming (MILP) model for the scheduling of flowshop plants with uncertain processing times in order to minimize the expected completion time. The uncertainty in processing times, modeled using discrete probability density functions, results in a combinatorially explosive state-space. Model sizes and solution times increase exponentially with the number of scenarios and mulfiperiod formulations cannot handle even modestly large problems. We use an activity network to represent a schedule, which enables us to derive an analytical expression for the expected completion time. The nonlinear optimization model is reformulated as an MILP using exact linearization techniques to yield rigorous optimal schedules. Numerical results show that the solution times for the proposed method are several orders of magnitude smaller than those for multiperiod models. 1. Introduction Production scheduling is an important area in chemical engineering that has received significant attention over the last few years (Reklaitis, 1991; Pinto and Grossmann, 1998). However, most of the scheduling models that have been developed assume that all the problem data are known in advance. These models are termed deterministic. In reality, however, there can be uncertainty in a number of factors like equipment availability, processing times and costs. Thus, the solutions generated using the deterministic model may not be very helpful. In this paper we address the problem of scheduling flowshop plants for minimum expected completion time given uncertainty in processing times. Uncertainty in processing durations can have undesirable short-term and long-term implications for economics and feasibility. To handle the very large combinatorial state space that arises ~om the discretization of uncertainties into scenarios, we use an analytical approach based on the representation of schedules by directed acyclic graphs. The paper is organized as follows. The first section reviews the relevant literature. This is followed by a definition of the problem ARer a discussion of the schedule representation and the derivation of an expression for the expected completion time, the initial nonlinear optimization model is proposed. Finally, this model is reformulated as a MILP and results fi'om several example problems are presented.
2. Literature Review While there are a large number of publications in the general area of deterministic scheduling problems, there is comparatively little work dealing with uncertainty in processing times. Furthermore, most of this work (stochastic scheduling) has involved restrictions on the number of machines - for example, single-machines (Hamada and Glazebrook, 1993) and the type * Author to whom all correspondence should be addressed.
80
of probability density functions (pdj~), usually exponential, used to represent the processing times. A general overview of stochastic scheduling can be found in Dempster et al. (1982) and Pinedo (1995). In the literature, a rule usually provides the optimal solution for a very specific class of scheduling problems. For example, the longest expected processing time first rule (LEPT) minimizes the expected makespan in the class of non-preemptive static list policies when there are two machines in parallel and the processing times are exponentially distributed (Pinedo, 1995). Although a number of the scheduling rules are based on the expected values of the processing times, rules based on the variance of the processing time distributions also provide optimal solutions in certain cases (Pinedo and Weiss, 1987). A more recent work (Kamburowski, 1999) presents a sufficient condition on the processing time distributions that stochastically minimizes the makespan in two-machine flowshops with unlimited intermediate storage. In the chemical engineering literature, Honkomp et al. (1999) have developed a reactive scheduling framework to deal with uncertainties in processing times and equipment breakdowns in batch processing. The authors use a deterministic MILP optimizer coupled with a Monte Carlo simulator to assess the quality of a schedule over multiple replicates of a simulation. An important consideration is the use of statistical criteria for stopping the replication process. Schmidt and Grossmann (1996) proposed a deterministic continuous-time MILP model for scheduling of tests in the new product development area that can be easily extended as a multiperiod model. Schmidt and Grossmann (1999) also developed a graph-based approach for evaluating the distribution function of the completion time when the durations of the tests are described by delta and polynomial functions. In this paper we use this idea as a basis to selecting an optimal stochastic schedule in flowshop plants. 3. Problem Definition Given are N products (A, B...) and M stages (1, 2 ...M). Each of these products requires processing in all of the M stages and follows the same sequence of operations (i.e. a flowshop plant). All the stages have one machine. The processing time of product i on stage j is a random variable T0. We use discrete (delta) probability density functions to capture the uncertainty in processing times. Using the notation employed by Hsu (1999), the p d f o f the duration of product i in stagej and for ke K scenarios is given by f(~/)= Ec~/k6(t-di/k), where Ciik denotes the probability k and d~k the duration of the k th scenario. The processing times T/j are assumed to be mutually independent and the tlowshop has no intermediate storage between stages (zero-wait). It is also assumed that changeover times can be neglected. The problem is to determine a schedule for the flowshop (i.e. sequence ofjobs) that minimizes the expected completion time. 4. Representation of the schedule The Gantt chart (Figure 1) is one of the common ways of representing a schedule. However, since the task durations are stochastic in nature, we would require different Crantt charts for each scenario. To circumvent this problem we use directed acyclic graphs to represent a schedule, since we are interested in representing a sequence regardless of the actual processing times. It can be seen that the operations in the Gantt chart of Figure 1 can be translated into the activity network shown in Figure 2 (Muth, 1979). The activity network is a directed acyclic graph with the arcs representing the time required to carry out certain activities and the nodes representing points in time at which certain events occur. The horizontal arcs of the network represent the task durations while the vertical arcs represent dummy activities of zero duration.
5. Completion time For the detel~nistic case, the total completion time of the activity network is given by the path in the activity network with the longest total duration. When the task durations ~i are
81
uncertain, and hence not fixed, different paths may possibly define the critical path depending on the particular realization of the task durations. Thus, in order to evaluate the expected completion time, it is necessary to consider all possible paths of the graph. Fortunately, we do not need to consider the entire graph simultaneously as we can use graph-decomposition procedures. Thus, two tasks X l and X2 in series can be replaced by a single task with duration X I +X2 through a convolution operation. Similarly two tasks, X1 and X2, in parallel can be replaced by a single task of duration max (X1, X2) through a maximization operation. A graph is said to be series-parallel if it does not contain the interdictive graph (Valdes, 1978). Hsu (1999) showed that the overall duration of a series-parallel network could be effectively computed through a series of addition and maximization operations. While the activity network representation of the flowshop schedule is not strictly series-parallel, the arcs that potentially give rise to interdictive graphs have zero processing time. The following example illustrates the formulation of an expression for the expected completion time of a flowshop with three products and three stages. For the schedule in Figure 2, the expected completion time is given by, E L 1= E[TA11+E[max (TA2,TBI)1+ E[max (TA3, TB2,TCI)1+ E[max (TB3,TC2)1+ E[Tc3I
(1)
This simplifies to
E[tc]=~`cA~kdA~k+y~`~A2k~cB~k2max(dA2k~dB~k2)+~`~cA3k~B2k2cC~k3max(dA3k~dB2k2~d~k3) k klk2 klk2k3 + ~ ~CB3klcC2k2max(dB3kl,dC2k2)+ ~kCC3kdc3k kl k2 St 1
6 A
St 2
I
5 B 6
4 C [ 5
A
13
C
A
B
St 3
[
~
Time Figure 1: Gantt Chart
T
B
(2)
3
12
Figure 2: ActivityNetwork
6. G e n e r a l F o r m u l a t i o n
The previous section showed how it is possible to develop an analytical expression for the expected completion time of a given schedule, when the pdf of a task is given by delta functions. If the schedule is not fixed, then each of the parameters in the expression for E[tr is a variable and takes on different values depending on the schedule chosen. In this section we derive a general MILP formulation for selecting the optimal schedule with minimum completion time. Let p be the position of the product in the schedule. Let yip be a binary variable that is 1 if product i is processed in the pth position of the schedule. Let/, J and P denote the sets of products, stages and positions respectively. Let Tpj denote a random variable that is a linear combination of the position assignments of the products processing times in stage j. The representation of the scheduling problem for a four-product, two-stage problem is shown in Figure 3.
82 Yip
A
_-.
B
Position 1
Tll
T12
Position 2 Position 3 Position 4
Figure 3: Choosing a sequence for a four-product, two-stage problem. By generalizing (1), the expected completion time is given by, M +N-L
e[,c]= e[~i,]+ Zelm~(r, jvp+ e,j+ a Ip+/= s)]+e[ruMl
(3)
s'-3
Let there be/.,, combinations ofp andj such that p+j=s. Then, for delta functions, (3) yields, E~nax(Zpj )]= Z Z ' " Z CpljlklCp2J2k2 ""Cplsjt~kls "max(dplJlkl'd p2J2k2 ""dplsJlskL, ) (4) kl k2 kts where the probabilities are given by cpjk and the durations by dpik. Expressing these in terms of the 0-1 variables yip as linear combinations of the assignments of the products in the corresponding positions leads to the following, cplJlkl : ECijlkfYipl ..... cplsJz, kl., = Zcijl.,klsYipls and dplJlkl = E d i j l k l Y i p , ..... dplsJlsk, s = EdijlsklsYipls i i i i
(5)
The product Cp~/,kCp:hk~ ...Cp,,i,~k,~ in (4) is as shown below, ~_,E...Eci,,+,%=~, il
i2
<6)
...ci,/,,k,,yi, p~yi2p~ ...yi,,p,,
its
We can specify in (6) that il, i: . . . . iz,. are all different, thereby reducing the number of terms. This is justified because of the assignment constraints. With this, the problem of selecting a sequence of products (i.e. a schedule) to minimize the expected completion time may be posed as follows,
&]-EEc;..<..y;. i
+
k
i
k
~_~"'Z~_.Z"'~_.C,,j,k,C,.hk2""%j,.,k,.Y,2p, Y,.p.."'Y,,~p,..max(d,,j,k,, .... d.~.j,.,..) ~ kI k,=
.'~=3
s.t. Z Yo, =1
kt.~.
iI
i
2
Vp~ e
(Pl)
"
,
Z Ye =1
i
V i ~ l,
Yip ~ {0,1}
p
Using standard linearization techniques for the products of the binary variables (Glover and Woolsey, 1974) and the max terms, (P 1) can be reformulated as the MILP model (P2) shown below. Min E[,cl = E Z CilkdilkYil + E E CiMkaiMkYiN i k +
i k
ZZ...ZEZ...ZCilJIkICi2J2k2
s=3 ~,kI k2 kls i1 i2 sJt. Z Y i p = l i
gp~P
...CilsJlskls~Sklk2...klsili2...il.,.
ils ZYip=l p
Vi+I
Zsili2...il s >_YilPl +Yizp2 +...+yiL,.pl s -(/s-l) Zsili2...ils < Yill~ ..... Zsili2...its < yiL~.pls
Vs, il,i2...ils,Pl,P2 .... Pls,il # i2 4:... r Vs, il,i2...ils,Pl,P2 .... Pls,il * i2 4=... * ils
MAXsklk2...kl s > Z dijlkl Yipl ..... MAXsklk2...kls > Z dijl,,.klsYipl.,. Vs, kl,k2...kls i i
83 [
ZMAXsklk2...klsili2...il s >- MAXsklk2...kl s - U ' ( l - Z s i l i 2 . . . i l s Yip e
)
Vs'il'i2""ils'kl'k2 .... kls,i 1 # i 2 ~ ... ~ ils
{0,1}
(P2)
z,~i2...#, >_o, M~',klk2...kZ~ _>0, ZMAX,klk2...kZ,~I~2...iz~ >_0
Here ZMAX are positive variables used to linearize the products of the Z and MAX variables. The number of variables in this formulation is O(KU.~C+M).(M+N)), where C denotes the combinatorial symbol. It is interesting to note that in contrast to the MILP model (P2), the scheduling problem may also be formulated as a multiperiod MILP model (Balasubramanian and Grossmann, 2000) in which each scenario is replicated by a time period. The main drawback with this approach is that the size of the model increases rapidly since the number of time periods is equal to the number of scenarios. In fact, the number of equations, and the number of variables, is O(KUU.M. N). 7. Results This section focuses on the computational study of the MILP reformulation of the stochastic flowshop-scheduling problerlL Two test problems with different number of products and stages were modeled using the methods of the previous section. The effect of the total number of scenarios on the model size and solution time is compared with those of the multipefiod MILP formulation.
7 a. Example 1 - Four products, Two stages: The results obtained using CPLEX 6.5 on an HP9000/Cl10 to solve (P2) are shown in Table 1. In comparison with the multiperiod model (MMIP), our graph-based MILP reformulation (P2) has a much smaller number of equations and variables, and orders of magnitude smaller solution times. Table 2 presents the results for the (MMIP) models. It is worthwhile to note that the solver cannot even generate the multiperiod model when there are 4 scenarios per task (total number of scenarios 65536) while the graph-based model for the 12-scenarios-per-task case (total scenarios ~ 4.3"108) is solved to optimality in 1 minute. Furthermore, the solution time of the 12-scenarios-per-task graph-based model is two orders of magnitude lesser than the time required to solve the 3-scenarios-per-task multiperiod model. The (MMIP) models' solution was stopped when the relaxation gap was below 10 %. 7 b. Example 2 - Five products, Four stages: The results of the MILP (P2) are shown in Table 3. Once again, the solver is unable to generate the multiperiod models as a result of the size of the fonuulation. However, the graph-based model is solved to optimality even for the case where the total number of scenarios is 3"10 9. As mentioned earlier, the number of equations and variables is O(Ku .(~tC+M).(M + N)). In this problem, M is 4 and hence, even for the case of K=2, the number of variables required is fairly large and comparable to the case of K=12 for the four-product-twostage problerrL Table 1. Results for the Four product, Two stage problem - Graph-based model (P2) K 2 3 4 6 8 12
, ,Scenarios ....... Equations ~, Variables 256 290 214 . 6561 5OO 409 65,536 794 682 1,679,616 1634 1462 16,777,216 2810 2554 429,981,696 6170 5674....
Binaries .
.
.
16 16 16 16 16 16
c P u sec. .......Nodes 0.56 42 1.07 46 1.50 43 3.30 44 9.92 46 42.66 46
84
Table 2. Results for the Four product, Two stage problem - Multiperiod model (MMIP) K 2 3
Scenarios 256 6561
Equations 4617 118107
Variables 4113 104993
Binaries
16 16
CPU sec. 21.81 9903.35
Nodes 1 1
Table 3. Results for the Five product, Four stage problem - Graph-based model (P2) K 2 3
Scenarios 1,048,576 3,486,784,401
Equations 6971 25705
Variables 5770 24788
Binaries
25 25
CPU sec. 47.86 920.00
Nodes 234 238
8. Conclusions The key concept presented in this work is that of an MILP model based on an analytical expression for the expected completion time of a schedule when the task durations are described by delta distributions. The analytical expression was derived from a graph representation of the schedule, which was used as a basis to derive the MILP model. The graph-based models for two example problems were considerably smaller in size when compared with the traditional multiperiod formulations. Furthermore, the solution times for the graph-based models were several orders of magnitude smaller than their multiperiod counterparts. References Dempster, M.A.H., Lenstra, J.K., Rinnooy Kan, A.H.G., Eds. Deterministic and Stochastic Scheduling; D.Reidel: Dordrecht, Holland, 1982 Glover, F.; Woolsey, E.. Oper. Res. 1974, 22, 180. Balasubramanian, J.; Grossmann, LE. Manuscript under preparation 2000. Hamada, T.; Glazebrook, ICD. Oper. Res. 1993, 41,924. Honkomp, S. J.; Mockus, L.; Reklaitis, G. V. Comput. Chem. Eng. 1999, 23, 595. Hsu, A. PD-type functions: Tractable system of probability distribution functions. Workingpaper 1999. Kamburowski, J., Eur. J. Oper. Res. 1999, 112, 304. Muth, E. J., Manage. Sci. 1979, 25, 152. Pinedo, M. Scheduling: Theory, Algorithms, and Systems; Prentice Hall: Englewood Cliffs, NJ, 1995 Pinedo, M.; Weiss, G. Oper. Res. 1987, 35, 884. Pinto, J. M.; Grossmann, I. E. Annals of Operations Research 1998, 81,433. Reklaitis, G. V. Perspectives on scheduling and planning of process operations, presented at the 4th International Symposium on Process Systems Engineering, 1991, Montebello, Canada. Schmidt, C. W.; Grossmann, LE. Ind. Eng. Chem. Res. 1996, 35, 3498. Schmidt, C. W.; Grossmann, I.E., to appear in Eur. J. Oper. Res. 1999. Valdes, J., Ph.D. Thesis, Dept. of Computer Science, Stanford University, Stanford, CA 1978.
European Symposiumon ComputerAidedProcess Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
85
Automatic Re-Weighting of Maximum Likelihood Functions for Parameter Regression Yu Xin, Victor R. Vasquez, and Wallace B. Whiting Chemical Engineering Division, University of Nevada, Reno, Reno, NV89557-0136 USA An inside-variance estimation method (IVEM) for regression of the kinetic parameters in kinetic models and binary, interaction parameters in thermodynamic models is proposed. The new method substantially improves the model predictions when compared with traditional least squares regression methods.
Introduction Maximum likelihood regression functions are widely used for obtaining parameters of nonlinear models. Classical applications include the estimation of kinetic and thermodynamic model parameters [1,2]. Once the objective function is selected, most strategies include weighting the objective function by pre-selected values, usually are based on experimental error estimates (i.e., standard deviation). When this is done, the regression problem is effectively converted into a traditional weighted least squares minimization. The model parameter estimation process consists of the definition of the objective function and the use of an optimization technique to determine its extremum. Even if the optimization method guarantees finding the minimum or maximum, the parameters obtained are still not the optimal ones if the objective function is not appropriate. To achieve an accurate prediction, the effects of all errors must be incorporated appropriately in the procedure. The focus of this work is on the objective function definition, which is shown to have a substantial impact in defining the optimal parameters for prediction purposes. In this work, we present a method (IVEM) that re-weights automatically the regression function [3,4]. The method requires the same number of constraints as weights being calculated in order to guarantee proper convergence and stability. Some nonlinear regression study cases in the fields of kinetics and thermodynamics are presented to illustrate the method and to show its effectiveness under specific circumstances. The results show substantial improvement in the model predictions when compared with traditional least squares regression methods. We believe that the concepts and methods presented can be extended to solve a wide variety of nonlinear regression problems in chemical engineering.
Inside-Variance Estimation Method (IVEM) The focus of the IVEM technique is on the objective function definition, which automatically reweights the objective function based on variance estimation during the opti-
86
mization process. Traditionally, the vector 0 of unknown parameters in a given model with n measured quantities and m experimental runs is obtained through: min~ ,0
i=t .1=1
where f
(ziJO_~ij)2 "3
s.t.
f(zs 0 ) = 0
(1)
is the model, z is the n x m matrix of experimental measurements, and ~ is the
n x m matrix of estimated "true" values. The assumed standard deviations of the errors, cri ,
apriori,
are set, based on the experimental error. Now, we still assume that we have random normally distributed residuals between measured and true values (assuming no bias error and no modeling error). Then, the likelihood function is defined as:
(zi~176
_lnL = m___nln2~+m ln~o.~ +1 , 2 2 = 2 ~=1j--~
(2)
o'~
where ~ is estimated for each iteration, automatically re-weighting the objective function. Herein lies the difference between traditional approaches and IVEM. Traditional approaches are based on estimation of the experimental precision. Unfortunately, such an approach is internally inconsistent because these estimated o'~ are not (in general) equal to
a priori
the final values obtained from the residuals at the conclusion of the regression procedure. However, IVEM estimates ~ from each iteration in the regression procedure. Thus, at the end of the optimization process, it will guarantee that the most likely values for the O~are found. Kinetics Examples
GeneralCSTRModel: The proposed method was evaluated using simulated measurements for a common chemical engineering system: a steady state, adiabatic CSTR with an irreversible, first-order reaction [5]. The simulation for this system involved five simulated measurements (inlet temperature, outlet temperature, inlet concentration of A, outlet concentrations of A and B), two parameters (ko and E), and three algebraic constraints derived from a differential equation model for the system. The reaction rate constant was expressed as
The IVEM objective function (F) here is defined as:
minF=l- 2yre(As_As} 7 7 22"s~. (TJ-!!}2-+ + 2.=
o"A
.=
o"r
87 where Aj is the outlet concentration of A from run j and Tj is the outlet temperature from run j. The inlet temperature and inlet concentration of,4 were considered to be error-free. The simulated noise-free measurements are considered as the true data. Four different noise levels ((ya=5, Oa=0.05; OT=5, Oa=0.1; Oa=10, era-----0.05; CYT=10,era=0.1) were chosen to create experimental data sets of different precision. To examine qualitatively the goodness of fit for the resulting parameter estimates, plots of outlet concentrations vs. outlet temperature were generated for the four noise levels. Figure 1 shows the fit obtained using the IVEM and the traditional technique at the noise level (oT=10, (YA=0.1). For the traditional technique, we used Equation 1 with oi s equal to those of the chosen noise level. The IVEM results generally follow along the curve representing the noise-free data, labeled "true values." The sum of squared residuals can also be used to estimate the regression result. It is calculated as" m
t
j=l
j=l
where the * denotes the original noise free data. For the four noise levels, the sum of squared residuals are (17.58, 12.66, 5.74, 102.8) from IVEM and (17.57, 12.54, 8.84, 136.9) from the traditional method. Notice that, at the lower noise level, good agreement was found between the traditional method and the new IVEM approach. This shows that when traditional objective functions produce good results, so does the new approach. However at higher noise levels, the parameter set obtained with the new approach produces better results.
Kinetics of the Air Oxidation of l,2-Dichlorobenzene The products of the oxidation of 1,2-dichlorobenzene (o-DCB) are 2,3-dichloromaleic
anhydride (DCMA), 2-chloromaleic anhydride (MCMA), maleic anhydride (MA), and carbon dioxide. 1.0 ..... " 0.9 E <
~
o
estimationby IVEM
x
estimation by traditional o
o.8
9 Traditional o IVEM
4.0
e
|
9
e 2.0
,-
0.7
I
~176
o
8
o
U
0
o
8
-2.0 <
o
'
0
-5
0.6
9
t
~6
8 E
6.0
,o
true values
= measurements
, -4.0
0.5
640
660
680
700
720
740
760
Outlet Temperature (K)
Figure 1. Reconciled data for noise level
(~T=10, (SA=0.1)
~,o
~o
~o
6;o
'
620
Temperture (K)
Figure 2. Error predicted for different regression methods
640
88 The oxidation of o-DCB was studied at temperatures of 285, 300, 320, 330, 340, and 355~ and at different feed flow rates by Atlaly and Alpay[6]. They defined the reaction rate for o-DCB as ro-DCa = kmol o-DCB reacted per second / kg catalyst The overall oxidation scheme of o-DCB considered was: ' 4.5o~" >C4C12Q + 2C02 + 2HzO C6H4CI2 =
(6)
402 )C, HCl03 + 2C02 + HC/ + H20 3%
>C4H203 + 2C02 + 2HCI
6.502
>2HCI + 6C02 + 1120
Reaction rate expressions derived according to this redox mechanism were applied on the basis of the above-mentioned reaction scheme for parallel reactions. The kinetic model considered is represented by the relation ro_ocB =
(kl + k2 + k3 + k4 )Po_DcsksPo 2
(7)
(4.5k, + 4k 2 + 3.5k3 + 6.5k4)Po_Dc8 + ks Po2
where kl, k2, k3, and k4 are kinetic constants for formation reactions of DCMA, MCMA, MA, and CO2, respectively. The IVEM objective function is: m n((ro~ 2 2 2 ) 1 ~'(rMA--rMA)2j minF =-~ 1 -DCS "CrDCMA"Cr~c~ "Cr~ +-~ L , "_~--"
l~(ro + O ~ z . i=1
"- j = l
^ J: + _ I _DCS_--ro-DC,) _2 ()o-DCB
~
2 i=1
OMA
^ )j + _1~" (rDCMA--rDc~ 2
t:rSc~
z_J
2 i-q
(rMCMA_ _- ^rMCMA~
(8)
_2
O MC~
To examine qualitatively the goodness of fit for the resulting parameter estimates, plots of the difference between experimental and predicted reaction rates of o-DCB for different temperatures is shown in Figure 2. It is observed that the predictions made with the IVEM regression approach are substantially better than those from the traditional maximum likelihood method (with t~UA= 6o-DCB= t~DCUA= ~ MCMA),especially at the higher temperatures.
ThermodynamicsExample The inside-variance estimation method coupled with the maximum likelihood method is used for regressing binary interaction parameters of thermodynamic models for liquid-liquid equilibria predictions for a ternary system in this example. The approach to define the objective function is based on the minimization of the distances between experimental and estimated mole fractions. The objective function is defined as:
89
(X[j f(;~-t
minF = 1
Where
+
X~j is the mole fraction of component k in phase i for tie linej.
The main algorithm starts with an initial guess for the parameters, usually taken from the literature. A reasonable first approximation for the parameters is required because a binodal curve has to be generated in order to estimate the mole fractions for each iteration of the regression procedure. If a good initial guess is not available, the regression should be started using an objective function based on the minimization of activity differences, which is less sensitive to the initial guess of the parameters and has a faster convergence [7,8]. One nonlinear regression case using the UNIQUAC activity coefficient model is shown, involving liquid-liquid equilibria predictions for the ternary system methanol + 2-propanol + hexane [9,10]. LLE Methanol(1)-2-Propanol(2)-Hexane(3) at 5~
The equilibrium data used for this system are reported at 5~ [10]. The values of the objective function F resulting from three regression techniques (ASPEN, DECHEMA[9], and I V E M ) are - 2 . 9 7 8 3 , 4.94038 and - 2 0 . 2 8 1 6 separately. For each technique, F is calculated at the stopping point using the r s calculated at that stopping point. The IVEM technique results in a significantly better (i.e. smaller) value of the objective function. Figure 3 presents the experimental and predicted tie lines using the binary interaction parameters reported by S~rensen and Artl [9]. On the other hand, Figure 4 shows the experimental and predicted tie lines using the new regression approach based on the insidevariance estimation method (IVEM). 10
.................. ., .
~,
.
.......
. ...........
Experimental
-DECHEMA --- DECHEMA
~.8 v
~ 8
,6.
Experimental
-._.
IVEM
o c
~ P
6
o
4
"8 6 o ~
4
2
20
40
60
80
1O0
Mole Percent of (3)
Figure 3. Predicted phase diagram with BIPs reported by D E C H E M A
0
0
20
40
60
80
100
MoDe Percent of (3)
Figure 4. Predicted phase diagram with BIPs regressed using IVEM method
90 In general, the accuracy of the tie lines prediction is substantially improved with this new method, and so is the prediction of the phase envelope, in particular for the left phase.
Conclusions A new approach (IVEM) for regressing kinetic parameters in kinetics models and binary interaction parameters in thermodynamic models was developed. The method, based on the maximum likelihood principle, involves the recomputation of the variance for each iteration of the optimization procedure, producing better results than traditional least squares regression procedures. The automatic re-weighting of the objective function always gives more emphasis to those measured quantities with more error in their estimation. It was shown that traditional least squares based methods sometimes overlook the optimum found by the new proposed method, but the IVEM procedure reduces to the traditional regression methods when the latter work well. The improvement in the regression results obtained using the new approach for the objective function allows a better quantification of the kinetic and thermodynamic modeling errors present.
Acknowledgement This work was supported, in part, by U.S. National Science Foundation, grant CTS-9696192.
References: 1. Anderson, T.F., Abrams, D.S. and Grens II, E.A., AIChE Journal, 24 (1978) 20-29. 2. Prausnitz, J.M.,Reid, R.C. and Poling, B.E., The Properties of Gases and Liquids, 4th Edition, McGraw-Hill, New York, USA (1987). 3. Vasquez, V.R. and Whiting, W.B., Fluid Phase Equilibria (2000) in press. 4. Vasquez,V.R., Ph.D, Dissertation, University of Nevada, Reno (1999). 5. Kim. I., Liebman, M.J. and Edgar, T.F., AIChE Journal, 36 (1990) 985-993. 6. Atlaly, S. and Alpay, H.E., Ind. Eng. Chem. Res., 26 (1987) 2212-2215. 7. S~rensen, J.M., Magnussen, T., Rasmussen, P. and Fredenslund, A., Fluid Phase Equilibria, 3 (1979) 47-82. 8. Novak, J.P., Matous, J. and Pick, J., Liquid-Liquid Equilibria, Elsevier, New York, USA (1987). 9. SOrensen, J.M. and Arlt, W., Chemistry Data Series, DECHEMA, Frankfurt/Main, Germany, 5 (1980). 10. Radice, F.C. and Knickle, H.N., J. Chem. Eng. Data, 20 (1975) 371-376
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
91
E n e r g y cost m i n i m i z a t i o n in an e n e r g y i n t e n s i v e i n d u s t r i a l p l a n t : an MINLP approach J. Vin a, M.G. lerapetritou a'l, P. Sweeneyb and M. Chigirinskiy b aDepartment of Chemical Engineering, Rutgers University, Piscataway, NJ 088.54
l(marianth@~sol.rutgers. edu) bThe BOC Group, Murray Hill, NJ 07974
1. PROBLEM DEFINITION This work addresses the problem of determining the optimal operating schedule to minimize the operating cost in an energy intensive process separation plant. The difficulty arises from the fact that the energy cost is subject to high fluctuations, thereby, Significantly increasing the total production cost in the plant. To deal with such cost variability the plant can operate in three different modes (modes A, B and C)that vary with respect to operation efficiency and energy requirements. Mode A is the normal mode of operation. When the energy cost increases the plant can shitt to the alternative mode of operation B which is less expensive but also less efficient since it requires the consumption of one of the plant products. In some cases the pIant shifts to the mode C in which there is no production but the plant continues to satisfy the demand utilizing the storage capacity. Hence given a forecasted trend in the energy cost, it becomes important to determine the optimal operating schedule that would minimize the total operating cost of the plant over a given time frame and also meet the customer demand. Such a schedule would determine the mode of operation at each time step together with the amounts of products that must be produced. It is essential to keep the computation time within a reasonable time limit, since the problem needs tobe solved fast to be used for further management decisions. The focus of this work is to propose an efficient mathematical formulation and solution approach to obtain the optimal operating schedule using mathematical programming techniques. 2. MATHEMATICAL MODEL OF THE SEPARATION PLANT The proposed mathematical formulation involves the following constraints: Objective Function: minimization of cost OB = E (C(i) * Pvov~ ) + (Penalty) * ~st,i Sst(i) (1) The objective of the problem is to minimize the total cost of operation within the time horizon under consideration. In order to take into account the uncertainty in the storage levels, due to uncertainty in product delivery, an additional penalty term is added in the objective function to minimize the violation of storage safety levels. The process involves the separation of feed (f) into t!~ree different products, two liquids (sl={ sll, s12}) and one gas (sg). The following equations correspond to the material balances aroundthe process for modes A, B.
92 Material Balances for mode A: X .,,, d (i) = R ,4 * X A (i) sll f
(2)
X,,]z (i) + X .,g A (i) = R A a2 , X l (i) = X-4.,.,1(i) + w,a2 , X,.,2 A (i) Total Feed Material Balance ,4 B Xttotal(i) = X f (i) + X I (i)
Xliq(i)
(5)
(3)
Material Balances for mode B X B .a3(i) = R Bs13 * X Bf (i) XB ~ a2(i) + X , Bg ( i ) = R sl2 * X fB (i)
(4)
X ~l (i)
(7)
.a2(i)
= Wsl2 * X B
(6)
(8)
where X ~ (i) correspond to the amount of material (s) at time period (i) when the plant operates in the mode (j), R~ corresponds to the recovery of product (s) when the plant operates in mode (j), and ws corresponds to the relative cost of energy to produce the different products (s), These equations reflect the mass balances in the various processes that take place in the modes A and B. Capacity constraints for material (s) in mode (j) *l~(i) _< X~ (i) _< 13s* ~(i), Vs ~ S, Vj e {A,B} (9) These constraints are derived from the capacity limits of the equipment used in the two modes. Storage Constraints Lsl(i+l) = Lsl(i) + { X~r (i)- X,.~(i) - Qsl(i)- Vs~ }, sic {slt, sl2}, i e I, i r 1 (10) Lsl (i) = Lsl-i,utial, i= 1 (10a) These constraints connect time step (i+ I) with the previous time step (i). According to these constraints the level of liquid (sl) in the storage tank at time step (i+l) (Lsl(i+l)), equals the level of liquid (sl) in the tank at the previous time step (Lsl(i)), adjusted by the amounts produced and delivered (Q~a(i)L during the time step (i). A small correction term that reflects material loss during distribution is also added, (Vsl.). Storage tank level bounds A sl high , , sic {sll,sl2} , i 6 I, (11) S.,Imm + "--^,4t~ - Ssl(i+l) -< Lsl(i+l), Lsl_initia I _< S s T " ,-These constraints express the requirement for a minimum level of liquid (sl) in the storage tanks (S ~ ), to enable trailer filling from the tank. The maximum level constraints (S.,.~x) arise from the storage tank capacity limits. Remark 1" Note that the A parameters used in equations (11), have been derived by considering a stochastic chance constraint programming fommlation (Charnes and Cooper, 1959) with a subsequent transformation to a deterministic problem, The use of stochastic modeling was required to account for uncertainty in the distribution of liquid (sl) over a time step (i). Remark 2: Slack variables Ssl(i)are introduced to allow the minimum safety levels in the storage tanks to be dynamically violated for short time periods, and ensure the problem is feasible in the uncommon case when total demand exceeds plant production capability.
93 Logical Constraints; pA(i) + pB(i)+ pC(i)= 1 ; (12) This constraint ensures that only one mode of operation is selected at an~' time step. For example, if the plant operates in A operation mode, then pg(i) = 1 and pD(i) = 0 and pC(i) =0 and constraints (9) for mode A correspond to the upper and lower bounds on the production rates for the various products, whereas constraims (9) wri~en for modes B and C result in. zero production variables for the corresponding production variables. Switch variable constraints: The variables SwD(i) and SwC(i) are introduced into the system to account for the switching between the operating modes. Swa(i) takes a value of 1, whenever the mode of operation shifts from either A or C at time step (i-1) to B at time step (i) (which means that there is a 'switch to B mode) and is 0 otherwise. Similarly swC(i)-takes a value of 1 whenever the mode of operation shifts from A or B at time step (i-l) to C at time step (i) and is 0 otherwise. Notice that although Sx~ and Swc are not defined as binary variables, they take the values of either 0 or 1 from the way they are defined by the following constraints. SwB(i) > pg(i-1)+ pC(i-t) + pB(i)---1 (13) SwB(i) <_ pa(i) (14) SwS(i) _< pA(i-1) (t5) SwC(i) >_ pA(i-1)+ pn(i-1) + pC(i)--1 (16) SwC(i) _ pC(i) (17) SwC(i) <_ pA(i- 1) + pB(i-1) (18) Equations (13)-(15) define the value of SwB while equations (16)- (18) define the value of
swC.
.
.
.
.
Constraints to satisfy specific operating conditions Based on the requirements of the plant, two operating critefiahave been introduced imo the model. These are: * If the plato switches imo B (C) mode of operation, then it must remain in that mode for a minimum number of time steps, NTb .('NTc). The condition for operating mode B is satisfied using the following condition" i+ N'l;, -1
~.(pA(i,).+ pC, (i')) < 2*NTb [l-Sw"(i)]
(19)
i'--i
If the. mode. of operation switches to B at time step ' i , i.e SwB(i) = 1, then the LHS of equation (19) becomes zero, forcing all pA and pC variables for the next NTb time steps to be zero. This ensures that the plant remains in B mode for the next NTb time steps. If SwB(i)= 0, then the condition is redundant. The requirement for operating mode C is satisfied by a similar condition: i+NTr-1
~ ( p A (i') + p~ (i')) _ 2*NTc.[1- SwC(i)]
(20)
i'-i
If SwC(i) = l, this forces all p variables on the LHS to be zero, thus ensuring that the plant remains in C mode for the next NTc time steps. If SwC(i)= 0, the condition is relaxed.
94 Energy Definitions PrOTAL(i) = Pl(i) + P2(i) + [ 1-pC(i)] *(Pn~sc) (2 i) Pl(i) = oq *[ 1-pC(i)] - ~1 *Xn,,t~l(i)+ '~1 *(Xn,,t~'(i))2 (22) P2(i) = o~2 *[ 1-pC(i)] + 132*XHq(i)+ T2 *(XHq(i)) 2 (23) where p variables are binary, X variables are continuous and positive, mad (x,13,7 parameters correspond to the energy consumption per unit material processed. The model proposed corresponds to a Mixed Integer Nonlinear Programming (MINLP) problem. The nonlinearities in the problem arise only in the objective function. Quadratic relationships are chosen for this study to assure model convexity.
3. MINLP SOLUTION APPROACHES Two algorithmic procedures are used for the solution of the proposed MINLP model presented in the previous section, namely the Generalized Benders Decomposition (GBD) algorithm and the Omer Approximation (OA) solution approach (Floudas, 1995, Biegler et. al., 1997). For this purpose two modeling environments are used GAMS and MINOPT. It must .be noted here that since the proposed formulation is convex ,on all continuous variables and separable in continuous and binary variables, both algorithmic procedures guarantee to obtain the global optimum solution. Table 1 shows some results for computational times for the MINLP using GAMS (for different scenarios) and a comparison between OA and GBD algorithms. Table 1" Comparison of MINLP solution approaches GAMS/DICOPT (0A_alg_orithm)___ -. . . . . . . Time Initial storage levels as Major CPU time as Horizon percentage of maximum Iterations multiple of tank capacity . . . . . . . . . . . . . . allotted time 24 time steps Lsil.initial=65%, Ls12.initial=97.5 40(limit) 4x 48 time steps Lsll_tmt~=83%, Lsiz_mi~al=97.5 40(limit) 19 X 72 time steps Lsll_initial= 100%, Ls12_initial=97.5 > 30X ,, MINOPT (24 time steps, Ls!!.iaitiat= 65~ Lm-ialtia!7' - 97%)_ Major Iterations CPU time OA GBD
31 18
0.9 X 1.4 X
4. MILP APPROXIMATION OF THE MODEL As shown in the previous section the proposed MINLP formulation performs well in terms of computational efficiency if time horizon is considered up to 24 time steps. However, when longer time horizons are considered, the problem becomes too expensive to solve. For comparison an alternmive Mixed Integer Linear Programming (MILP) model is described in order to solve the problem in an efficient way. As shown above, the only nonlinearities in the formulation occur in the objective function in the form of quadratic expressions at equations (22), (23).
95
Using the capacity constraints we can linearize the quadratic terms between the bounds. The MILP approximation thus-, has the same constraints, and objective function, except the fact that all quadratic terms in the power function have been linearized. The MILP problem was solved using GAMS/CPLEX as the solution algorithm and the results are presented in the next section and compared to the MINLP solution. 5. RESULTS AND DISCUSSION MILP problems were solved to optimality using GAMS/CPLEX. MINLP problems were solved using GAMS/DICOPT and MINOPT. All computations were carried out using a SUN'ULTRA 60 Workstation. The 'real time optimization requirement imposes a time limit on the computational time for solving the optimization/scheduling problem. In this work, time is expressed as multiple of this allotted time. A typical case that has been solved is shown in Figures 1 and 2. In particular Figure 1 shows the demand of liquid products that must be met, and Figure 2 shows the energy cost variability together with the predicted mode of operation.
~ool 90 80 7O 6O 5O
!
~ Qsll(i) 9Qsl2(i) , ,
204~ ltllltlli i l,l 0
Time Step
Fig. 1" Product Demand
I
)
. Scaledprice
ii
Ii
li
i'
I ~
ll, l
::::::::::::::::::::::::::::: t' '~ 1'I~......|i~~ ..... i
::::::z:___. -~ti.:!._
Time steps Fig. 2:Power costs and operating modes
A few of the computational results are tabulated in Table 2. As seen from Table 1, the MINLP approximation becomes computationally very expensive for 72 time steps and beyond. The MINLP model was also found to be sensitive to the input parameters to the problena- initial levels of liquid products in tank, distribution of product demand and power cost variation with time. All cases in Table 2 refer to 72 time steps. It is seen that the MILP approximation solves well within the allotted time for most of these cases (even for a 120 time steps, the computational time remains within the allotted time). Also, the values of the optimum obtained by the MILP approximation are very close to those found by the MINLP. For the case when L~H and Lm are 79% and 73% of their naaximtma values respectively, the MhNLP was solved to optimality within the allotted time. This corresponds to the Global Minimum. We see that the solution obtained by the MILP (34905) is extremely close this Global Minimum (34904). It is worth noting that the MILP took about one fortieth the time to solve compared to the MINLP. This is also shov~aa in the case when Lsn and L~2 are 79% and 97% of their maximums. In this case the MILP is solved to optimality within the allotted time (31608)
96 however the solution yielded by the MINLP (32114) within this time is suboptimal compared to this (the MINLP was allowed to proceed to 25 iterations i.e about 30 times the allotted time, but could not yield a solution better than 32114). These results show that for most cases the MILP formulation yields satisfactory results both in terms of computational times and also in terms of being very close to the Global Optimum. To verify this, an analysis was done to evaluate the average and maximum difference between the objective function values as calculated by the MILP versus MINLP approaches and this showed that on an average, this difference would be approximately 7%. This means that the MILP results in a solution that is very close to the Global Optimum.
Initial levels (Lsli, Lm) % of max 100% 97% 79% 97% 100% 73% 79% 73% 100% 88%
Table 2: Comparison of MILP to MINLP approach . . . . . Objective Time taken for Objective function after function MILP problem running as MINLP for the from MILP (multiple of allotted allotted time time) 15461.53 5.9 X No solution within time limit 31608..4
0..06.3 X
32114.04
22195.16
0.0045 X
34905.8
0.014 X
No solution found within time limit 34904.74
22319.6
0.0082 X
No solution found within time limit
6. CONCLUSIONS It has been shown that the problem of Power Consumption Optimization and Scheduling c,an be solved efficiently by using Mathematical Programming techniques. The formulation leads to a MINLP problem which is guaranteed to yield the Global Optimum solmion. However the MINLP formulation results in very high computational times. Since this might not be efficient in practice, due to the relative small time frame of the decision making process an alternative MILP approach was developed which was very efficient computationally and yielded solutions very close to the Global Optimum.
References Biegler, L.T., I.E. Grossmann and A.W. Westerberg, Systematic Methods of Chemical Process Design, Section A.3.4, Prentice Hall, 1997. GAMS Solver Manual 9 GAMS Development Corporation, November 1998. Charnes, A. and W.W. Cooper, Chance-constrained programming. Manag. Sci. 5:73-79, 1959. Floudas, C.A., Nonlinear and Mixed Integer Optimization: Fundamentals and Applications, Oxford University Press, 1995.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
Generic object-oriented dynamical systems
modelling,
simulation
97
and
optimization
of
Dipl. Phys. T. Wack, Dr.-Ing. G. Deerberg and Dr.4ng. S. Schltiter Fraunhofer-Institute for Environmental, Safety, and Energy Technology, Osterfelder Strasse 3, 46047 Oberhausen, Germany 1. INTRODUCTION Dynamic simulation is an important tool for the design, optimization and control of nonlinear dynamic or complex systems as they appear in chemical engineering. Such simulations are based on detailed mathematical models, which are much more effective than conventional models (i.e. perfect mixed reactors) regarding a precise description of reality. The editing of such highly detailed models belongs usually to the most laborious and tedious tasks of technical and chemical engineering. Objective of the presented work is the development of a general modelling tool, that is able to generate model components automatically, which are documented as well as they are provided with a self-generated GUI interface for model parameterization. Intended starting point for this automated model generation is an OOA (object-oriented analysis) of the process to map, which itself is supported by interfaced knowledge-bases and exchangeable expert system instances. There are two major advantages of this architecture. On the one hand the models are very transparent because of their implicitly simultaneous formulation on mathematical side, on the documentation side and also within the context-sensitive layout of the graphical user interface (GUI). On the other hand there results an abstract model formulation, which ensures a high degree of reuse for single components. Furthermore the exchangeable expert system component offers the possibility to use the same modelling tool for mapping different complex systems (as regards the particular requirements of different disciplines), without decreasing the degree of specialization. This aspect leads directly to the opportunity of hybrid model formulation. In general there results a significantly less time-consuming generation of models.
2. MASSIVE RAPID PROTOTYPING To realize this approach it is necessary to develop a problem-related strategy which takes into account modern software engineering techniques, i.e. object orientation, rapid prototyping etc. to minimize developing time without decreasing run time quality and stability of the modelling system. This newly created concept of class generation we labeled Massive Rapid Prototyping (Figure 1).
98
Figure 1. Concept of Massive Rapid Prototyping as a new method for fast model development by using a reduced set of objectoriented techniques. The basic advantage of this concept is the high degree of abstraction in the lower layers. That ensures the implicit implementation of the basic functionality.
The software GenOOM is provided with a set of superclasses to ensure that the user is able to build up models only by connecting the components. The MRP concept has the advantage that the user must not be familiar with the complete method of object oriented modeling because he uses object frameworks that are filled with the mathematical knowledge. For the description of the components on this mathematical level we developed an equation-editor. This step is supported by an interfaced expert system. On the one hand the formula knowledge is stored in relational databases as binary trees and on the other hand the inference mechanism takes place through docked algorithms which work on the database structure. In the first step the user has to determine the model complexity. Therefore he navigates through decision trees as introduced by Marquardt et al. [1], [2] and selects the correct balancing equations for mass, energy and momentum: d d t Iv ~(r, t)dv =--~FJ(r, t) 9nF(r)df + ~vq(r,t)dv dE
P
(1)
q
dt - E F j E j - E F k E k j=l k=l
+W P
d M _ M(i) M(0) + ~ Fk dt k=l _
dm i P q dt - ~ f i , j - E fi,k + gi j=l k=l
(2) (3)
(4)
The second step is the formulation of constitutive relations and closure constraints e.g. mass transfer rate, energy transfer, reaction rates and component properties as presented in the work of Jensen [3]. These equations are imported from the knowledge database and therefore
99 easily expansionable and customizable regarding the special tasks. This second step is completely assisted by the expert system. The resulting modular submodels are merged in a flowsheet editor. The connections between them give information about coupling variables. This is necessary to link different DAE (differential algebraic equation) systems of the submodels to an aggregated DAE-system which is solved numerically.
Figure 2. Interaction of the several clients of the GenOOM-framework Figure 2 shows the workflow through the several components of GenOOM. Gray boxes without shadow represent hidden clients which work in the background without interaction with the user. 3. STRUCTURE OF THE SOFTWARE GENOOM
GenOOM's architecture is based on the concept of distributed applications which at the moment is limited to DCOM clients in a local area network and TCP IP clients on wide area networks. A future perspective is the expansion to ActiveX on wide area networks which means that flowsheeting can take place via W W W or internet on small clients and model aggregation, generation and finally simulation is done on a high performance computing machine. Currently implemented components are: flowsheet editor which uses modem software technologies (i.e OLE, drag and drop of components from a tool box, multi layer concept and component aggregation and segregation by group and ungroup functionality) DAE-system solver (DASSL, LIMEX, LIMEXS, LSODI, RADAU5, RADAUP, RADAU, RODAS, ROS4, SDIRK4, SEULEX, SODEX)
100 component properties database (ASPEN PROPERTIES PLUS, Data Compilation Table, DIPPR) formula knowledge database knowledge based equation editor regression tool PAREG for offiine parameter estimation model, code (FORTRAN 90, C++, java) and documentation generator (HTML, MATHML) Figure 3. Structure of the software GenOOM: Distributed clients communicate via DCOM or TCP IP in a local or wide area network. The service clients (optimizer, expert system, solver , compounds and other distributed components) provide the framework (in the middle) with data and functions which the model developer or the model user are able to use in their projects
One important part in the model generation is the model ~>debugging~ where the user can gather information about the consistence of his model. At the moment this debugging step consists of: dimension analysis of the used equations analysis of the degrees of freedom of the whole aggregated matrix analysis of the incidence matrix index analysis analysis of time constants and eigenvalues A very important aspect is the integrated environment the user works with from problem definition over model construction until model validation. Only in the background this integrated environment decomposes into several tasks on different machines where clearly defined component interfaces ensure an exchange or expansion of single parts.
101 4. PROTOTYPE AND FUTURE PERSPECTIVES
The data exchange with several simulation programs (Speedup, gProms etc.) is planned. This presupposes the development of translation-clients which work between GenOOM's mathematical formulation of models and the solver-specific problem representation. Additionally the support of formulation of Partial Differential Equation systems (PDE) is planned.
Figure 4. Flowsheeting environment of the GenOOM-prototype 5. REFERENCES
[1]-
[2] [3]
Marquardt, Towards a process modeling methodology RWTH Aachen 1993 R. Bogusch, B. Lohmann, W. Marquardt, Ein System zur rechnergesttitzten Modellierung in der Verfahrenstechnik Jahrbuch der GVC, VDI Verlag 1997 Anne Krogh Jensen, Generation of Problem Specific Simulation Models within an Integrated Computer Aided System Ph.D. Thesis Technical University Of Denmark 1998
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European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
103
Detecting outliers in multivariate process data by using convex hulls J. P. Bamard and C. Aldrich Department of Chemical Engineering, University of Stellenbosch, Private Bag X1, Matieland, Stellenbosch, 7602, South Africa Email: [email protected] (secretary of Prof. Aldrich)
In this paper the problem of near-real-time detection and removal of up to 15% radial outliers from large data sets (10000 or more records) is investigated and a practical solution demonstrated. The proposed method was compared to the Rocke and Woodruff algorithm in an example of elliptically arranged random data and applied successfully to test data recorded of a flotation plant. The technique has the benefits of low computational cost with minimal operator input, and can be implemented as a real-time outlier detection tool. 1.
Introduction
Each radial outlier lies in a different direction, with its mean offset from that of the good data. There are several classes of outliers. Shii~ outliers, i.e. outliers that has a mean that is offset from that of the good data, but with the same covariance matrix, are the most difficult to detect. On the opposite end, radial outliers are easiest to detect. It often occurs that an outlier is not revealed by inspection of the individual components (limit checking) of a multivariate data vector, since for each component the extreme points are within acceptable limits. However, in a two-dimensional phase plot, the outlier may become apparent when the vector containing the outlier components protrud.es significantly from the neighbourhood of data vectors. Human judgement has often proved to be subjective and quite inconsistent, hence many mechanistic means have been proposed so far to detect multiple outliers in multivariate process data. Amongst these methods are, for example, the interpretation of principal component residues [ 1], the projection of data onto two- or threedimensional co-ordinates by principal component analysis [2]; cluster analysis, without necessarily visualizing the data. Direct statistical approaches such as probability plots can facilitate discovery of outliers that distort location, scale and correlation estimates of data. The efficiency of some of these techniques can be severely limited when high-dimensional data are considered. In order to distinguish between valid observations and outliers, it is required to estimate characteristic parameters of the distribution of observations as well as calculate a test statistic. Often the probability distribution of the data is unknown, and consequently one attempts to find a robust estimator of the location and shape of the multivariate data. Examples are search algorithms to find the minimum volume ellipsoid [3] or minimum covariance determinant [4]. Most methods are affine equivariant, which means they are invariant in terms of outcome under linear transformations. Rocke and Woodruff [5] constructed a generalized, two-phase method that estimates the location and shape (phase one) and determines outliers by applying a X2 outlier criterion (phase two). In the first phase the
104 location and shape are initially calculated by sequential point adding, using the minimum covariance determinant as initial estimation of shape. Finally shape is estimated using bi,weight M estimation. In the second phase a cutoff point is determined by simulation and the location and shape are updated using the (l-c~) fraction of points falling within the cutoff region. Observations whose Mahalanobis square distance, using the updated location and shape, is larger than ~ 2p,l-,~ (P is the dimension of the data) are rejected as outliers. Arguing that shift outliers are the hardest to locate, they showed successful results based mainly on this class of outliers. A practical limit to the ability of this algorithm, in terms of required data and processing time, is a maximum of about 35% outliers in 20-dimensional data. However, linearly extrapolating the results in table 5 of [5] suggests that computational cost of this method may be prohibitively high for application to large data sets (size of order 10 000 records) often found in industry. Detecting radial outliers in data does not require the sophisticated search for location and shape of generalized outlier detection algorithms. Therefore, it is suggested here that a convex hull can indicate the position of radial outliers and that the true location and shape of the data can be estimated after removing the hull from the data. The indicated outliers are qualified as true or false using a test statistic based on statistical properties of the retained data. This method is compared to the Rocke and Woodruff algorithm [5] in an example of elliptical random data and applied successfully to test data recorded on a flotation process. The technique has the benefits of low computational cost with minimal operator input and can be implemented as a real-time outlier detection tool.
2. Detecting radial outliers using convex h u l l s Formally, the convex hull of a point set P is the smallest convex set that contains P. If P is finite, the convex hull defines a matrix A and a vector b such that for all x in P, Ax+ b < - [0]. A convex set can be defined as the intersection of a set of half-spaces, that is the set of points on one side of a plane. The convex hull with maximum possible volume is co-planar with the outer data points in the set, and is by definition convex everywhere"along its surface. For a detailed treatment of convex hulls, the reader may refer to [6]. Suppose one can visualize the data space, X ~ 9~m, then outliers tend to stand out from the local trend in the data. A convex set of the data space will be coplanar with at least some of the outlier vectors. The authors suggest that for data that are grouped into a number of distinct clusters, and fall along a piecewise smooth manifold, it is possible to improve the success of outlier detection by constructing the convex hull on the first difference of the data. Inspecting first differences of such data, the incongruity of outliers with normal data should be more pronounced than when inspecting the data itself. On the other hand, the detection of radial outliers in Gaussian data does not respond well when applied to first differences of the data. Detecting outliers in data requires a criterion by which an outlier can be qualified as a true outlier. Removal of valid data is undesirable. A convex hull does not distinguish between outliers and valid data along the perimeter of data space. Therefore, such a criterion will be defined in the procedure described in the following section.
3. Procedure for radial outlier detection method Assume a data set, X~9~ m, with radial outliers.
105 1. Scale the data by standardizing to zero mean and unit standard deviation. Then normalize each component Xi to have unit length (llXill = 1). Standardization and scaling ensure that all data components are of the same order of magnitude, otherwise a multidimensional outlier detection algorithm would disregard outliers in components of comparatively small magnitude. 2. Depending on the class of data, calculate the first difference, X', of the data, X. 3. Construct a convex hull Q0 around X' (or X). The algorithm and code by Barber et al. [ 1996] can be used for this purpose. Let d 2 be the squared distance of each vertex on the convex hull to the location of the remaining data, and dl 2 the mean square distance of the remaining data to its location/z. 4. Remove Q0 from X' (or X) to give X1. 5. Calculate the Mahalanobis distance from vertices of Q0 to the mean of X~. The Malahanobis distance between points x and y in ~}~m is defined as d~(x, y ) - ( x - y ) r f2-1 ( x - y ) , where the metric f2, is the p x p covariance matrix of the data. 6. A hull vertex qualifies as an outlier if the following inequality is satisfied:
d~ (Qoi,/zl) > sd~ (X1,/z 1), where d(.) is the Mahalanobis distance normalized with the covariance matrix of the remaining data, Q0;e 9~m a vertex of the convex hull, X1 e 9V' the remaining data and s a sensitivity factor.
4.
Demonstration of outlier detection method
The outlier detection method is demonstrated in two examples. The first is a random data set containing known outliers. The second example is a data set recorded of flotation process, without a priori knowledge of possible outliers. The results from the gocke and Woodruff algorithm were taken as comparative standard. Visualization was acfiieved by projecting the data space onto the first two principal components of the data.
4.1. Random data containing outliers To enable an objective evaluation of the outlier detection technique, a two-dimensional Gaussian data set was intersected with data describing an ellipse to produce a data set of 533 records. Six outliers were manually created by moving six arbitrary points in an A'T phase plot to outside the elliptic boundary. None of the outliers could be detected as such from inspection of the X and Y components of the data set, since they were within the range {-2.5, +2.5 } spanned by each component. However, in a phase plot the outliers were clearly visible. As a first approach, a convex hull was constructed around first differences of the data and then removed. After two iterations only two out of the six outliers were removed from the first differences of the data. The authors therefore conclude that for random data, it appears that extreme values in first differences of data generally do not coincide well with extreme values in the data themselves. When applied to the data, instead of first differences of the data, the outlier detection scheme correctly detected four out of the six outliers, with the two least severe outliers remaining after one iteration. The cost of outlier detection was five false outliers identified at the ends of
106 the long axis of the ellipse. This is due to the spherical outlier criterion applied to a data set that is very non-spherical in terms of distribution. A default sensitivity factor, s = 1, was used in this example. However, since the motivation behind the convex hull method is fast detection in large data sets, this cost is very low at 0.9% of sample size. By comparison, the Rocke and Woodruff algorithm detected the same outliers as our technique, also failing to detect the two least severe outliers. Their algorithm incurred zero detection cost by not indicating any false outliers. Finally, the computational cost in terms of time required to process 10000 elliptically distributed random data points was investigated. The convex hull technique took 1.0 second to complete one iteration of detection while the Rocke and Woodruff technique needed 660 seconds, running ANSI C code for both algorithms under Microsoft | NT4 on an Intel | PII 400 Celeron TM processor with 256 MB RAM. 4.2.
Flotation process data
As a second example, the convex hull technique was applied to data recorded of a flotation process, used for the recovery of base metals. Five independent variables representing visualization data of the process as well as the percentage lead recovery were combined into a six-dimensional data space, containing 1425 data points. Table 1 Summary of results of outlier detection on flotation process data. Algorithm Convex hull Rocke and Woodruff
Outliers indicated 115 60
Common Outliers 35 35
Relative detection cost 5.6% 0
computation time [s] 3 36
A principal component analysis was performed first. Plotting the first versus the second principal components indicated two distinct clusters. The Rocke and Woodruff algorithm identified 60 outliers in the data, and took 36 seconds to run. These outliers coincided with data lying beyond the 95'th percentile in terms of Hotelling's T-squared statistic, calculated from the data. The convex hull outlier detection procedure was iterated once with a detection sensitivity, s = 3, and indicated 115 outliers. Figure 1 shows the outliers identified by the convex hull algorithm as well as the Rocke and Woodruff algorithm. The results are summarized in Table 1. 5.
Conclusions
In this paper the proposed outlier detection method for the near real-time detection of radial outliers was effectively achieved by way of an algorithm that incorporates the construction of a convex hull and the removal of points supporting the hull. The method was demonstrated successfully on artificial random data, as well as real process data. Our investigation into computational cost showed the convex hull technique to be up to two orders of magnitude faster than the generalized Rocke and Woodruff algorithm. Compared to the Rocke and Woodruff algorithm, the cost in terms of number of false outliers detected was higher on the random data, owing to the less sophisticated shape estimator in the convex hull technique.
107 015
01
,~+,
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• - convex hull algorithm + - Rocke and Woodruff
9
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~ 0.05
0 ~.1
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Figure 1 Projection of flotation data onto first and second principal components, showing outliers detected by the convex hull algorithm, and Rocke and Woodruff algorithm.
The results for the flotation process data were satisfactory in that the indicated outliers, common between the two algorithms, were the major ones along the perimeter of the data space. [ 1] D.M. Hawkins, Identification of outliers, Chapman and Hall (London), 115, (1980). [2] R. Gnanadesikan, Methods for statistical data analysis of multivariate observations, Wiley and Sons, 292, (1977). [3] F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw and W.A. Stahel, Robust Statistics: The approach based on influencedfunctuions, John Wiley (New York), (1986). [4] P.J. Rousseeuw, A.M. Leroy, Robust regression and outlier detection, John Wiley (New York), (1987). [5] D.M. Rocke, D.L. Woodruff, J. Am. Stat. Association, 47(435), 27, (1996). [6] J. O'Rourke, Computational geometry in C, Cambridge University Press, (1994).
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European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
109
M I N L P O p t i m i z a t i o n of S e v e r a l P r o c e s s S t r u c t u r e s for t h e S e p a r a t i o n of A z e o t r o p i c T e r n a r y M i x t u r e s D. Brusis a, Th. F r e y a, J. Stichlmair a, I. Wagner b, R. Duessel c, F.-F. Kuppinger c a L e h r s t u h l fuer F l u i d v e r f a h r e n s t e c h n i k , Technische U n i v e r s i t a e t M u e n c h e n , B o l t z m a n n s t r . 15, 85747 Garching, G e r m a n y Email: [email protected] b W e l C h e m GmbH, B o l t z m a n n s t r . 15, 85747 Garching, G e r m a n y c D e g u s s a - H u e l s AG, P a u l - B a u m a n n - S t r . 1, 45764 Marl, G e r m a n y
Abstract - For m a n y separation tasks in chemical i n d u s t r y there exists a variety of different processes to meet the desired product specifications. This paper presents a comparison between four different process alternatives for the separation of the t e r n a r y mixture methanol/2-propanol/water. To create a reliable basis for the evaluation of these different processes a rigorous M I N L P optimization is carried out.
1. INTRODUCTION The separation of multi component mixtures is of great interest in i n d u s t r y as well as in scientific research. To achieve high product purities distillation often is the appropriate unit operation. In doing so, the separation of azeotropic m i x t u r e s is m u c h more complicated t h a n the separation of zeotropic mixtures, as azeotropes form barriers t h a t cannot be crossed by distillation. Often, there exists a variety of different processes to perform the separation but with rising n u m b e r of required fractionations the n u m b e r of suitable unit operations and their possible combinations rises significantly. Due to a lack of quantifying methods, up to now the decision for a particular process is often based on the design engineer's preferences and experiences or on heuristic rules. This s t r a t e g y can lead to significant losses in either operating or i n v e s t m e n t s costs. This paper presents a comparison between four different process alternatives for the s e p a r a t i o n of the t e r n a r y mixture methanol/2-propanol/water. Because of the m i n i m u m azeotrope between w a t e r and 2-propanol and the resulting distillation boundary, the recovery of all pure components by distillation is difficult (see Fig.l). Potential process alternatives discussed in the paper are pressure swing distillation, azeotropic and extractive distillation and p e r v a p o r a t i o n . To create a reliable basis for the evaluation of these processes a rigorous M I N L P optimization of each process is carried out. Depending on the feed composition, the best process can be determined with respect to the a n n u a l
110 methanol (a) 64,5 ~
water (c) 100 ~
-
IF-
.
.
.
.
l u
80,7~
-.~
2-propanol(b) 82.6 ~
Fig. 1. T r i a n g u l a r diagram of the mixture methanol/2-propanol/water costs, which are calculated using industrially applied cost functions. As this m a t t e r is of special interest for the industry, this work is a cooperation of L e h r s t u h l fuer Fluidverfahrenstechnik, TU Muenchen, Degussa-Huels AG and WelChem GmbH.
2. S U P E R S T R U C T U R E S
AND MODELS
A basic r e q u i r e m e n t for a MINLP optimization is the creation of an appropriate process superstructure. To g u a r a n t e e t h a t the superstructure comprises all relevant process structures, it is developed on the basis of t h e r m o d y n a m i c information t h a t can be retrieved from the phase equilibrium of the m i x t u r e to be separated. Only in this way the variety of imaginable processes can be limited to a reasonable n u m b e r with respect to ensure t h a t all relevant v a r i a n t s including the optimal one are contained. Thus, it is obvious t h a t each of the four process alternatives t h a t are discussed in the paper requires a different method to create an adequate superstructure, which is presented in the following. Azeotropic or extractive distillation is often used for the separation of azeotropic mixtures. Both have in common t h a t an additional e n t r a i n e r is necessary to fulfill the separation task. For azeotropic distillation the e n t r a i n e r m u s t be immiscible with one of the components of the original mixture [1]. Extractive distillation is an essence combination of physical absorption and distillation. Thus, a high boiling e n t r a i n e r is required which does not form additional azeotropes and which selectively absorbs one of the components of the original
111 mixture. The s u p e r s t r u c t u r e s for azeotropic and extractive distillation t h a t are applied in this paper are based on those t h a t have been developed by Bauer [2] and Glanz [3]. Bauer and Glanz restricted their s u p e r s t r u c t u r e s to the separation of b i n a r y m i x t u r e s with an additional entrainer. To apply these s t r u c t u r e s to this t e r n a r y mixture an additional column is required to s e p a r a t e the low boiler methanol. This leads to a binary m i x t u r e for which a s u p e r s t r u c t u r e can be determined according to the rules of B a u e r and Glanz. This methodology has two inherent advantages: First, the p r e s e p a r a t i o n of the low boiler reduces the a m o u n t of feed to the subsequent process which leads to smaller column diameters. Second, it is much easier to find an appropriate e n t r a i n e r for the separation of the r e m a i n i n g binary mixture. The s u p e r s t r u c t u r e for the azeotropic distillation is shown in Figure 2. It applies for any e n t r a i n e r t h a t forms a miscibility gap with water. The question which e n t r a i n e r is best suited for the separation can only be answered by a rigorous optimization of the whole process. Feasible entrainers are toluene, cyclohexane or benzene [4]. The s u p e r s t r u c t u r e of extractive distillation is shown in Figure 3. Appropriate e n t r a i n e r s would be ethyleneglycol and dimethylformamid. Besides the first two process alternatives t h a t require an e n t r a i n e r there exist also processes without any entrainer. The third possibility to be studied is the so called generalized process [5] which includes the v a r i a n t of a pressure swing distillation shown in Figure 4. The last process to be considered is a hybrid pervaporation process. The s u p e r s t r u c t u r e is shown in Figure 5. Here, only one column is necessary to separate the mixture in three pure components. Methanol is s e p a r a t e d at the top of the column and 2-propanol is w i t h d r a w n at the bottom. The high boiler w a t e r is drawn off as a side s t r e a m t h a t is split within a pervaporation unit into pure w a t e r (permeate) and a r e t e n t a t e which is recycled into the column. Polymeres or anorganic m a t e r i a l s can be used as m e m b r a n e s . The m a s s t r a n s f e r through the m e m b r a n e depends on t e m p e r a t u r e . Therefore, the feed m i g h t be passed through an additional h e a t e r to achieve a constant p e r m e a t e flow and to supply the h e a t required for the vaporization of the permeate. D1 (a)
F a+b+c
r
I
I
|I I I i I i
I
'~
I
I
c
I I
I
r" ~
1
1~
D3
entrainer makeup
e
F L2
B2 (c)
E-1
II B3 (b)
Fig. 2. S u p e r s t r u c t u r e of the azeotropic distillation
B1U
B2T
I
e
Fig. 3. S u p e r s t r u c t u r e of the extractive distillation
112
a
D1
D3
F
~
F ~ a+b+c
P+CL~
L ~ c
b
b
Fig. 4. S u p e r s t r u c t u r e of the pressure swing distillation process
Fig. 5. S u p e r s t r u c t u r e of the m e m b r a n e process
3. R E S U L T S OF T H E M I N L P O P T I M I Z A T I O N
For each of the four processes a M I N L P optimization with respect to the a n n u a l cost h a s been carried out. During the optimization the total n u m b e r of stages, the feed stages, the recycle s t r e a m s as well as reflux and boil-up ratios are determined. The results for a feed s t r e a m of 10 mol/s with a composition of 9.4 mol% water, 11.3 mol% 2-propanol and 79.3 mol% methanol are shown in Figures 6-9. The specified product purities are 99.4 mol% for 2-propanol as well as for m e t h a n o l and 99.9 mol% for water. The columns in the figures are scaled r e g a r d i n g to t h e i r height and diameter. D1 (a)
I
~ "[.776
(: D2
C-1
~ 19
38 ~ 7 7 7
I
2~
I
KW
C-2 ~ i
287kW ID~
~
KW
E-1 L1-0,59F
C-3
~_6KW L2=0,1F 26
B2(c~
12 ... ~.
7 KW
IJ. B3(b)
Fig. 6. Optimized process structure for the azeotropic distillation
113
D2 ~ b " ~ 8 9 kW 3~
~D1 -- -766kW F 15
-
~
46 777 kW 1
c
I 16T:
E=0,12/~ C-2 ~.~-15 kW
C-1
38 T
entrainer makeup
93
_
kW
B2
Fig. 7. Optimized process structure for the extractive distillation
S=2,4F
~
~D
a
.~-816 kW
7
F
C-1 C-2 1 bar
15
29 38 ~
777 kW
31
c
C-3 3 bar ~
C-1
kW
F ,.. 2422 kW
40 2519kW
Fig. 8. Optimized process structure for the pressure swing distillation
30 ~ L ~J
855 kW
F2=0,26 F
Fig. 9. Optimized process structure for the membrane process
A comparison of the total costs, operating costs and annualized investment costs can be seen in table 1. The values include the costs for the column shells, the heat exchangers and the costs for the steam and coolant. The results show that the azeotropic and extractive distillation are mutual comparable in their specific costs and more favourable than the other two processes. The pressure swing
114 Table 1 Comparison of the four different processes Azeotropic distillation
Extractive distillation
Pressure swing Membrane distillation process
Operating costs [TDM]
47.9
38.6
276.9
96.7
Investment costs [TDM]
65.3
59.14
144.6
86.8
Total costs [TDM]
113.2
98
421.5
183.5
distillation needs high recycle streams due to the small variation of the azeotropic composition by the pressure change. This leads to much higher operating and investment costs due to larger columns and higher steam flows. It is obvious that this process is not competetive with the other ones. The pervaporation process is more expensive due to the higher costs for the coolant which is needed for the permeate and due to higher investment costs which include additionally the costs for a vacuum pump. In case the price for the special coolant could be reduced the total costs will become more comparable with the other processes. Nevertheless, as the concentration of methanol in the feed was very high, the total mole flow within the azotropic and extractive process was very low. It is assumed that the costs of these processes will rise and approach the costs of the membrane process if the concentration of methanol is smaller. In contrast the total costs of the pervaporation will be r a t h e r equal as long as the amount of water is approximately the same. Then the costs for the single column and the operation costs for the membrane will not change dramatically.
RERERENCES
[1] Stichlmair, J.; Fair, J.R.: Distillation - Principles and Practice, Wiley-VCH, New York 1998 [2] Bauer, M.H.: Synthese und Optimierung nichtidealer Rektifizierprozesse, PhD thesis TU Muenchen; 1997 [3] Glanz, S.: Synthese und Strukturoptimierung von Prozessen zur Trennung heterogener Fl~ssigkeitsgemische, PhD thesis TU Muenchen, 1998 [4] Pham, H.N.; Doherty, M.F.: Design and Synthesis of Heterogeneous Azeotropic Distillation - I I I Column Sequences; Chem. Eng. Sci.; 45 (1990) 7; 1845-1854 [5] Stichlmair, J.: Distillation and Rectification; in Ullmann's Encyclopedia of Industrial Chemistry; B33; 4; 1-94; VCH Verlagsgemeinschaft; Weinheim; 1988
huropean Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
115
MINLP Optimization of Reactive Distillation Columns Th. Frey and J. Stichlmair Lehrstuhl ffir Fluidverfahrenstechnik, TU Mfinchen, Boltzmannstr. 15, D-85748 Garching, Germany, email: [email protected] Reactive distillation, i.e. the simultaneous implementation of reaction and distillation in a counter current column, has proven to be quite advantageous for several reasons. The whole capacity of reactive distillation and its potential for chemical processes can only be shown at optimized reactive distillation processes. This requires a rigorous simulation and optimization of reactive distillation processes which leads to a Mixed Integer Non Linear Programming (MINLP) problem. This MINLP model is based on the concept of equilibrium stages. In addition to the rigorous formulation of the thermodynamic equilibrium it is necessary to develop an appropriate superstructure for the process to be optimized. To guarantee that the superstructure contains all physical relevant process alternatives, the definition of the superstructure is based on a thermodynamic background. The MINLP optimization can be performed with respect to minimum energy requirement or minimum total annual costs. The results are presented for the synthesis of methyl acetate. 1. I N T R O D U C T I O N In the last decade the interest in reactive distillation has continuously grown. The reasons therefore are quite advantageous synergy effects that arise from the simultaneous implementation of reaction and distillation in a single apparatus and the inherent superimposition of chemical equilibrium upon the vapor liquid equilibrium. Some of these effects are [ 1]: 9 Continuous withdrawal of products from the reaction section can lead even to total conversion of the reactants. 9 Undesired side reactions can be suppressed. 9 Heat of reaction can be utilized for the simultaneous distillation. Furthermore, it must be stated that conventional distillation boundaries, as azeotropes or border distillation lines, can be overcome by reactive distillation. As a negative side effect the superimposition of two equilibria may lead to so called reactive azeotropes [2]. In analogy to conventional distillation these reactive azeotropes form a barrier for reactive distillation. Thus, one of the necessary conditions for process synthesis and for the synthesis of appropriate superstructures for the MINLP-optimization is the determination of the location of reactive azeotropes and, in turn, the determination of possible products within a reactive distillation column. Until now, most of the works dealing with reactive distillation apply so called transformed compositions for the representation of the thermodynamic equilibrium [2,3]. The advantage of these transformed variables is that reactive distillation can be described in close analogy to conventional distillation, because the dimension of the valid concentration space is reduced by the number of chemical reactions. But the use of transformed compositions implies a severe disadvantage as a lot of information about the vapor-liquid equilibrium is lost. In an early stage of process design, as it is for the definition of a process superstructure for a MINLP-optimization, the use of transformed compositions may lead to structures that do not
116 comprise the optimal process configuration. Thus, a novel method for the determination of reactive azeotropes has been determined that is based on the collinearity of distillation and reaction steps [4]. Knowledge of the location and the type of reactive azeotropes is a necessary condition for the determination of possible products within a reactive distillation column and can be used for the design of appropriate superstructures. In previous works the methodology has been presented in detail for equilibrium and non equilibrium reactions [4,5]. In this paper the method is applied to the synthesis of methyl acetate. 2. D E T E R M I N A T I O N OF PROCESS STRUCTURES
The determination of the location and the type of potential reactive azeotropes leads to the definition of a process structure since possible products can easily be determined. This is to be presented for the synthesis of methyl acetate [6], which is one of the most important reactive distillation processes in industry and thus, experimental results are available and can be compared with the results of an optimization. Nevertheless, the method is generally applicable to multicomponent mixtures and even to kinetically controlled reactive distillation processes [5]. Figure 1a shows the tetrahedron of the mixture methyl acetate (a), methanol (b), water (c) and acetic acid (d). The letters a to d denote the substances in the order of increasing boiling temperatures. The mixture contains two minimum azeotropes, one between the methyl acetate and methanol, the other one between methyl acetate and water. Between these azeotropes there exists a distillation boundary. This distillation boundary together with a tangential pinch at the water edge impede the synthesis and purification of methyl acetate in a sequential process. Figure 1A also presents the chemical equilibrium saddle plane for the esterification of methanol and acetic acid towards methyl acetate and the by-product water. With the considered phase equilibrium the condition of collinearity between stoichiometric lines and tangent to residuum lines leads to no lines of possible reactive azeotropes. This condition is met only at the methanol/methyl acetate minimum azeotrope, that lies on the chemical equilibrium surface, i.e., this non reactive azeotrope is also a reactive azeotrope. All reactive distillation lines run on this equilibrium surface. They start at the high boiler acetic acid and end at the reactive minimum azeotrope. It can be seen that the desired products methyl acetate (a) and water (c) are no starting or ending points of reactive distillation lines. In Figure 1B this can be emphasized using the transformed compositions mentioned above [3]. This transformation is equivalent with a projection of the equilibrium plane along the main stoichiometric line on a two dimensional plane. Thus, the saddle surface of Figure 1a transforms into a square in Figure lB.
117
Figure 2: Superstructure for the synthesis of methyl acetate. In Figure 1B the course of the reactive distillation lines upon the chemical equilibrium plane is obvious. The desired product methyl acetate (a) and the byproduct water (c) do not lie on the same distillation line. Thus, the separation in a single feed reactive distillation column is not possible, as the reactive distillation line represents the liquid concentration profile of a reactive distillation column for total reflux. In conventional distillation, this limitation can be overcome by introducing a second feed stream into the column, as it is done in extractive distillation. In contradiction to the usual temperature profile within a distillation column, the high boiler is fed near the top of the column and the low boiler near the bottom. Between the two feed stages a physical absorption takes place and, in turn, products can be reached that do not lie on the same distillation line. The same can be done in reactive distillation. Figure 2 shows the structure of the column for the synthesis of methyl acetate. The high boiler acetic acid is fed near the top of the column. The low boiling reactant methanol is fed near the bottom of the reactive column, preferably as vapor. Within the reactive column, three different sections are necessary: a reactive section between the two feed stages, a conventional rectifying section above the acetic acid feed to separate methyl acetate from acetic acid and a conventional stripping section beneath the methanol feed to separate water from methanol. The degree of freedom that remains for a process optimization is still considerably high. There are three operating parameters that have to be optimized (condenser and reboiler duty together with the temperature of the liquid feed) and five geometric variables (total number of stages, number and location of reactive stages together with two feed stages). This problem has been formulated as a MINLP-problem. 3. M I N L P - M O D E L L Until now there are only a few works dealing with the MINLP-optimization of reactive distillation columns. Ciric and coworkers presented a MINLP-model for the optimization of the ethylene glycol production [7], but under the assumption of ideal vapor-liquid equilibrium.
118
Figure 3: Model of a reactive equilibrium stage. The MINLP model is based on equilibrium stages with the use of efficiencies when required. The model for the distillation sections has been taken from [8] and includes the MESH-equations on every stage with a rigorous description of the vapor-liquid equilibrium using the Wilson approach. The thermodynamic equilibrium upon a reactive stage is not calculated using a simultaneous approach as proposed in [9]. As we use an OuterApproximation/Equality-Relaxation/Augmented-Penalty algorithm for the solution of the M1NLP-problem [ 10], it is necessary to define a maximum number of possible stages, as well tbr reactive as for non reactive sections. Thus, there must be several stages within the column that have been defined as possible reactive stages but as a result of the optimization they turn out to be conventional distillation stages. A different description of a single phase equilibrium and a combined phase and chemical equilibrium would lead to a large number of equations on these trays that have to be solved during optimization although they are not relevant for the solution and therefore only deteriorate the convergence. Thus, the combination of two different equilibria occurring on a reactive stage has been split in two sub-problems (Figure 3). First the liquid xn-l,i from the stage above undergoes the chemical reaction. The equilibrium composition x i~q" that can be reached in this stage is calculated with respect to all nonidealities, i.e. also including the activity coefficients: K,~ = l-[ (x2'*' 9y,)"
(1)
i
In total, the number of equations that have to be solved for the chemical equilibrium section including the calculation of the value of temperature dependant equilibrium constants and the heat of reaction is 4n + 5. For a quaternary mixture this means that 21 additional equations have to be solved on each reactive stage. The liquid stream that leaves the reaction stage is calculated according to: iHI (2) X ,,I __ X , - l , i + breac.t,i 9 ,.tx~,, equ - x,_,j) where b,-~aa,, denotes the integer variable that represents the chemical reaction. If the integer variable b,.~aa,, equals 0, the term in brackets is not considered, i.e., no reaction occurs on this stage and, in turn, this stage is treated as a conventional distillation equilibrium stage. If the integer variable b,.ea~t,, equals 1, the equilibrium composition x2q" after chemical reaction is reached and, in combination with the following distillation section, the stage represents a reactive distillation stage. A sequence of these reaction and distillation sections combined in a reactive stage represents the model of a reactive distillation line as mentioned above and therefore is appropriate to describe the concentration profiles within a reactive distillation column.
119 A second advantage besides the decrease of the number of equations is, that this model can also be used to optimize kinetically controlled reactive distillation columns. Therefore, equation (2) must be rewritten into int
equ
x ,,,, = x ,_~,, + b ..... ,,, 9rl " ( x , , ,
- x
~,, )
(3)
The parameter r/represents the ratio of residence time of the liquid upon a reactive stage to the reaction rate. This ratio can also be expressed by a Damk6hler-Number. r/=
Da ~
(4)
Da+l
In contrast to a Damk6hler-Number, 77 is only defined for values between 0 and 1. In addition to the calculation of the equilibrium constant and the equilibrium composition, expressions for the reaction rate of the reaction have to be included. 4. RESULTS Figure 4 presents the results of an economic optimization of the superstructure presented in Figure 3. The objective function is the minimum of the total annual costs of the process, with the use of industry relevant cost functions [8]. It is assumed that a heterogeneous catalyst is applied, i.e., the reaction section can be located anywhere within the column. Each reactant is fed in an amount of 10 mole/s. The distillate contains 99 mole percent methyl acetate, the bottom product contains 96,3 mole percent water. The total annual costs of the column are 430 000 $. The total number of stages is 45. The reactive stages are located between the 10th and the 40 th stage. The condenser and reboiler duties are 788 kW and 606 kW, respectively. The liquid acetic acid is fed on stage 6 at a temperature of 349 K. It is quite advantageous to feed the vapor methanol at several stages between the 36 th and the 40 th. Looking at the concentration profiles published by [6,11] there are always two points of intersection between the water and the methanol concentration near the bottom of the column, i.e., on several stages near a single methanol feed the amount of methanol is higher than the amount of water. This ratio changes if one approaches the bottom of the column, as the whole methanol is vaporized and moves upwards the column to react with the acetic acid and the bottom products is almost pure water. This disadvantage can be overcome by multiple vapor feeding. Figure 4 presents the optimal solution which includes five methanol feed stages. Limiting the number of feed stages to two leads to similar results as shown in Figure 4. The methanol is introduced at the 35 th stage with an amount of 4.6 mole/s and at the 40 th stage with an amount of 5.4 mole/s. The total annual costs of the process increase by less than one percent. Similar to the results presented by [6,11 ] there exists an extractive section between the liquid acetic acid feed on stage 6 and the reactive section beginning on stage 10 (Figure 4a). Looking at the concentration profiles there is nearly no change in concentration on these four stages. In order to determine whether or not this extractive section is necessary for the separation or only advantageous as the costs for a reactive stage is higher than the cost for a conventional distillation stage, an energetic optimization has been carried out where the number of stages has been fixed to 45. As a result of this optimization the number of reactive stages is increased and the reaction section is located between stage 5 and stage 40. The liquid acetic acid is fed at stage 5 and 6, the methanol is fed between the stages 33 and 38. This implies that the extractive section in usual column designs is not necessary. Due to the multiple feeding of the reactants the energy demand is decreased by 11 kW to 595 kW.
120
Figure 4: Results of a rigorous economic optimization for the synthesis of methyl acetate together with the concentration profiles OUTLOOK As mentioned above, the model is formulated to consider further chemical reactions or reactions that do not reach equilibrium but are limited by their reaction kinetics. In this case the liquid holdup, i.e. the residence time of the liquid in the column plays a important role for the calculation of reactive distillation processes. This is to be shown for several kinetically controlled reactive distillation processes in near future. REFERENCES
1.
Doherty, M. F.; Buzad, G.; Reactive Distillation by Design; Trans I Chem E, Vol. 70 (1992) Part A, pp. 448-458 2. Barbosa, D.; Doherty, M. F.; The Influence of Equilibrium Chemical Reactions on VaporLiquid Phase Diagrams; Chem. Eng. Sci. 43 (1988), pp. 529-540 3. Ung, S.; Doherty, M. F.; Vapor-Liquid Phase Equilibrium in Systems with Multiple Chemical Reactions; Chem. Eng. Sci. 50 (1995) 1, pp. 23-48 4. Frey, Th.; Stichlmair, J.; Thermodynamische Grundlagen der Reaktivrektifikation; Chem. Ing. Tech. 70 (1998) 11, pp. 1373-1381 5. Frey, Th.; Stichlmair, J.; Reactive Azeotropes in Kinetically Controlled Reactive Distillation, Chem. Eng. Res. Des. 77 (1999) A7, pp. 613-618 6. Agreda, V. H.; Parin, L. R.; Heise, W. H.; High-purity methyl acetate via reactive distillation; Chem. Eng. Progr. 86 (1990) 2, pp. 40-46 7. Ciric, A. R.; Gu, D.; Synthesis of Nonequilibrium Reactive Distillation Processes by MINLP Optimization; AIChE Journal 40 (1994) 9, pp. 1479-1487 8. Bauer, M. H.; Synthese und Optimierung nichtidealer Rektifizierprozesse, PhD thesis, Technische Universitaet Muenchen, 1997 9. McDonald, C. M.; Floudas, C. A.; GLOPEQ: A New Computational Tool for the Phase and Chemical Equilibrium Problem, Comp. Chem. Eng. Vol. 21 (1997) 1, pp. 1-23 10. Viswanathan, J.; Grossmann, I. E.; A Combined Penalty Function and OuterApproximation Method for MINLP Optimization, Comp. Chem. Eng., 14 (1990) 7, pp. 769-782 11. Perrv. R. H-: Green-DW: Perry's Chemical Engineers' Handbook ,-]th ed McGraw-Hill
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rights reserved.
121
Batch Distillation Optimization with a Multiple Time-Scale Sequential Approach for Strong Nonlinear Processes M. Wendt, P. Li and G. Wozny Institut ftir Prozeg- und Anlagentechnik, Technische Universit~it Berlin, Sekr. KWT 9 Strage des 17. Juni 135, 10623 Berlin A multiple time-scale sequential strategy is proposed to solve batch distillation optimization problems possessing strong nonlinear properties. Large time intervals are allowed in the optimization layer for the sensitivity computation, while small intervals are adjusted to make sure the convergence of the Newton method in the simulation layer. This makes it possible to optimize batch distillation processes with an abnormal phase equilibrium behavior. The column pressure policy, which is usually a variable during the batch operation in the industrial practice, can also be optimized by the multiple time-scale framework. The results of an industrial batch distillation show the effectiveness and efficiency of the approach. 1. I N T R O D U C T I O N Batch Distillation is one of the most common processes in industry for separating liquid mixtures. Mostly batch distillation columns are still run by conventional policies derived by heuristic rules. Nevertheless, many investigations on new optimization strategies for batchdistillation have been done recently. Usually a large-scale dynamic optimization problem will be formulated for such processes. Methods to solve such a problem can be classified into simultaneous (Cervantes & Biegler, 1998) and sequential (Vassiliadis et al., 1994) methods. In the simultaneous methods, there is mathematically no real distinction between the state and the control variables. The model equations, which reduce the degree of freedom, are defined as equality constraints. In the sequential methods, the entire algorithm is divided into one layer for optimization and one layer for simulation .The model equations are integrated in the simulation layer, so that the state variables can be computed by given controls. The control variables are computed in the optimization layer as the only decision variables. In Li et al. (1998) a sequential approach is proposed, how to apply it in batch distillation optimization. SQP is used for the optimization layer, while the orthogonal collocation combined with the Newton method is used for the simulation layer. The control variables are chosen to be piecewise constant. Thus the length of those time intervals are considered as decision variables as well. Those time intervals computed by the optimization layers are equal to the time interval for the discretization in the simulation layer. This makes the sensitivities easy to compute as well. For that algorithm, convergence of the Newton method must be guaranteed even for large time intervals. For strong nonlinear processes, e. g. if the physical characters of the components of the mixture to be separated are very different, convergence can be only guaranteed for very small time intervals. This leads to an increase of decision variables and thus the scale of the
122 optimization problem will be enlarged. Additionally, the optimized policies with very small time intervals for the control parameters are not practicable for the implementations of such policies in industry. That could be overcome by setting the control parameters in a certain number of neighbored time intervals equal, which will lead to many equality constraints in the optimization layer. This makes the optimization problem unnecessarily complex and the number of iteration steps will be increased. Moreover, the CPU-time of each simulation will be increased, since the time-consuming computation of the sensitivities has to be done at each time interval. In this study, we propose a multiple time-scale strategy to overcome this problem. The large time intervals should be long enough for the practical realization as well as for the reduction of the computation time in the sensitivity calculation. The small time intervals are adjusted in the simulation layer and their length will be kept more flexible to guarantee the convergence in the Newton iteration. This approach is applied to solving the time-optimal problem of an industrial batch distillation process which has an abnormal vapor-liquid equilibrium (VLE) behavior. Furthermore, besides the reflux ratio, we have also included the column pressure as a decision variable into the optimization problem. 2. T H E E X I S T I N G S E Q U E N T I A L A P P R O A C H A general dynamic nonlinear optimization problem can be described as (P1) in Fig. 1, where x and u are the state and control variables with given lower and upper bounds, g and h the equality and inequality constraints, respectively, x 0 is a known initial state. Using collocation on finite elements the whole time period will be discretized into time intervals (1 = 1,..., N L ) . In each interval the variables on the collocation points ( j = 1,..-, N C ) will be computed. Now problem (P1) has been transformed to (P2) in Fig. 1, which is usually a large-scale nonlinear programming (NLP) problem. To solve it with a sequential framework, the equality constraints will be eliminated by means of an extra simulation step, i. e. sequential integration of the model equations with the Newton method. This leads to a small optimization problem described as (P3) in Fig. 1, which includes only the control variables and the inequality constraints. A solution approach to this problem is proposed by Li et al. (1998). The successive quadratic programming (SQP) is used to optimize the independent variables in the optimization layer, while the discretized state variables will be solved in the simulation layer. The independent variables include the controls which are assumed in piecewise constant form in each time interval as well as the lengths of the intervals. min J ( x , u , t ) s.t. g(/~, x, u, t) = 0
min J(xl,i,ul,~) s.t.
g ~,~(x l,~, u ~,~) - 0
h(:~, x, u, t) _>0 (P1)
Xmi n ~ X ~ Xma x
(P2)
h l i (xli, u l i ) >- 0 X rain -~ X/,i '~ X max
U rain ~-- U ~__U max
X(to) : Xo
min J ( f (ut,;), ut,, ) (P3) s.t.
h~,z(f(u~,,:),ut,~)_> 0 Umi n ~ Ul, i ~ Uma x
U rain ~ U 1,i ~ U max
Fig. 1. Problem transformation: from dynamic optimization to reduced NLP.
123 The gradients of the objective function and the inequality constraints to the controls will be computed inside one interval. These gradients will be transferred from an interval to the next interval through the continuity conditions for the state variables. It should be noted that the gradient or sensitivity calculation causes the major computation costs. Although this approach has been successfully applied to the optimization of several batch distillation processes (Li et al. 1997; Li et al., 1998), it has the drawback that the convergence of the integration of the model equations for large time intervals must be guaranteed. This requires that the process considered has a weak nonlinear property. However, many processes (in particular batch processes) possess strong nonlinear behaviors. For instance, if the physical characters of the components in the mixture are very different from each other, the VLE behaves in an abnormal form, which leads to a strong nonlinear process. Another case of strong nonlinear behavior comes from the drastically changing pressure during the batch operation, which happens very often in industrial practice, to remove a heavy component from the mixture. In these cases, the integration step or the time interval should be very small to make sure the convergence of the Newton iteration. Thus the above optimization approach will be inefficient, since the gradients must be computed for each time interval. 3. THE M U L T I P L E T I M E - S C A L E STRATEGY In this study, we propose a new strategy to overcome this problem by dividing each large time interval computed from SQP, into small intervals for simulation. In those large time intervals, the control parameters are still set to be constant, thus they should be long enough for the practical realization. The small time intervals are adjusted in the simulation layer and their length has to be kept more flexible to guarantee the convergence in the Newton iteration. For this purpose, conditions for nonconvergence have to be integrated in the Newton algorithm. In case of nonconvergence, which will be detected during the Newton iteration, a step length adjustment will be activated to reduce the step length until convergence is achieved. Another purpose of the step length control is that the last collocation point of a small time interval must be one of the collocation points of the large time interval. This ensures that we can approximate solutions of the state variables right at the collocation points of the large intervals. Simulation studies have shown that one advantage of the collocation method is the solutions of state variables at the same time point are almost independent of the step length. Thus, the solutions at the collocation points of the small intervals can be approximated as solutions of the large step integration. Therefore the state variables at those large intervals can be used to compute the sensitivities. This means, the gradient calculation has to be done only at the end of one large time interval. As a result, both the number of decision variables and the computation time for the sensitivity calculation can be significantly reduced
jj
]
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to
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Xk
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1,2
Xk 1,3 = Xl+ L()
Atl+l uz+l "
~
[ l Xr 1,3
SS
tr
Fig. 2. The multiple time-scale strategy for discretizing the dynamic system.
124 Fig. 2 shows the multiple time-scale strategy. The time period considered (t ~ [to,t r]) is divided into large time intervals (l = 1,...,NL). For the continuity of the state variables, we use the last collocation point of a interval as the initial point of the next interval. In each large interval the sensitivities of the state variables to the piecewise controls u t as well as the length of the interval At l will be computed according to the values of the state variables on the collocation points x~,i (here we use the three-point-collocation, i. e. j = 1,2,3 ). To compute the states on the collocation points, these large intervals are again divided into small intervals, which are the integration steps of the model equations. Note that the lengths of the small intervals are adjusted to ensure the convergence of the Newton iteration and meanwhile to coincide the last collocation point of a small interval with one of the collocation point of the large interval. With this strategy the performances of the approach described in the last section will be improved and thus it can be applied to the optimization of strong nonlinear processes. 4. A P P L I C A T I O N TO AN INDUSTRIAL B A T C H D I S T I L L A T I O N P R O C E S S The multiple time-scale sequential approach is used to optimize the operation policies for an industrial batch distillation process. A packed column is operated to separate a fourcomponent-mixture, with A, B, C, D representing from the lightest to the heaviest component. Three main cuts (fractions A and C from the top of the column as well as one fraction D from the reboiler) will be obtained during the batch. An off-cut mainly containing B will be also received from the distillate. It is desired to minimize the total batch time in order to enhance the throughput of the process, under the constraints of the specifications of the 4 fractions. The heaviest component D has no vapor phase and remains in the reboiler during the batch. The VLE relations of the other three components (A, B, C), which will be distillated through the top of the column during the batch, show an abnormal behavior, especially those involving the least volatile component C. With "normal" we mean the x-y diagram of a binary system has the form of (a) and with "abnormal" the form of (b), as shown in Fig. 3. The relation between component B and C in the mixture has the form of (b), from which one can imagine how drastically the state will change, when component C appears and goes up through the batch column.
2/
X
(a)
~
X
)
(b)
Fig. 3. x-y diagram of a binary system: (a) normal and (b) abnormal.
125 In addition, because component C is much more heavier than A and B, the column pressure has to be decreased within the period of the distillation of fraction C. This means, beside the reflux ratio, the policy of the column pressure should be considered as a decision variable for the optimization. Until now, column pressure has been considered as a fixed parameter in previous studies on optimization of batch distillation. As we know, the variation of column pressure leads to a strong nonlinearity of the entire process and thus causes more sever convergence problems in the simulation. Since the convergence problems can be overcome by the multiple time-scale approach, using the policy of column pressure as decision variables becomes possible. The optimization of column pressure is of interest, because an increase of the column pressure allows an increase of the total mass flow of vapor stream at the same F-factor (vapor load term), which is critical to the separation effect in distillation columns. On the other hand, an increase of column pressure also causes a decrease of the relative volatility, that necessitates a higher reflux ratio to fulfill the purity constraints of the distillate products. To describe the packed column we use a detailed dynamic tray-by-tray model. The number of the theoretical trays is calculated corresponding to the height of the packing. The holdup of each theoretical tray is computed with the correlation model proposed by Engel et al. (1997). The vapor load from the reboiler to the column is restricted by the F-factor of the column as well as the heating capacity of the plant. Thus the equality constraints of the time-optimal problem consist of a large set of differential algebraic equations (DAEs). The inequality constrains are the purity specifications of the 4 fractions. The objective function is defined as the total batch time. The operating policies to be optimized are the reflux ratio and the column pressure profiles within the batch, which are restricted by the real plant. The initial amount and compositions of the mixture charged to the reboiler are known. As a result, a large-scale strong nonlinear dynamic optimization problem is formulated. Fig. 4-7 show the optimization results from the solution of this problem using the multiple time-scale approach. The reflux ratio, as shown in Fig. 4 should be increased during the first fraction to keep a high composition of component A in the distillate, which can be seen in Fig. 6. During the second fraction (off-cut) it should be lowered to accelerate the removal of component B. After that there should be no reflux so as to pull component C out of the reboiler as quickly as possible, since component D will not vaporize at all. The column pressure, as shown in Fig. 5, should be high during the first fraction so that the column will have a large vapor load, because there is a large amount of component A in the reboiler at the beginning of the batch.
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Fig. 5. Optimal pressure policy.
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Fig. 7. Optimal reboiler compositions.
After that the pressure should be decreased, since the effect of separation is more and more important as the mixture in the reboiler becomes heavier. This result illustrates the compensation between the amount and purity of the distillate by regulating the column pressure. The three fractions received from the distillate can be clearly seen from the composition profiles shown in Fig. 6. From Fig. 7 it is shown that only component D remains in the reboiler at the end of the batch. The total batch time resulted from the optimized policy is about 8 hours, which is only 50% of the batch time needed for the conventional operation. 5. C O N C L U S I O N S We propose a multiple time-scale strategy to optimize strong nonlinear batch distillation processes. This improves the performances of the existing sequential approach by means of adjusting the small time intervals for the model integration while allowing the large time intervals for the sensitivity calculation. Operating policies for an industrial batch distillation with an abnormal VLE have been optimized. The results show that the multiple time-scale framework is necessary to make the optimization successful. Furthermore, the policy of the column pressure is taken as a decision variable. The results illustrate that the inclusion of the column pressure as a decision variable leads to a considerable reduction of the total batch operation time. A meaningful future work on batch distillation optimization is to consider the uncertainties both from the initial charge and the model parameters. Thus a robust optimal policy can be developed for the real plant operation. Stochastic optimization techniques will be used to address such problems. R E F E R E N C E S
1. A. Cervantes and L. T. Biegler, AIChE J., 44 (1998) 1038. 2. V. Engel, J. Stichlmair and W. Geipel, IChemE Symposium No. 142 (1997) 939. 3. P. Li, H. Arellano-Garcia, G. Wozny and E. Reuter, Ind. Eng. Chem. Res., 37 (1998) 1341. 4 P. Li, G. Wozny and E. Reuter, IChemE Symposium Series No. 142 (1997) 289. 5. V. S. Vassiliadis, C. C. Pantelides and R. W. H. Sargent, Ind. Eng. Chem. Res., 33 (1994) 2111.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
127
Non-Linear Constrained GRG Optimisation under Parallel-Distributed Computing Environments Gustavo E. Vazquez a, Rainiero Rainoldi b and N61ida B. BrignoleC aDepartamento de Ciencias de la Computaci6n, Universidad Nacional del Sur, 8000 Bahia Blanca, Argentina - e-mail: [email protected] bLehrstuhl Industrielle Informationstechnik- Branderburgische Technische Universit~it Cottbus Universit~itsplatz 3-4 - D-03044 Cottbus, Deutschland - e-mail: [email protected] CPlanta Piloto de Ingenieria Quimica, UNS-CONICET, Bahia Blanca, Argentina Phone: 54 0291 4861700, Fax: 54 0291 4861600, e-mail: [email protected] We have designed and implemented a parallel version of the Generalised Reduced Gradient optimisation method (GRG), especially devised for efficient processing on heterogeneous NOWs. The core parallel routines deal with simultaneous constraint evaluation and the calculation of gradients for both the objective function and the constraints. A hybrid model for task scheduling that minimises idle time and considers the heterogeneous nature of the processors was proposed. As to performance comparisons, a modified speed-up metric that takes into account heterogeneity was employed. Significant time improvements were obtained for both academic and industrial examples corresponding to process-plant units. The best results were attained for large-scale problems or when the functions to be evaluated were costly. 1. INTRODUCTION Scientific and technological progress is nowadays supported by intensive use of efficient computing tools. In this sense, parallel processing is a fundamental resource that enables significant reductions in execution time. In the parallel paradigm [1], a bigger problem is subdivided into minor tasks and their execution is assigned to different processors. In this way, the computing time taken by a given algorithm on a computer with a single processor may be reduced up to P times in the ideal case when the work is distributed among P processors. The traditional approach was to employ parallel computers [2]. Nevertheless, its applicability is limited by the need for expensive equipment and the lack of manufacturer standards. Presentday trends are to "migrate" the concept of parallel processing towards heterogeneous computing environments [3]. In a distributed configuration, the multiple processors that constitute a parallel computer are workstations connected by a local data-communication network that allows information transfer among them. This approach allows the efficient use of existing resources, offering minimal start-up budget and easier scalability, not only to boost the speed of computation but also to accommodate larger problems in a distributed memory environment. Specific tasks can be assigned to the most adequate workstation according to individual features
128 such as file-access speed, calculation power and graphical performance. Besides, system scaling simply implies the connection of a new workstation to the data-communication network. Previous publications on parallel processing refer almost exclusively to parallel machines, whose availability implies high investment costs. In contrast, the implementation of algorithms under distributed environments is a novel topic, not yet investigated in detail. In this respect, it is important to remark that the philosophy behind the developments for parallel machines is not applicable to heterogeneous environments in a direct way because key distinguishing features of this architecture, like communication overhead and processor heterogeneity, are not considered. Optimisation of industrial-scale chemical processes is an area that can benefit greatly by the use of parallel techniques because these problems are typically big and computationally intense. Sequential GRG [4], in particular, is a traditional optimisation tool that succeeds in solving a huge amount of non-linear constrained problems. Therefore, its judicious parallellisation is attractive for the treatment of complex rigorous problems. Besides, the gains in computing power can be exploited to improve the accuracy of the results or achieve real-time capabilities. In this paper we describe the design and implementation of a parallel distributed algorithm based on GRG. We firstly implemented and analysed a sequential GRG. The parallel version was built on its basis and several test problems were used for performance assessment. 2. GENERAL DESCRIPTION OF THE M E T H O D
The GRG formulation employed for non-linear minimisation with equality constraints is:
Minimisef(x)
s.t. h~ (x) = 0,
k = 1...,K
where K is the number of problem constraints, x =(Xl,X2,...,XN) contains the optimisation variables and N is the number of problem variables, where K < N. The method consists in solving the system of non-linear equations that represent the constraints for a subset of K variables, afterwards generating a new reduced problem as a function of the remaining N - K variables. Then, the reduced gradient Vf(x) of the objective function is calculated and a line search in its direction is carried out. The procedure is repeated until [IVy(x)ll2 becomes small enough. The vector x is partitioned into two sets of variables: ~ e 9~x which contains the so called basic variables, and Y e 9~ N-x , with the remaining non-basic variables. Then, the gradient Vh(x)is partitioned accordingly into Vh(x)=J(x) and V-h(x)-C(x). There are different ways of choosing which N - K elements in x will be non-basic. Each choice corresponds to a different tangent subspace at Xl along which the line search is carried out. For an initial feasible point x 0 that satisfies all the constraints within a tolerance ~2, a tolerance 81 for algorithmic ending, another tolerance ~3 for Newton convergence, a line search parameter cz0 and a reduction factor ~,, the implemented procedure involves these stages" 1. Search for a partitioning of the optimisation variables for the current approximation x t = (2t[~,) so that J(x,) is non-singular. Then, calculate V f ( x , ) for that partitioning. 2. If [IVy(x,)l < ~1, then E N D ' a solution has been found. Otherwise, build a search direction as follows" d = -(Vy(x, )) v ," cl = - J - ' (x,)C(x, )d , that is, d = (did) v
129 3. Carry out a line s e a r c h in that direction by calculating v, : x, + a d . If Ihk (v,)l < e , , V k , the step is too large. Newton's method is employed to bring the point back within the limits imposed by the constraints according to ~3. If Newton fails to converge, then do a = y a and go on with the next v~+~. If Ihk(v;)l
> f(vl).
If so, xt+1 = v~ and go back to step 1; otherwise, a = 7' a
and
go back to 3. 3. THE PARALLEL I M P L E M E N T A T I O N The method was parallelised on the basis of the sequential version described above. We first determined that the parts that would benefit most from the parallelisation were the evaluation of the constraints and their gradients as well as the line search procedure. For greater flexibility, the parallellisation was carried out at procedural level to enable the introduction of new different basic numerical routines easily and directly without affecting the parallellisation code. For many applications, the amount of variables is high and the constraints are numerous, frequently involving expensive procedures for individual function evaluations. Most process plant optimisation problems are typical examples of this kind because the equations that enable accurate evaluation of thermodynamic properties are usually complex. Therefore, we found it convenient to divide the constraint evaluation task among all the available processors. As to the line search, this phase implies frequent calls to Newton's method, which is rather costly. So, this routine was parallelised by distributing the evaluation of the constraints and their gradients. In other words, the evaluation of h~(vi) k= 1..... K , was distributed for a given a. In fact, this strategy is not unique. Another alternative would have been to make each processor deal with a different a, afterwards choosing the best one. Nevertheless, we do not recommend this strategy because it implies many superfluous evaluations to calculate intermediate points with unnecessary accuracy. As regards load balancing, there are two basic strategies: static and dynamic scheduling. The purely static approach plans task allocation in advance, thus requiring prior knowledge of processors' computing powers and task duration. Both these aspects are typically unknown at run-times. In contrast, dynamic scheduling avoids these disadvantages because task assignment is demand-driven. A dynamic policy involves, however, the passage of a higher number of messages. This becomes a drawback when the time required for individual task completion equals the time taken to pass a single message. This bottleneck is normally encountered for distributed computing environments with standard communication networks. The task distribution among the different processors was carried out so that each one dealt with equivalent loads. For optimisation problems running on homogeneous distributed environments where the evaluation of all the constraints and constraint gradients had similar time consumptions, a static policy was adopted. For C P U processors and K constraints to be evaluated, a range h~(vi) k = p ..... q was sent to each processor, where q - p + 1 < [ K / C P U ~ . In contrast, for heterogeneous processors and/or problems with unbalanced constraint evaluation times, which is most often the case in process plant optimisation, we found that a hybrid approach resulting from a convenient combination of static and dynamic scheduling was the best strategy. The dynamic approach was employed at the beginning in order to get typical computing times for each evaluation, following a static policy afterwards. Though the first run
130 involves more message-passage expenditures, it allows the proper definition of the best task distribution that minimises idle times in the following iterations. For task-distribution implementation, a client-server scheme was adopted. Two kinds of servers were introduced: Constr. and Jacob., which carry out the evaluations of the constraints and the constraint gradients respectively. The sequential code was slightly modified so that the evaluation requests were directed to these servers. While the program is running under a distributed environment, all nodes execute a copy of each server and a processor executes the main program. Simultaneous service of both servers is never required. After the main program asks for evaluations, it has to wait for the results, which implies that the processor is set free. Therefore, the node that contains the main program also executes one copy of each server in order to make efficient use of its resources. In short, for CPU processing nodes, there are CPU Constr. servers, CPUJacob. servers and one main program, that is the master of all processes. As to parallel implementation, we used the PVM message-passing library [5], which features heterogeneous platform support. The program was implemented in C and runs under several heterogeneous platforms, including UNIX (Solaris, OSF/1, LINUX, SCO) and Windows NT. 4. P E R F O R M A N C E ANALYSIS The sequential version was the basis for performance comparisons. A fundamental topic for sound parallel performance evaluation is the definition of a fair metric that determines the parallelisation gains with reasonable accuracy. Speed-up measurements, in particular, set up the relationship between the time required to solve a given problem using only one processor and the time employed to solve the same problem by means of a parallel implementation. Nevertheless, the existing speed-up formula was originally designed for parallel machines, not being applicable when the processing units have different computing powers. For systems of homogeneous multiprocessors, it is still possible to use the traditional metric, while for heterogeneous environments it cannot be applied because the time taken by a uniprocessor differs for each individual processor. If the sequential time corresponding to the most powerful workstation were used as numerator, the comparison would not be fair because the resulting speed-up would be an underestimation of the real value. Conversely, the choice of the slowest processor would lead to an overestimated speed-up. Therefore, so as to enable fair comparisons under heterogeneous environments, we defined a weighted speed-up (WSU) as follows: WSU
=
WST,
,
W S T i : STt * C P F I
i=1
where the subscript i refers to the i th processor, WST~ is the Weighted elapsed Sequential Time; TM, the elapsed Time for Multiprocessor computing; p, the number of processors; STy, the elapsed Sequential Time and CPFi, the Computing Power Factor. This definition agrees with the classic metric because when all the processors have the same computing power, their WSTi are equal and coincide with the elapsed time for a uniprocessor in the traditional formula. 5. CODE TESTING Homogeneous and heterogeneous environments were employed for parallel performance assessment. The homogeneous environment was made up of eight 200 MHz Pentium/LINUX
131 workstations, connected through a 10 Mb Ethernet network. In turn, the heterogeneous runs were carried out on a similar network made up of the following machines: a 400 MHz Pentium II/LINUX, a 133 MHz Pentium/LINUX and a 150 MHz DEC ALPHA/OSF/1. First of all, the sequential version of the code was tested with several examples from the literature [4,6,7]. Then, the parallel program was run for the same set of problems in order to test the accuracy of both programs. The same final results were obtained and the execution times for the parallel version were greater than the corresponding sequential ones, which was the expected result in view of the small size of these problems. Due to communication overhead, parallel solving under distributed environments is not convenient in these cases. Time savings are achieved as the systems of equations increase in size and complexity. Therefore, parallel performance analysis makes sense for big and/or complex problems.
6. TEST PROBLEMS 6.1. Scalable Geometric Case Study We designed the following optimisation model: T
Minimise ~-'~x~2
s.t 9 h,"
(Xo_Xloo+t)2
_~_X 2 t+l - X
2lO0+t
"]j -'- (~
t=l
A thousand FLOPS were added to each constraint and the objective function to increase the computing load artificially, thus enabling the simulation of costly working conditions.
6.2. Process Optimisation Case Study The objective of the case study is the search for optimal operating conditions for the reactor of an ammonia synthesis plant [8] that uses hydrogen and nitrogen feedstocks from a coal gasification plant. The ammonia plant consists of a medium-pressure synthesis loop with water absorption, followed by distillation for ammonia recovery. Hydrogen is recovered by means of membrane separation. The model for the synthesis loop section, which was used to test the parallel code, involves 8 compounds, 12 process units and 30 streams. It constitutes the set of constraints for the optimisation problem, consisting of about 150 equations and 180 variables. The key optimisation decision was the choice for the most convenient synthesis pressure and reactor conversion so that operating costs were minimised. In this respect, there is a trade-off between production and expenditure. Higher pressures lead to larger equilibrium conversions and lower recovery costs, while lower pressures imply cheaper compression stages. The test example was posed so that the objective function represented operating costs while the set of constraints corresponded to the steady-state rigorous model that represented plant behaviour. 7. PERFORMANCE ANALYSIS Significant run-time improvements were achieved through GRG parallelisation. In Table 1, we present the speed-ups and efficiencies for the geometric case study with 80 constraints (T-80), where nhosts stands for the number of Pentium 200 MHz Workstations employed in the parallel virtual machine. The sequential run-time for this example was 483 sec. For 2 and 4 processors, the performance is satisfactory. However, as the number of processors augments, the results gradually worsen due to the increasing communication overhead.
132 Table 1 Parallel Performance for Homogeneous Distributed Processing Nhosts Parallel Time Speed-up Efficiency
2 281 s 1.71 85%
4 175 s 2.76 69%
8 101 s 4.78 59%
Heterogeneous runs were also carried out for both examples with satisfactory results. The weighted sequential time for the geometric problem was 105 sec, while the parallel version required 42 sec, the corresponding weighted speed-up being 2.5 (83.3% efficiency). The load-balancing policy is problem dependent. The static approach for task distribution proved to be suitable for the geometric problem because it was clear from its formulation that all constraints would require similar computing efforts. This policy is also applicable for those mathematical models of process plants that only involve mass balances and/or highly simplified formulae to calculate thermodynamic properties such as densities and enthalpies. The hybrid approach became indispensable to achieve efficient results for industrial examples, like the ammonia synthesis plant, where the amount of computational effort for the calculation of the constraints differs significantly. Since the first iteration yielded representative computing times for each constraint, this information was used to distribute the tasks appropriately among the different processors. This model basically contains two kinds of equations: the mass balances were quick to evaluate, while the energy balances involved costly enthalpy calculations. The use of complex thermodynamic relationships could not be avoided in order to ensure reasonably accurate results because the feed constitutes a strongly non-linear mixture. So, enthalpies were calculated as functions of pressure, temperature and composition with SRK equation of state and a T-0 approach was used to obtain equilibrium constants. 8. CONCLUSIONS A parallel distributed GRG optimisation algorithm suitable for heterogeneous environments was designed and implemented. Careful consideration was given to design aspects like load balancing and performance metrics and significant time savings were achieved. REFERENCES
Brinch Hansen P., "Parallel Programming Paradigms", Prentice Hall, 1995. Anderson T.E., Culler D.E. & Patterson D., IEEE Micro, 15, 1, 54-64, 1995. Khokhar A., Prasanna V., Shaaban M.E. & Wang C., IEEE Computer 26, 18-27, 1993. Murtagh B. A. & Saunders M. A., Math. Prog. Study 16, 84-117, 1982. Geist A., Beguelin A., Dongarra J., Jiang W., Manchek R. & Sunderam V., "PVM: Parallel Virtual Machine. A Users Guide and Tutorial for Network Parallel Comp." MIT Press, 1994. 6. Reklaitis G.V., Ravindran A. & Ragsdell K.M., "Engineering Optimisation", J. Wiley, 1983. 7. Hock W. & Schittkowski K., "Test Examples for Nonlinear Programming Codes", SpringerVerlag, New York, 1981. 8. Bike S., "Design of an Ammonia Synthesis Plant", CACHE Case Study, Dept. Chem. Engng. Carnegie-Mellon University, 1985.
1. 2. 3. 4. 5.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
133
A Bi-Index Continuous Time MILP Model for Short-Term Scheduling of Single-Stage Multi-Product Batch Plants with Parallel Units Chi-Wai Hui and Avaneesh Gupta Chemical Engineering Department, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong. Email: [email protected], [email protected]
ABSTRACT This paper presents a mixed integer linear programming formulation for the short-term scheduling of single-stage multi-product batch plants with parallel non-identical production units. This scheduling problem is highly combinatorial in nature especially because of the sequence-dependent changeover constraints. To formulate this type of problem, tri-index discrete decision variables, i.e. (order, order, unit), are commonly applied to represent the order assignments. This approach requires a large number of discrete decision variables that consequently make the model very time consuming to solve. To overcome this problem, the proposed formulation instead applies bi-index discrete variables (order, order). This greatly reduces the overall number of discrete decision variables while still keeping the generality of the model. For handling large-scale problems, pre-ordering heuristics were imposed to further reduce the solution time. Examples with various numbers of units and orders illustrate the effectiveness of the formulation both with and without the pre-ordering constraints. 1. INTRODUCTION This paper presents a general formulation for short-term scheduling of single-stage multiproduct batch plants with non-identical parallel units. A review of this assignment problem can be found in Pinto et al. (1998). The proposed formulation applies three sets of bi-index discrete variables to handle sequence-dependent constraints both with and without imposing preordering heuristics. The main advantage of this formulation compared to other recent formulations is the significant reduction in the number of binary variables and consequently the shorter solution time, makes it more suitable for handling large problems.
2. PROBLEM DEFINITION A fixed number of production units are available to process all customer orders. Each order involves a single product that requires a single processing step, has a predetermined due date, and can only be processed in a subset of the units available. The production capacity of a unit depends on the order processed. The size of an order may be larger than the size of a batch, so several batch jobs may be required to satisfy an order. Batch jobs of the same order are processed consecutively by the same unit. A production unit processes only one batch job at a time. The batch time of an order is fixed and production unit dependent. During the transition between production orders, time is required to prepare the unit for the changeover. This preparation time is sequence or production unit dependent. The objective of the scheduling is to minimize the total tardiness or schedule makespan by assigning orders to units while satisfying all the above constraints.
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3. TRI-INDEX FORMULATION To formulate the problem of short-term scheduling of multi-product single-stage batch plants, traditional approaches have relied on the application of tri-index variables, such as Xiju, to represent the assignment of order j after order i to unit u. One example of such an approach is the MILP formulation proposed by Cerda et al. (1996). The comparisons in the following sections refer to this formulation.
4. BI-INDEX FORMULATION In this section, a new MILP mathematical model for the short-term scheduling of single-stage multi-product batch plants with parallel non-identical production units is presented. The proposed model is continuous time model of the time domain uses three sets of bi-index decision variables, X O, Wi,,, Siu, to represent an order i that has a succeeding order j, is assigned to unit u, and is the first order of unit u. Wiu can naturally become binary at the optimum solution and therefore can be treated as a set of continuous variables. (a) Assignment of consecutive orders in a unit This is the most important constraint in the formulation. The logic applied in formulating the constraint is: if order i and order j are consecutive orders, and order i is assigned to unit u, then order j is not processed in any other unit than unit u. This constraint assures the assignment of consecutive orders to the same unit.
Wiu + Z W j v + Xij + x j i - 2 < o,
v i ~ i,j ~ Psi,u ~ U i
(1)
vEU j l:=~U
(b) Each order has at the most one unique successor Each order has a unique successor provided it is not the last order to be processed in the unit. ] ~ X U_
(2)
jESU i
In the case that order i is the last order to be processed in the unit the inequality holds. (c) Each order has at the most one unique predecessor A unit processes exactly one order. This order is either the first order to be processed in a unit or it is preceded by a unique order. Z x ji + Z siu = 1, i~.PR i
,v,i ~ i
(3)
u~U i
In the case that order i is the first order in unit u, a variable Siu is equal to 1. When order i is not the first order, and order i is preceded by a unique order j, a variable Xji is equal to 1. (d) Each unit has a unique starting order It is assumed that each unit processes at least one order. This is accomplished by assigning a unique first order to each unit.
ZSiu
~
"
1,
Vu~ U
ie 11,
(e) Each order is processed Each order must be processed by one of the processing units.
(4)
135 ~Wiu =1,
(5)
Vie I
uEUi For each order i, the summation over the elements of Ui is equal to 1. (f) Relation between the starting times of the consecutive orders in a unit During the transition of production orders, time Cij is required for the preparation of the unit for the changeover.
(1-Xu)*M
+ T f >_Ti" -t- E(Wiu*Liu)-l-Cij,
V i e I, jE SU i
(6)
ueUi
In the case that order j follows order i, the starting time of order j must be greater than or equal to the summation of the starting time and the processing time of order i, and the changeover time from order i to order j. The overall processing time Liu of order i in unit u is obtained by multiplying the number of batches required, NBiu, with the batch processing time, TPiu. This constraint is enforced only when order j is followed by order i. If this is not the case, a large M is added to the starting time of order j to make sure that the constraint is feasible in all cases. (g) Starting time of the first order in the unit The starting time of the first order in a unit can be expressed by the following constraint:
Tis > Y__,Wiu*(Maa((RTU)u,(RTO)i)), ~v i
Vie I
(7)
In the case that order i is assigned as the first order to unit u, the order starting time is the latest of the release time of order i and the release time of unit u. In the case that the release time of both the unit and order is zero, the constraint is not required. (h) Relation between variables Wiu and Siu Variable W/u is always greater than or equal to Siu. In the case of consecutive assignments, variable Siu is equal to 0, and Wi,, is equal to 1. In the case that order i is the first order in unit u, the equality holds. The mathematical form of this constraint is written as follows.
(8)
Wiu >- Siu, Vi E l, ue U i This is a crucial constraint that forces the continuous variable W/u to be binary. (i) Tardiness in the completion of order Tardiness indicates the delay in the completion of the order at the due date.
Oi >- f TiS + E W i u * L i u } - T i
, ViE
(9)
The expression in the brackets on the right hand side of the constraint is the completion time of order i. (j) Schedule makespan The makespan of a schedule is denoted by the symbol H and is defined by the following constraint:
f
]
H>-~r/ + ZW~u*L,.~
l
u~Ui
J
-
Ma4(RTO),. Mi4(RTU~.]]. ueUi
Vi e I
(10)
136 The negative term on the right-hand side represents the latest release time of the order i and the earliest release time among the units available to order i. (k) Objective function The objective of the scheduling problem is to minimize the overall makespan or the total tardiness given by the following expressions:
Min ~ Di
(Obj. 1- Minimizing Tardiness)
i
Min H
(Obj. 2- Minimizing Makespan)
5. WORKING EXAMPLES Two examples (Examples 1 and 2) with various problem sizes demonstrate the capabilities and effectiveness of the bi-index model. These example problems are solved by applying both the tri-index and the bi-index models. The models are formulated in GAMS (Brooke et al., 1992) and solved by OSL (IBM, 1991) on a 300MHz Pentium PC.
Example 1: Data such as batch sizes, processing and changeover times are shown in Tables 1 and 2. Due dates and order sizes are given in Table 3. In the examples of this paper, both the order and unit release times are assumed to be zero. Examples la and lb involve 4 units, and 8 and 10 orders respectively. The results, shown in Table 4, indicate that the number of binary variables in the tri-index model is nearly 3 times as many as in the bi-index model in each example. The number of binary variables of the two models can be calculated using the following formulas: Tri-index model: Bi-index model:
(NI * NI- NI) * NU + NI*NU (NI * NI- NI) + NI*NU
In the case of minimizing tardiness, finding the global optimum is relatively easy for these small problems. In the case of examples la and lb, both the bi-index and the tri-index models reach the optimum of 0 tardiness within 5 seconds of computing time. The bi-index model finds the solution faster than the tri-index model in both these example problems. Finding the minimum makespan is much more difficult. In the case of example l a, the bi-index model reaches the optimum of 14.0 within 6 seconds of computing time. The tri-index model reaches the optimum not within 103 seconds. In the case of example lb, both models cannot reach the optimum within 100,000 iterations. Again, the bi-index model found a better solution.
Example 2: To shorten solution times, heuristics are often applied to reduce the number of decision variables by eliminating some unlikely combinations of order assignments. In this example, a simple heuristic is applied to force orders to be processed in a sequence of increasing due dates - i.e., an order with an earlier due date is processed first if assigned to the same unit. With this heuristic, some order assignments and sequences are eliminated and consequently a reduction in the number of decision variables and a shorter solution time is accomplished. By reducing the solution space with the heuristic, the optimality of the original problem cannot however be guaranteed. Examples 2a and 2b involve 4 units, with 8 and 10 orders respectively. The bi-
137 index and tri-index model results are compared in Table 5. By applying the pre-ordering heuristic, the number of binary variables of the two models is further reduced by 25% to 40%. Both models were solved by limiting the number of iterations to 100,000. In example 2a, both the models reach the same optimum of 15.2 in the case of minimizing the schedule makespan. This solution is not as good as the optimum of 14.0 found in Example la. In Example 2b, the solutions of minimum makespan are improved for both models by the application of the heuristic in comparison with example lb. Both models were however unable to reach the optimum within 100,000 iterations. The bi-index model required approximately one-third of the number of binary variables used by the tri-index model. For this reason, the performance of the bi-index model is better than the tri-index model. 6. CONCLUSIONS The bi-index model requires considerable fewer binary variables than the traditional tri-index model and thus requires a shorter solution time. For large problems, a pre-ordering heuristic was used to further reduce the number of binary variables, resulting in better solutions within a shorter time. However, the optimum cannot be guaranteed, since the pre-ordering heuristic might eliminate some order sequences from the original problem that might be required for the optimum. NOMENCLATURE Indices i, j, k = Order u, v = Unit
Problem Sets I = Orders to be processed Iu= Orders to be processed in u U = Units available Ui = Units available to process i U/j = Units available to process i with successor j PRi = Feasible predecessors of i SUi = Feasible successors of i PSi = Orders processed either just before or immediately after i Parameters NI = Number of orders N U = Number of units Qi = Size of i (RTO)i = Release time of i zi = Due date of i (RTU)u = Release time of u NBiu = Number of batch orders of i processed in u TPiu - Processing time for a batch job of i in u Liu = Total processing time of i in u Ciju = Changeover time for the transfer from i to j in u
Biu = Batch size of i processed in u Binary Variables Xij = Assignment o f j after i Win = Assignment o f j to u Sin = First assignment of i to u Continuous Variables T i s = Starting time of i in the unit Tif= Finish time of i in the unit Di = Delay in the processing of i H = Makespan
REFERENCE: Brooke, A.; Kendrick, D.; Meeraus, A. GAMS - A User's Guide (Release 2.25); The Scientific Press: San Francisco, CA, 1992. Cerda,J.; Henning, P., Grossmann, I.E. A Mixed Integer Linear Programming Model for Short-Term Scheduling of Single-Stage Multiproduct Batch Plants with Parallel Lines. Ind. Eng. Chem. Res. 1996, 36, 1695-1707. IBM. OSL (Optimization Subroutine Library) Guide and Reference (Release 2), Kingston, NY, 1991. Pinto, J.M., Grossmann, I.E. A Continuous Assignment and Sequencing Models for the Scheduling of Process Systems. Annals of Operations Research, 81, 1998, 433-466.
138
I1 12 13 14 15 16 17 18 19 I10
U1 U2 100/1.7 200/1.2 150/1.2 180/2.1 140/1.25 210/1.3 130/1.6 120/1.7 100/1.8 90/1.4 280/2.4 210/1.8 240/1.9 300/1.5 130/2 140/2.1 200/1.6 210/1 250/2.6 270/1.9
U3 300/2.1 210/0.9 170/1.1 140/2.2 110/0.95 120/1.2 390/1.05 110/2.1 190/2.1 240/1.6
Model
U4 100/1.8 200/0.95 160/1.02 150/2.01 130/0.85 260/1.1 290/1.65 120/2.1 220/1.23 260/2.42
Example 1a (N-index) 88,49,237 NI=8, NU=4 88,41,236 (tri index) NI=8, NU=4 Example 1b (bi-index) NI=10, NU=5
T a b l e 1" B a t c h size / Processing time I1 I1 .00 12 1.8 13 1.0 14 1.2
12 1.0 .00 .15 .02
13 .15 1.10 .00 .10
14 1.10 1.30 1.20 .00
15 .10 .20 .30 .30 16 1.4 .80 .30 .70 17 1.2 1.8 1.30 .90 18 1.3 19 2.1 I10 1.5 T a b l e 2: Order I1 12 13 14 15 16 17 18 19 I10
15 2.00 1.40 1.50 .05
16 .65 .90 2.1 1.6
17 .30 .20 .30 1.20
18 1.2 1.2 1.8 2.0
19 .85 .4 1.6 1.2
I10 .40 .30 .20 .50
.00 .70 .90 .60 1.0 .90 2.00 .00 .90 1.2 1.2 1.6 .85 .80 .00 .45 1.2 1.3
1.4 1.50 1.40 1.20 1.3 1.65 .00 1.3 .80 2.0 1.25 1.35 1.45 .80 1.60 .80 .0 .65 1.2 .60 .75 .50 .40 .90 .60 .7 .00 C h a n g e o v e r time (Cij) Qi 550 850 700 900 500 1050 950 850 450 650
, ,
T a b l e 3: O r d e r size and D u e date
T
0
0.5
587
M
14.0
6.6
7691
256,25,133
T
0
3.5
4167
256,17,132
M
130,61,355
T
130,51,354
M
1 4 . 3 103 100000 0
1.8
1338
18.75 273 100000
(tri-index) 400,31,185 T 0 7.3 7439 NI=10, NU=5 400,21,184 M 1 9 . 3 220 100000 Table 4: Results o f E x a m p l e 1 Model
Binary vars., Obj. Opt. CPU Cont. vars., Rows
Itera.
Sol.
time (sec)
T
0
0.4
416
M
15.2
6.7
8793
144,25,104
T
0
0.6
528
144,17,103
M
15.2
19.6
19887
88,61,313
T
0
0.6
199
88,51,312
M
Example2a (N-index) 60,49,208 NI=8, NU=4 60,41,207 (tri index) NI=8, NU=4 Example 2b (N-index) NI=10, NU=5
'l;i
10 22 25 20 28 30 17 23 30 21
Binary vars., Obj. Opt. CPU Itera. Cont. vars., Sol. time Rows (sec)
1 8 . 2 272
(tri-index) 232,31,143 T 0 0.5 NI=10, NU=5 232,21,142 M 1 8 . 8 102 Table 5: Results o f E x a m p l e 2
100000 220 100000
!
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rightsreserved.
139
Importance of parameter selection in classification systems using neural networks Ordieres j.(a), Ortega F.(b) (a) Project Engineering Area. Mechanical Department. University of la Rioja. 26004 Logrofio. Spain (b) Project Engineering Area. University of Oviedo. 33004 Oviedo. Spain 1. ABSTRACT In this contribution we will try to set some "general" guidelines to select the criteria (and best suited parameters) based on many test raised on an industrial example, which is a heating regulation taking place without feedback in a power plant owned by a medium size electrical company located in the north of Spain. Also we will tray to analyze from an experimental point of view the ability of NN technology for modeling the process considering a low percentage of total patterns for training, just as a measure of its tolerance to the noise. 2. I N T R O D U C T I O N It is well known that the multilayer feedforward network (MFN) is the most widely used neural network model for pattern classification applications, mainly in applications related industrial tuning of processes[ 1],[2],[3]. This is because the topology of the MFN allows it to generate internal representations tailored to classify the input regions that may be either disjointed or intersecting [4], [5]. The hidden layer nodes in the MFN can form hyperplanes to partition the input space into various regions and the output nodes can select and combine the regions that belong to the same class. Backpropagation (BP) [6] and its variants such as QuickPropagation, Backpropagation with Momentum, Resilient Propagation[7], etc. Are the most widely used training algorithm for the MFN networks. Recently researchers have begun to examine the use of Radial Basis Functions (RBF) [8] for solving function approach and pattern classification problems. RBF are also well suited for these problems due to their simple topological structure and their ability to reveal how learning proceeds in an explicit manner. In the classical approach to RBF network implementation, the basis functions are usually chosen as Gaussian and the number of hidden units is fixed a priori based on some properties of the input data. The weights connecting the hidden and output units are estimated by linear recursive mean square (LMS) or recursive least square (RLS). However, in all cases and starting in the training process, it is necessary to make choices about the kind of neural network to use, the best topology, the training strategy most suitable in terms of errors, the criteria for halt the training and some other parameters. It is true that there are some more or less theoretical analysis about the convergence properties of some topologies and training strategies[9][11], but, unfortunately the estimations carried out from those strategies are sometimes under the experimental requirements. Obviously in an ideal environment with lots of patterns we could think that, following theoretical results[8][12][14][16], we can supply a NN with a high number of neurons and we need only train it. Unfortunately even when the theory stay that as possible, there are other problems
140
(overtraining, complexity, pattern's spatial distribution and so on that make it no useful. From here we try "to see" how is the answer of several topologies with a different number of neurons and a relatively low number of patterns. 3. M E T H O D O L O G Y The methodology consists in normalize the data sets, and to define the boundary conditions (topologies, training strategies, parameters, etc.) and carry out several tests looking for empirical validations and cross correlation making guarantee not overtraining occur. In our real example we have 9000 patterns with 7 inputs and 1 output variable. We use the NN shell provide by the University of Stutgart[ 10] and in order to avoid the dependence from patterns we choose them randomly. From the point of view of topology we will analyze one hidden layer networks with 7,14 and 21 neurons into and two hidden layer networks with 7x7, 14x7, 7x 14 and 14x 14 neurons into. There are results [13] showing that networks with one hidden layer can approach any continue function, but there are some problems making interesting to use two hidden layer architectures [ 14][ 15][ 12]. In this particular aspect we will analyze the error surface considering architectures, size of pattern sets and training technique. The result shown here correspond to more than 20000 simulations carried out and with parameters stored on a database for its analysis. 4. RESULTS We can find in the figure 1 how is the evolution of validation error considering architecture against learning parameter when a backpropagation technique is selected and considering several training cycles. We see how the problem is nonlinear and how the topology with 14 hidden neurons (H14) works better than the topology with H7. Also we can see how, due to the low number of patterns used for training the topology H21 carry out significant low results, this can be read as if we have not many patterns we could get a mistake by suggesting big topologies. Also we note that the use of two hidden layer requires doubling patterns and cycling to get similar results (H7x7 against H14) more or less. We find a non symmetric comportment by making H14x7 and H7x 14. In this case the conclusion is close to the idea that we get better change when we have more neurons in the first hidden layer. Fig. 1.- Maximum error against topologies and learning parameters, considering several cycles of training. Probably it could be possible to establish a link with the weight's actualization law. In any case in this approach we get that the number of patterns is really significant and also we can conclude that the symmetric topology catch better than the non symmetric ones the model of the real process (H 14x 14 against H7x 14 or H 14x7).
141 In the previous study we look the validation error, because the training error is lower, but it could be interesting to analyze the importance of choice one training strategy among others. 0.12-
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F i g . 3.- Training and validation Error when the strategy BPM is used on a network ( H 1 4 x 1 4 )
F i g . 2 .- Training and validation Error when the strategy BP
is used on a network ( H 1 4 x 1 4 )
In the literature we take several results showing that the trainig process can be reduced (in time) by strategies such as Backpropagation with momentum (BPM) instead the classical backpropagation but, in our analysis we found that the validation error is bigger using BPM than when we use BP, even when the time exhibits the announced comportment, see figure 4. CYCLES BPM BP 1000 22 0 2000 4000 8000 12000
103
0
315
0
575 707
2 4
F i g . 4.- Number of cases where the training error was reduced under 0,0001 before to get the limit of cycles.
F i g . 6 .- Training error when the strategy BPM (mom=0) is
used on a network (H14)
F i g . 6 .- Training error when the strategy BPM ( m o m = 0 , 0 5 )
is used on a network ( H 1 4 )
An interesting problem is the estimation of the whole set of parameters involved in the training process and especially when we use BPM. In order to get some empiric knowledge about that before try to model it mathematically, we plot the combination of parameters making good estimations. The idea is to carry out some kind of regression helping us to choice good parameters.
142
BackPropagation
with Momentum
N ~ training patterns" ".....
1000
N ~-validation patterns:
>8000
N ~ eproch:
12000
P3
0 0:05 0:10 0:15 0:20 0:25
PI(0)
0,578 0,524 0,438 0,395 0,336 0,247
slope
-13,721 -17,204 -18,184 -16,188 -14,134 -7,748
We can observe that as far as the parameter P3 of BPM increases more selective is the technique, so, a better estimation is required, i.e. less choices have quickly convergent networks. Also we see the monotonic character of the slope for the maximum likelihood line, i.e. bigger values for P3 have associated lower slopes.
Finally we would like to finish our analysis taking into account other strategies for training probably less used than BP but we found interesting from the point of view of their approach to the validation error. The literature show us that strategies such as Back Propagation Batch (BPB) (For "e rror_validacion" < 0,0015) are unsuitable because they make training process extremely slow. These kind of theory are quickly verified but we observe that, from the point of view of stability they are the best, probably because here before change the weights vector we need to hear all the patterns and BP change that vector each time it process one pattern.
143
144 We see now how the error structure is not maintained from training to validation. It must be taken into account that most of previous results are remarked by the short number of patterns used in training, in order to get a better simulation of real environments. REFERENCES [ 1] Yang J. "Classification of acoustic emission signals via Hebbian feature extraction". Int. Joint Conf. Neural Networks, Seattle, WA, vol. 1, 1991, pp. 113-118. [2] Haykin S. And Deng. C. "Classification of radar clutter using neural networks". IEEE Trans. Neural Networks, vol 2 pp 589-600 1991 [3] Ordieres J.; Men6ndez C.; Ortega F. "Predicci6n del ancho en productos planos laminados." Revista de Minas. n~ 12. pp 35-41.2 ~ Semestre, 1995. Oviedo [4] Lippmann R., "An introduction to computing with neural nets", IEEE Acoust. Speech, Signal Processing Mag., vol 4, pp 4-22 Apr 1987. [5] Ordieres J.; Men6ndez C.; Ortega F. - "Comparison between different neural network topologies for non linear classification". Revista Informaci6n Tecnol6gica. La Serena. Chile. Vol 7(4). Pp. 109-115. 1996 [6] Fahlman S.E., "An empirical study of learning speed in backpropagation networks". School of Comput Sci. Carnegie Mellon Univ., Pittsburg, PA, Tech Report CMU-CS-88162, 1988 [7] Riedmiller M., Brau H., "A direct adaptative method for faster backpropagation learning: The rprop algorithm", in Proc IEEE Int Conf Neural Networks, San Francisco CA, Apr 1993. [8] Broomhead D, Lowe D. "Multivariable functional interpolation and adaptative networks". In Conf. Rec. 27 th Asilomar Conf. Signals, Syst. Comput. Pacific Grove, CA 1993 pp 401-405 [9] Falhman, S.E. (1.998) Faster learning variations on backpropagation: an empirical study, in T.J. Sejnowski, G.E. Hinton and D.S. Touretzky (Eds.) Connectionist Model Summer School, Morgan Kaufmann, San Mateo, CA. [ 10] Stuttgart Neural Network Simulator User Guide. ftp://ftp.informatik.uni-stuttgart.de [ 11 ] Menendez C., Ordieres J. and Ortega F. "Importance of information pre-processing in the improvement of neural network results" Expert Systems and Neural Networks. May 1996, vol 13, n~ 2, pp 95-103; [ 12] Schiffmann W., Joost M., Werner R., "Optimization of the backpropagation algorithm for training multilayer perceptrons". University of Koblenz. Institute of phisycs. Rheinau 3-4. W-5400 Koblenz [13] Sontag E.D., "Feedback stabilization using two hideen layer nets". Preceedings, American Automatic Control Conference. Boston 1991 pp 815-820 [ 14] Baum E.B. "On the capabilities of multilayer perceptrons". J of Complexity 4. 1988 pp 193-215. [ 15] Chester D. "Why two hidden layers are better than one". Proceedings, Inter. Join Conference on Neural Networks. Washington D.C. 1990 pp 1.265 - 1.268. IEEE Publications, 1990 [ 16] Cybenko G. "Approximation by superposition of a sigmoidal function". Mathematics of Control, Signals and Systems, 2, 303-314.
EuropeanSymposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V.All rightsreserved.
145
A two dimensional conceptual model to support data integration in process plant operations A.-D. Yang ~, H.-S. Lib and M.-L. Lub aDepartment of Chemical Engineering, Louisiana State University, Baton Rouge, Louisiana 70803-7303, USA *+ bAigis systems, Inc., Bloomfield, New Jersey 07003, USA Data integration is very critical in developing an integrated software system for real time process operations. To support this, this paper presents a conceptual data model that considers the data used in process operation activities as the combination of two dimensions: "Domain" and "Variability". The former classifies objects of the process operation domain such as process plant, its components, and operation activities. The latter focuses on describing the variability characteristics of those objects. The conceptual model renders flexibility in allowing different ways of combining two dimensions to meet application requirements. The model provides a conceptual basis for developing detailed data models or class libraries. 1. INTRODUCTION In recent years it has been recognized that process plant operations (PPO) can benefit most from software technologies by integrating a set of tools to support most of process plant operation tasks E~1.Such integration, however, has to be accomplished on the basis of information or data integration that enables data sharing and exchange between these tools E~jt21.Without exception, a common data model is indispensable for realization of data integration. The data model provides unified data structures and clearly represented relationships between different data items; applications intended to share data with others can then generate and access data following the same data model. Obviously, the quality of the data model is critical to the database configuration and the database performance which is especially important for process plant operations. In the domain of data modeling for process plants, significant effort has been made within ISO 10303, which is also well known as STEP (STandards for Exchange of Product data) p]. Currently there are several application protocols (AP's) of STEP being developed for process engineering, examples of which include AP221 [4], AP227 [51, and AP231 [61. Besides the STEP community, an information modeling effort for design process management [7] and the development of modeling language VEDA [g] are also reported. However, major attempts of above work are to support process design activities but not for Present address: Lehrstuhl fiir Prozel]technik, RWTH-Aachen,Aachen, Germany + Financial support from Aigis Systems, Inc. (ASI) is gratefully acknowledged. The authors would like to thank Tetsuya Wada for his strong support in the ISO process plant operation PWI project, Taesuk Chang and SoyoungChao for their participating the data model applicationand validationproject.
146 process operations. Compared to process design, PPO has the following special requirements in data modeling: (1) For data created at a point in time, the data model should represent time-related information. (2) The data model should be able to capture a set of data created during a period of time. (3) Information related to the process operation tasks should be taken into account in the data model together with the information about the process itself. (4) The data model should support efficient data accessing to support real time operation tasks. So far there have only been very limited modeling efforts for PPO(e.g. [9][ 10]) and a data model addressing all aspects identified above has not yet been reported. In this paper, a conceptual data model is developed attempting to meet the requirements of PPO domain. The model divides the dimensions and presents the high level classification of the total set of PPO data. It can then be used to develop detailed reference data library and guide the development of data stores. In the rest of the paper, the modeling dimensions will be discussed first, then for each dimension detailed classification will be described. Finally, the two dimensions will be combined to form a complete data model. 2. MODELING DIMENSIONS In PPO domain, data items can be classified into: Constant - Continuously changing - Changing at certain times According to the different nature of data, different requirements on data accessing need to be met. For example, accessing to the frequently changing data must be more efficient than that to constant or steady data so that applications manipulating various kinds of data can be supported appropriately. To achieve this, all data involved in PPO are organized in two dimensions: - Variability dimension Domain dimension The "Variability dimension" represents the "time characteristic" aspect of all the process plant data. Since all the time related aspects are systematically covered by the "Variability dimension", the "Domain dimension" then focuses on data modeling in the perspective of the domain nature. The following sections discuss some details of the two dimensions after which usable model elements (classes) can be built using the classes defined in the two dimensions. 3. VARIABILITY DIMENSION This dimension comprises two categories: "Time measurement" and "Characteristics of change", as described in the following two sections. 3.1.
Time
Measurement
In general, time measurement includes two aspects: point in time and duration of time. According to this, in the model two classes, namely "Time_point" and "Time_duration", are defined. "Time_point" is composed of "Date" and "Time". The former is specified by
147 number of year, month in year, and day in month; the latter by hour in day, minute in hour, and second in minute. "Time__duration" describes an temporal interval in terms of the amounts of year, month, day, hour, minute and second. Further more, the value of a "Time__reference" is made meaningful by referring to a specific reference (e.g. US Eastern time) which is modeled by "Time_reference". A "Time_reference" has a name to be referred by "Time_point", and is characterized by an offset to another "Time_reference". The Greenwich Mean Time should be used as the base line for all instances of Time__reference", which itself is an instance of "Time_reference" by referring to itself with a zero offset. 3.2. Characteristics of change
In this category, data are classified by the frequency of change. Generally, any data item should have one of the following characteristics: - Existing instantly without a life duration, e.g. an event. - Existing constantly without any change during the whole lifetime, e.g. a thermodynamic constant. Possible to change, but usually in a very low frequency, e.g. equipment design parameters. - Possible to change, usually in a high frequency, e.g. temperature and pressure of a material stream. Corresponding to these, all objects are classified into "Instant__object", "Constant_object" and "Variable_object"; the last one is further classified as "Steady__object" and "Dynamic_object". With the "Variability dimension", time-related information can be expressed explicitly. More importantly, the classification based on variability potentially supports the data storage distribution based on the frequency of change. As the result, data with similar frequency of change can for example be stored in the same database file for efficient data access. Figure 1 shows the UML class diagram of the "Variability dimension". specifies
Time rneaurement I
I
I
Time_duration
Time__point
efers to --~me__re~
Time
Variability_classification
Date
Constant_object
~ffset
I
Constant_object
Variable__object base I
Dynamic_object
Static__object
Figure 1. UML class diagram of"Variability" dimension 4. D O M A I N D I M E N S I O N
While the "Variability dimension" represents the generic "variability view" of the process plant data, the "Domain dimension" classifies the data according to the domain
148 nature. This dimension is further divided into two categories: "Domain_object" and "Domain_object_descriptor", each of which is also the root class of that category. 4.1. Domain_object The objects in this category build up the backbone of PPO information system, including plant equipment, process materials, physical and chemical phenomena, and data related to control activities (in general sense) responding to those physical and chemical phenomena. These objects are classified as "Material", "Equipment", "Facility", "Process", "Control_loop", "Activity", "Performer" and "Event". A detailed description about these classes can be found in [ 11]. It is worth mentioning that, different from the others, the last three classes are for describing process operation activities, which is of great importance for operation managements. 4.2. Domain_object_descriptor Domain_object_descriptor is used to describe objects under "Domain_object" category and logically never exists independently. It can be classified into "Property", "State" and "Relation". "Property" is used to describe each characteristic of an object in "Domain_object". Temperature, pressure, and flow-rate are examples of "Property". "State" describes the state or mode an object is in. On/off, normal/abnormal, startup/steadysunning/shutdown are examples of State. "Relation" keeps the information that is related to more than one objects in "Domain_object" category. It describes the relationships between different objects. "Relation" is further classified into "Topological_relation" and "Logical_relation". Figure 2 shows the UML class diagram of the "Domain dimension". is_described_by
Domain_object
I
Process
Performer
Domain_objectdescriptor
I
I
State
Property
Equipment
Relation i
Topological_relation Material
I
Logical_relation
Control_loop Figure 2. UML class diagram of the "Domain" dimension
5. BUILDING USABLE CLASSES Each of the above two dimensions only models a partial view of the PPO data. To finally generate usable classes possessing "complete" information, class definitions in the two dimensions have to be combined together. There are two approaches to combination having been considered.
149
5.1. Defining "Attribute classes" "Domain_object_descriptor"
by
inheritance
from
"Variability"
and
Following this approach, complete usable classes could be defined in two steps: - Defining variability-based "Attribute classes" by deriving from "Variability" classes and "Domain _objecydescriptor" classes. - Defining domain object classes by adding attributes, the types of which are classes defined in the first step A domain object class defined in this way usually contains attributes in different variability type. Therefore, it is possible to decompose it into several smaller ones, each of which only contains attributes in the same variability type. As mentioned earlier in this paper, this enables the database system to be configured in the way that objects with similar changing frequency are stored together so as to access the data efficiently.
5.2. Combining "Variability" and "Domain_object" Using this approach, variability based domain object classes (not attribute classes) can be defined by deriving from both "Variability" and "Domain_object", which have two types of attributes: - Attributes inherited from "Variability" class holding information about time stamp, frequency of change, valid period, etc. - Other attributes, with "Domain_objecydescriptor" type, describing properties, states of the "Domain_object" and relations between different objects. This means that all the domain_objecydesriptor attributes of a class share the same variability characteristics, e.g. same time stamp, same changing frequency, same valid period, etc. The benefit of combining in this way can be seen, for example, by considering the circumstance of real time process data acquisition from DCS system where most data such as temperatures, pressures and liquid levels are collected simultaneously and therefore have the same variability characteristics. It is necessary to mention that to use this approach attributes of a sub-view of a "Domain_object" should have the same or at least close variability characteristics. Compared to this, the first approach could be used in any circumstance, but with the disadvantage that in some circumstances variability-related information have to be repeatedly stored. This suggests that an application data model be developed by choosing suitable approach of combining the two dimensions to meet the special requirement of the application.
5.3. Defining Historical Data So far, the model discussed considers only the "snapshot" of an object in PPO at certain point in time. However, many applications request historical data or in general the data accumulated in a period of time. To meet such a requirement, two classes, "Data_point" and "Historical_record", have been defined to represent historical data. A "Historical_record" object sequentially contains a list of "data point" describing the information of a "changeable" data item in a specific snapshot. A "Data_point" object could be either an attribute of a domain object or a domain object as a whole. In the first case, a history record can be generated for each attribute of a domain object only if the type of this attribute inherits "Variable_object" (not constant/instant), no matter what "Domain_objecydescriptor" type (i.e. Property, State or Relation) the attribute has. In the second case, a "Historical_record" object will contain a list of sub-view of a domain object
150 created by the combination of "Domain_object" and "Variability"(e.g. instances of "Dynamic_process_object). 6. CONCLUDING REMARKS
Conceptual data models are the foundation for data integration. As part of result of an ISO SC4 PWI initiative for process plant operation and maintenance t~lj, a conceptual data model has been developed by modeling process plant data in "Variability dimension" and "Domain dimension" and then combining them together. This model considers information for both process plants and operation activities, and expresses time-related information explicitly not only for data generated in a snap-shot but also histories. Attention has been paid to performance which is crucial to real time operation. The conceptual model provides a basis for more detailed data model development in the domain of PPO. For instance, part of the model has been implemented and verified in a prototype integrated information system for operation tasks in the simulated Tennessee Eastman process tl21. Since the principle of developing this conceptual model is quite genetic, the result of this paper can also be used in other similar domains. REFERENCES
1. V. Venkatasubramanian and G.M. Stanley (1994). Integration of process monitoring, diagnosis, and control: issue and emerging trends. Proc. FOCAPO'94, Austin, TX, 1994. 2. M.M. Valleur (1998). System integration improves refining. Hydrocarbon processing, May, 73-77 3. ISO TC 184/SC4. STEP Part 1: Overview and fundamental principles. 4. ISO TC184/SC4. (1997). STEP Part 221 Functional data and their schematic representation for process plant. 5. ISO TC184/SC4 (1997). STEP part 231 Process engineering data: process design and process specifications of major equipment. 6. ISO TC184/SC4 (1997). STEP part 227 Plant spatial configuration. 7. J.L. Roberson, E. Subrahmanian, M.E. Thomas and A.W. Westerberg (1994). Management of the Design Process: The Impact of Information Modelling. Proc. FOCAPD'94, Snowmass, Colorado, 1994 8. W. Marquardt, L. von Wedel, B. Bayer (1999). Perspective on lifecycle process modeling. Proc. FOCAPD'99, Breckenridge, Connecticut, 1999 9. M.L. Book and A. Sharma (1994). Information models for batch and real time chemical process data. Proc. FOCAPD'94, Snowmass, Colorado, 1994 10. G.S. Mannarino and H.P. Leone (1998). A Task-Resource Based Framework for Process Operations Modeling. Proc. FOCAPO'98, Snowbird, Utah, 1994 11. The project team(1998). ISO SC4 process plant operation and maintenance PWI project report, presented at ISO SC4 meeting, Beijing, October 1998 12. A.D. Yang, et al (1999). A prototype integrated information system for process plant operations". Proc. 8th Congress of Asian Pacific Confederation of Chemical E.ngineering, Seoul, 1999
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
151
Feedforward Control Based on Online Concentration Calculation of a Heat- and Mass-Integrated Distillation System K. L6we a and G. Wozny a aInstitute of Process and Plant Technology, Technical University Berlin, Sekr. KF 1, Stra6e des 17. Juni 135, 10623 Berlin, Germany Reduction of energy consumption as well as the better use of resources has been an important aspect in the design and operation of chemical processes. For most separations fully thermally coupled distillation columns are thermodynamically more efficient than conventional arrangements. The double column system is superior to a single column in energy saving but disadvantageous in dynamic operability. The aim of this study is the development and experimental verification of a feedforward control scheme for coupled distillation systems. This paper presents computational and experimental results for large changes in feedflowrate for a heat- and mass-integrated distillation system. 1. I N T R O D U C T I O N The use of integrated distillation systems can lead to a significant reduction of energy consumption in comparison with conventional distillation columns. With heat integrated distillation column systems energy savings up to 45 % can be achieved [1]. In the complex configuration of a heat- and mass integrated distillation system the condensing vapour from the top of the high pressure column (HPC) is used to heat the reboiler of the low pressure column (LPC). In spite of the superiority of double- effect distillation systems to a single column concerning energy saving these systems are not often used in the industrial practice, because of the disadvantages in dynamic operability. Interactions and time delays will lead to a more complicated controllability, so a double column system needs a higher expenditure in the design and the control systems than a single column. In the operation of distillation columns, the major disturbances occur in feed flowrate and composition. Through feedback control one is generally able to handle these disturbances, but control action cannot begin, before the effects of the disturbance are measured in the output variables. As integrated distillation column systems are characterized by large time constants, the concentration profile will move further away from the desired operation point in the meantime. Feedforward control, on the other hand, offers the theoretical possibility of perfect disturbance rejection. If the disturbances can be measured, physically founded values of manipulated variables can be computed from a suitable model. Depending on the accuracy of the chosen feedforward control model, these manipulated variables may even take into account the dynamics of the column and reduce control loop interactions. In this examination a feedforward control scheme for heat- and mass-integrated distillation columns is developed for large changes in feed flowrate.
152
2. PILOT PLANT
Experiments are carried out on a fully automated methanol/water distillation column system of pilot plant scale. In this fully thermally coupled distillation system .external heat is introduced only in the reboiler of the HPC. For material integration the so called LS/R (lightsplit / heat integration reverse) configuration is selected. Thereby the bottom product of the low pressure column is fed to the high pressure column. Boilup is provided by an electrically heated thermosiphon reboiler with a maximum duty of 30 kW. In addition to the heatintegration condenser/reboiler the HPC is fitted out with an auxiliary condenser and the LPC with an auxiliary reboiler. A cooling water condenser is used for the LPC. 20 (LPC) and 28 (HPC) bubble cap trays with central downcomers are installed. The column system is equipped with an extensive technique of measurement. All measurements are digitalized on a decentralized control system TELEPERM M, Siemens. The DCS is coupled to the local area network consisting of several VAX- and UNIX workstation as well as PC's to save experimental data. The pressure of the HPC is not controlled. The level in the heatintegrating condenser is controlled to a fixed value, whereby the corresponding pressure results. The column system is operated in the LV-configuration. Liquid levels are controlled by bottom and distillate flowrates. Product compositions are controlled by the two reflux flowrates ( Lnp, LLp ) and the reboiler duty ( QHp ). 3. S I M U L A T I O N MODEL
A rigorous model is used to simulate the double effect column system. The model is implemented within a SpeedUp environment. The model equations of each tray consist of three parts: mass and energy balances, phase equilibrium and tray hydraulics. The dynamic balances of mass, component and energy lead to a set of differential equations. The vapourliquid equilibrium is described by the Wilson- and Antoine- equations. Tray hydraulics are modelled with the Francis-weir-formula to correlate the tray holdup. Tray pressure drop is calculated by the gas and liquid fluid dynamics based on the tray geometric sizes. Tray efficiencies are estimated with measured data from experiments. 4. C O N C E N T R A T I O N C O N T R O L As the concentration is usually very difficult and expensive to be measured on-line, temperature measurements are often used for control in the industrial practice. Because small deviations in pressure are unavoidable for heat-integrated distillation systems, the temperatures cannot be used as control variables in the high pressure column. Therefore the concentrations on the topmost tray and in the reboiler must be known. As these variables cannot be measured online, a method for online calculation of the concentrations is developed. For this calculation the current measured values of the pressure and the temperatures are handed over to a FORTRAN routine, in which the current concentrations are calculated from the knowledge of the vapour-liquid equilibrium. But considering that changes in concentration are not as fast as changes in pressure, the calculated concentrations are filtered by PTl-elements. The time constants are determined in simple experiments. The concentration calculation is validated with experimental data over a wide range.
153
Fig. 1 shows the calculated (solid line) and measured (points) bottom product purity, the results for the validation of the distillate composition are given in Fig. 2. The investigated concentration range extends from = 95 mol% to = 99.5 mol%. A good correspondence
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between the calculated and the measured concentrations concerning not only accuracy but also in dynamic behaviour is recognizable. These calculated concentrations are used as control variables for conventional PID- controllers. In addition to this a feedforward control scheme for load variations in feedflowrate is developed. 5. F E E D F O R W A R D C O N T R O L OF COUPLED DISTILLATION SYSTEMS Steady state design of heat-integrated distillation columns has been explored by different authors. Studies about the dynamic and control of coupled distillation systems are rarely to find. Ding et al. [2] compared the dynamic controllability of low- and high purity separations. For the high purity separation, the LOF (light-out-firsO configuration gives much better load rejection as for complex heat integrated configuration. Mizsey et al. [3] investigated the controllability of heat integrated distillation schemes and the fully thermally coupled distillation column (FTCDC). It can be shown that the control performance of the heat integrated distillation system is superior to the FTCDC. Luyben et al. [4] applied a multiobjective, mixed-integer non-linear programming algorithm for the synthesis and design of a double effect system, but used steady state models and open-loop controllability measures. Bansal et al. [5] investigated the interactions of design and control of double-effect distillation systems. In this paper the clear economic advantage of double-effect distillation systems comparing to a single column is shown. L6we et al. [6] presented investigations about the startup-behaviour of a heat- and mass integrated distillation system. Experimental investigations about the dynamic behaviour and control of heat- and mass integrated distillation systems have not been published in literature. Even in single columns large time constants can appear, which will be increased in column systems. The advantage of a feedforward control is to come into action before the disturbance
154 effects a deviation in the product specification. For this reason, especially for coupled systems feedforward control can prove to be lucrative. Wozny et al. [7] realized an observer with a state controller for an industrial methanol/water column and tested it for a disturbance step in feed concentration and flowrate. Most feedforward control models described in literature are based on linearized short-cut models with simplifications [8,9]. Rix et al. [10] developed a feedforward control scheme for a binary high purity single distillation column. The experimental results for PID- and feedforward control prove, that feedforward control can improve the operation performance of single columns significantly. The developed feedforward control scheme, developed exemplary for the reverse Unmeasured connection, is outlined in Fig. 3. For Disturbance a better practical application, instead Dynamic Filter of running a dynamic column model u fl~, I [ online, the new steady-state manipulated variables are computed Controller ~ Pla I Y" from characteristic curves (fl,f2,f3), describing the variables reboiler f,P duty ( QHP ) and reflux ( LHp ,Lp ) as a function of feed flowrate for a Jill" Ix,,='<~"), " given product and feed composition. The parameters of the feedforward Fig. 3: Feedforward/Feedback control scheme correlations and the time constants of the dynamic filters are determined by simulation with the rigorous model of the column system under SpeedUp and validated with plant data. To compensate unmeasured disturbances and modelling errors, a conventional feedback controller is applied, thus combining the advantages of feedback and feedforward control. 6. S I M U L A T I O N R E S U L T S As an example the results for a load variation of feedflowrate from about 30 % (27 1/h 35 l/h) are presented. Thereby three different control schemes are compared: PID-control, direct setting of the manipulated variables and feedforward + PID-control. Fig. 4 shows the HPC's bottom product purity, which is mainly influenced by the disturbance, for the three cases. For PID- control the set-point deviation reaches 0.8 tool%. Return to set-point takes 170 rain. Using the direct setting strategy set- point deviation can be reduced to 0.5 mol%. For feedforward control the set-point deviation can be reduced further to 0.3 mol%. The HPC's distillate purity for the three cases is shown in Fig. 5. For PID-control the set-point deviation reaches 0.4 mol%. Direct setting inverts the sign of the set point deviation, because the reflux increases before the disturbance reaches the top of the HPC. The control
155 100
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. . .
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Time [mini Fig. 5: HPC's distillate purity for a load change in feedflowrate of 30%
performance gets worse. Feedforward control shows a small stable oscillation around the setpoint. Fig. 6 shows the LPC's distillate purity, which is not affected as much as the HPC. The set-point deviation for PID-control reaches 0.3 mol%. Feedforward control shows an oscillation with a maximum set-point deviation of 0.05 tool%. 100 -~ ,_. 9 99.5
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7. EXPERIMENTAL
" 98.5
9a
Feedforward + PID PID- control
97.5 97
300
0
20
40
60
80
100
120
140
160
Time [min]
Fig. 7" HPC's bottom product purity for a load change in feedflowrate of 30%
RESULTS
The simulation results are verified in experimental investigations and results for a load variation of feedflowrate of 30 % are presented. Fig. 7 shows the HPC's bottom product purity for PID-control and feedforward + PID-control. Because the disturbance mainly effects the HPC's bottom product only this concentration profile is shown. Just as in simulation results the maximum set-point deviation for PID- control is 0.8 tool%. Return to set- point takes 150 min. Using the feedforward control the maximum set- point deviation can be reduced to 0.3 mol%. Return to set- point takes about 70 rain. With the help of the presented
156 feedforward control strategy the control performance can be improved considerably, the decline period of control potential can be reduced to 46%. 8. CONCLUSIONS In this paper a feedforward control scheme for heat-and mass integrated distillation columns for load variations in feedflowrate is presented. The advantages of this feedforward algorithm is the low sensitivity to modelling errors and the low effort of realization. As concentration is often difficult and expensive to control, an online concentration calculation is developed and validated over a wide range. The control performance of the feedforward algorithm is compared with a conventional PID-controller and the direct setting of the manipulated variables. The computational and experimental results show, that feedforward control can improve the control performance significantly. In experimental investigations a time saving of 54% comparing to a conventional PID-controller could be achieved. ACKNOWLEDGEMENT
This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under the contract GE 394/4-1/WO 565/10-2. The authors thank DFG for the financial support. REFERENCES
[]1 [2]
[3] [4]
[5] [6]
[7]
[81 [91
[lo]
H.-C. Cheng and W. L. Luyben, Heat Integrated Distillation Column for Ternary Separations, Ind. Eng. Chem. Process Des. Dev., 24 (1985) pp. 707-713 S. S. Ding, and W. L. Luyben, Control of a Heat- Integrated Complex Distillation Configuration, Ind. Eng. Chem. Res., Vol. 29, No. 7 (1990) pp. 1240-1249 P. Miszey, N. T. Hau, N. Benko, I. Kalmar and Z. Fonyo, Process control for energy integrated distillation schemes, Computers chem. Engng Vol. 22, Suppl. pp.$427-434, 1998 W. L. Luyben, and C. A. Floudas, Analyzing the Interaction of Design and Control- 1. A Multiobjective Framework and Applicaton to Binary Distillation Synthesis, Comp. Chem. Engng Vol. 18 No. 10 (1994) pp. 933-969 V. Bansal, R. Ross, J. D. Perkins and E. N. Pistikopoulos, Interactions of Design and Control: Double- Effect Distillation Systems, IFAC 1998 K. L6we, G. Wozny and H. Gelbe, Startup of Heat- and Mass- Integrated Distillation Columns: an Experimental Study, PRES'99, 31.05 -02.06. 1999, Budapest, Hungary, pp. 415-420 G. Wozny, G. Fieg, L. Jeromin, M. K6hne and H. Gtilich, Design and Analysis of a State Observer for the Temperature Front of a Rectification Column, Chem. Eng. Technol. 12 (1989) pp. 339-344 A. Jafarey and T. J. McAvoy, Short-cut Techniques for Distillation Column Design and Control, parts I & II, Ind. Eng. Chem. Process Des. Dev., 18, (1980), pp. 197-210 A. Jafarey, J. M. Douglas and T. J. McAvoy, Steady State Feedforward Control Algorithms for Reducing Energy Costs in Distillation, ISA transactions, 19 (1980) pp. 89-99 A. Rix, K. L6we and H. Gelbe, Feedforward Control of a Binary High Purity Distillation Column, Chem. Eng. Comm. (1997) Vol. 159, pp. 105-118
European Symposium on Computer Aided Process Engineering - l0 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
Controllability Distillation Using the Element model
Analysis
of
Input-Output
157
in
Reactive
A D E s t r a d a - V i l l a g r a n a * , I D L Bogle*, E S F r a g a and R G a n i t *Dept of Chemical Engineering, University College London, U K t CAPE-Centre. Dept of Chemical Engineering, Technical U n i v e r s i t y of D e n m a r k A b s t r a c t . There are several tools from linear control theory to help in control system design. The aim of the present work was to establish if it was possible to determine the best control scheme for a reactive distillation column with current linear control tools. A study of the control properties for three control schemes of a MTBE column was undertaken to determine the best. The findings were tested using dynamic simulation. The results show that it was possible to select an input-output control scheme with the help of linear measures and that the results can be reliable.
1. I N T R O D U C T I O N One of the main advantages of reactive distillation is that it can favour reactions towards the products leading to high conversions with almost stoichiometric feeds. Furthermore, the energy liberated during an exothermic reaction may be used for the separation, and this can boost the reaction (Doherty and Buzard, 1992). Simulations of the reactive distillation process have shown that such columns may present multiple steady states and that such states are related with the configuration of the column (Hauan et al., 1990). Dynamic simulations of reactive distillation have shown the interdependency of the operation of a reactive distillation column and its control system design (Schrans, et al., 1996). Sneesby et al. (1998) modelled an ETBE production column and showed how the control scheme selection is related to the performance of the column and its multiple steady states. The objective of the present work was to perform a study of controllability of reactive distillation column for MTBE production to help in deciding of the control structure of such columns. The present work will help to determine if it is possible to apply common methods of control structure design to reactive distillation and whether the effort is justified or not. 2. R E A C T I V E D I S T I L L A T I O N M O D E L A model was developed to represent the behaviour of a reactive distillation column. Such a model is used to find the column steady state for the later controllability analysis. The model consists of material and energy balances for a specified number of physico-chemical equilibrium stages. The equilibrium was described with an element model approach (P6rezCisneros et al., 1997).
2.1. Element model For the present study we opted for a model in which the main assumption is that vapour-liquid equilibrium is achieved on every stage of the column (Hauan et al., 1995). This kind of model was chosen since our main interest is the control of the column and the simple model describes the dynamics of the column well.
158 In the element based model, the stoichiometry is rewritten by the use of a formula matrix that relates the elements that form the compounds. The solution of the chemical model equations and the criteria for equilibrium, i.e., equality of chemical potentials for the coexisting phases, provide the element phase compositions at equilibrium. The main advantages of this model are that the reaction term does not appear explicitly in the material and energy balance equations, there is no need for equilibrium coefficient calculation, and there is a reduction of the dimensionality of the problem. Such a model was developed by P6rez-Cisneros et al. 2.2. C o l u m n
model
The model developed in this work can describe a distillation column that can be either completely non-reactive, reactive or have distinct separation and reaction sections. The column consists of a fixed number, NS, of trays, total condenser(s) and partial reboiler(s) and a reflux drum. The column may also have product streams. In order to develop the model a number of assumptions were made. The first assumption is that the hold-up of liquid on all the stages is much larger than the hold-up of vapour. Consequently, the dynamics of the vapour phase are ignored. The second assumption is that physical and chemical equilibrium is achieved in all the stages. The third assumption is that reactions take place only in the liquid phase. Regarding the hydraulics of the column, it is assumed that entrainment is negligible. The dynamics of the downcomer are neglected. The hold-up of energy and mass in the reboiler and the reflux drum are considered. The final mass balances are written in terms of elements. The energy balances are written in terms of specific (with respect to elements) enthalpies. There are (M+I)• ODEs where M is the total number of elements for each stage and NS is the total number of stages. The full model was presented in Estrada-Villagrana, et al. 1999. In addition to the ODEs, the model includes the algebraic equations for the equilibrium. Thermodynamics can be considered as ideal or to calculate activities and fugacities with the Wilson and SRK equations respectively. Additional algebraic equations are used to calculate properties. The liquid flow rates are calculated using the Francis correlation. The vapour flow rate is calculated assuming the plates are sieve plates. Although the sieve plate assumption seems to be approximate, it is reasonable since axial diffusion is a major effect in this sort of column and we are not using a transport model. The system has both disturbances and manipulated variables. The disturbances are the feed flow rates and compositions. There are three control law equations included in the dynamic model. The ODEs were integrated with a fifth order Runge-Kutta-Fehlberg method. The simultaneous algebraic equations were solved with a quasi-Newton method. 2.3. L i n e a r m o d e l a n d t r a n s f e r f u n c t i o n
The analysis of controllability was made with the linearization of the system at the steady state. The ODEs were linearized with respect to the states, manipulated variables and disturbances. In the Laplace domain the model can be written as an input-output model (Skogestad and Postlethwaite, 1996) and scaled: Y(s) : G ( s ) u + G d (s)d
where G(s) = C(sI - A) -1 B + D and Gd (s) = - C ( s I - A)-IE + F .
159 3. I N P U T - O U T P U T C O N T R O L L A B I L I T Y A N A L Y S I S Input-Output controllability can be defined as "the ability to achieve acceptable control performance; that is, to keep the outputs (y) within specified bounds or displacements from their references (r), in spite of unknown but bounded variations, such as disturbances (d) and plant changes, using available inputs (u) and available measurements (Ym or d~" (Skogestad and Postlethwaite, 1996). This means that if a system is controllable, it is possible to design a controller to keep the system within certain boundaries even with the presence of bounded variations. Therefore, certain manipulated variables and certain measurements may be better than others to achieve the control objectives. To do the controllability analysis it is necessary to work with a minimal realisation of the system. Having a minimal realisation ensures that there are no hidden states in the analysis. This was ensured by test of s-controllability and s-observability proposed by Kaczorek (1992). To assess the controllability the following indicators were used: number and direction of poles and zeros in the fight hand plane (RHP), frequency analysis, singular value decomposition (SVD) and relative gain array (RGA). The analysis of RGA was done because it allows us to analyse the pairings of inputs and outputs. Large RGA elements indicate that the system is sensitive to input uncertainty. This was also checked with the RGA-number defined by Skogestad and Postlethwaite, (1996) 4. M T B E P R O D U C T I O N C O L U M N The process studied was the production of MTBE. It has been shown that a good representation of the phenomena involved in this process is not possible with an 'equilibrium' model (Higler et al., 1998 and 1999). However, the results obtained with our model are similar to those obtained by Hauan et al. (1995) and Schrans et al., (1996). Furthermore, this representation may be enough for process control. The reaction is written as follows: +
cH
oH +
c -
H
+
<=>
COCH
The isobutene is normally a component of a mixture of various C4 compounds that are inert to this reaction. For this study the inerts were represented by n-butane. The column consisted of a total condenser, 3 rectifying stages (including a reflux drum), 8 reactive stages and 6 stripping stages and a partial reboiler. The feed of the column was fixed on stages 10 (pure methanol) and 11 (C4 mixture). The main features of the column are similar to Hauan (1995). The analysis was made for the high conversion steady state. There were three control objectives. The first two were to keep the steady state level both in the reflux drum and in the reboiler. The third objective was to maintain the purity of MTBE in the bottom stream. Three control schemes were analysed (Table 1). Control schemes (1) and (2) are commonly used in conventional distillation. The second scheme intends to relate the concentration of MTBE in the product to the temperature close to the reaction zone. Table 1. Control schemes analysed = : _ _
1 2
. . . . . . . . . . . . . . . .
_ .
.
.
.
.
.
.
.
.
Inputs D, B, Qreboiler D,B,~r ..................................
_.
.
.
.
.
.
.
.
.
.
.
.
.
. ...........................................
=
Outputs M1, M17, T17 (417 K) M , M ,T ~ ~
3 ............Reflux r 0 w rat e, B, Qreboikr........M1., M!=7,T_~7=(4.!7K) .....
160 The difference between the first case and the third is that the first one manipulates an external flow (distillate flow), while case 3 manipulates an internal flow (reflux flow). Cases 1 and 2 are different because of the selection of the output that would be used to control the column. 5. R E S U L T S The problem will be illustrated using the high conversion state for the MTBE column. The control objectives are to keep the levels in the reflux drum and reboiler and to keep the purity of MTBE in the bottom stream. The problem was linearized and scaled at the steady state. The selection of inputs and outputs was done with the help of input-output controllability analysis. The main results are shown as follows. None of the schemes had hidden states. The three schemes proposed were functionally controllable. In general, there were similarities between schemes (1) and (3). In contrast, results for scheme (2) were different and this scheme presented several problems. At the steady state, the transfer functions were very different for schemes (1) and (2). As expected, the third output (T12) is not strongly influenced by any of the inputs. Consequently, the condition number for scheme (2) was two orders of magnitude higher (Table 3). In all cases, although none of the RHP-zeros was pinned, such zeros were close to the origin. This means that there might be control problems. One point of concern is that scheme (2) has two additional RHP-zeros. Furthermore, inverse response is expected since there are two RHP-zeros in all cases. In general, there were similarities between schemes (1) and (3). In contrast, results for scheme (2) were different and this scheme presented several problems. At the steady state, the transfer function was very different for schemes (1) and (2). As expected, the third output (T12) is not strongly influenced by any of the inputs. Consequently, the condition number for scheme (2) was two orders of magnitude higher (Table 2). Table 2. RHP-zeros, minimum singular value and condition number for the cases of study. I/O
RHP zeros <1
cy(G(0))
7(G(0))
1 D,B,Qreboiler/M1,Ml7,T17 4 17.799 1.1 * 101 2 D,B,Qreboiler/M1,Mlv,Tl2 6 1.029 3.3" 10 3 . 3 ........ R,B,Qre~~M~,M~z,T17 9 ...................4 .........................17.404..... 4.4"100 In all cases, although none of the RHP-zeros was pinned, such zeros were close to the origin. This means that there might be control problems. One point of concern is that scheme (2) has two additional RHP-zeros. Furthermore, inverse response is expected since there are two RHP-zeros in all cases. Frequency response analysis was also used to analyse the characteristics of the different schemes. At steady state scheme (2) has large RGA elements. Moreover, the analysis of the RGA-number at different frequencies (Figure 1) unveils that schemes (1) and (3) have better diagonal dominance and that at certain frequencies, scheme (2) has very large RGA numbers. This was repeated at the crossover frequencies. Again, this index confirms that scheme (2) is poor for control purposes. At this point, it is certain that scheme (2) has more disadvantages compared to (1) and (3). However, it is not simple to decide between (1) and (3) and we have chosen not to distinguish between them. Experience suggests that for non-reactive distillation the group of inputs D-BQreboiler has more influence on the compositions. This fact could be supported by the high
161 condition number of scheme (1). This scheme has the highest maximum singular value and this could be used as a criterion to select this scheme. 5.1. D y n a m i c s i m u l a t i o n s To compare the selected and disregarded control schemes dynamic simulations were performed. Proportional controllers were used to control the mass in the reflux drum and reboiler. A proportional-integral controller to maintain T17 o r T12 at its steady state value, using schemes (1) or (2) respectively, as a way to control the purity of MTBE in the bottom stream. In both cases proportional-integral controllers were used. The simulations start at the steady state. At t=0 a step change in the flow of methanol is introduced. The feed flow rate is increased by 12.5%. In figures 2, 3 and 4 it is possible to compare the results for control schemes (1) and (2). In both cases, there is inverse response as expected because of the presence of RHP zeros. The simulations confirm this for the scheme (2). The system response is too slow in comparison with scheme (1). It is possible to see that in the second case, the fraction of MTBE drops to 0.82. This is not explained in terms of conversion. In fact, for both cases the conversion increases slightly. In figure (3) it is possible to see that the change in Yl2 is small. This leads to small changes in the heat duty. Most of the methanol ascends the reactive section since the temperature in tray 12 is still high. At the beginning of the process the excess methanol reacts with the unreacted iso-butene that was accumulated in the reactive section. Later, the excess methanol leaves the column by the distillate stream. Since the distillate flow rate is related with the reflux level and not with the composition, it does not increase substantially. Therefore, the excess of methanol in that stream makes the n-butane (non reactive) move down the column. There, again the bottom flow rate is changed according to the reboiler accumulation. Hence the bottom flow rate contains more n-butane. 6. C O N C L U S I O N S Control structure design (CSD) has been applied before for non-reactive columns. The interest of this work was to find whether it was possible to investigate CSD of a reactive distillation column by using linear control tools. The controllability analysis that was used to help in the selection of inputs and outputs led us to conclude that the control scheme D-B-Qreboile r offered more advantages than the others. This selection was confirmed by dynamic simulations. The efficiency of schemes D-B-Qreboile r and R-B-Qreboile r c a n be explained because of the fact that the inputs are closer to the outputs (M1, M~7, T~7). Dynamic simulations were done with both the selected scheme and the rejected one following a step in the MeOH. Results confirm that for the second scheme it takes more time to see the effect of the control in the mole fraction of MTBE. Although these results confirm that for the selection of scheme (1) is preferred, it would be interesting to test the system under pulse changes. Although the present study does not analyse the influence of disturbances in the selection of inputs and outputs, the results show that it is possible to use linear control for CSD. Moreover, this analysis does not differ from the normal procedure for non-reactive distillation. Future work will aim to determine whether these results are widely applicable. This will include the analysis of the influence of disturbances on the control scheme selection. In addition, it may be important to analyse additional problems and differences in the column configuration. Furthermore, it would be interesting to determine whether the controllability
162 indices change with the use of other approaches to simulation of the reactive distillation process. ACKNOWLEDGEMENTS The authors would like to acknowledge support for A. Estrada-Villagrana from CONACyT 140
361
120
360
100
0.96 0.94 0.92
359
0.9O
..-.,
8O 60
~358 357
40 20
356
0
355
0.00001
0.001
0.1
10
Frequency (rad/s) ---rgan-S2
1000
8 w-
O.88 -6 E 0.86
, , , , T~2 __ ......
0
~m3E~ 0.84 0.82
I
|
50O
l(II)
1500
Urre(s)
Figure 3. MTBE purity in the bottom tray after a step of 12.5% of the MeOH feed. Control Figure 1. RGA numbers for the three schemes scheme (2) proposed .............rgan-Sl
--rgan-s3
1.0
0.96
417
0.9
0.94
4t6
0.92
~" 415
0.90
0.8 8
0.88 ~
414 ,,,,, T17
0.86 0.84
413 412
I 0
500
1000
Ume(s)
--
XMTBE
1500
~
0.82
0.7 8 0.6 ~0.5 -~ 0.4 0.3 0.2 0.1 0.0
,
1 2
3 4
5
6
7
8
,w
,i,
,w
,~,
,~
,!
9 10 11 12 13 14 15 16 17
stage
Xiso-butene .--n-Xn-butane
--x-Xmtbe
-m-Xme~anol
Figure 2. MTBE purity in the bottom tray after a step of 12.5% of the MeOH feed. Control Figure 4. Composition in the trays at t=1800s. scheme (1) Control scheme (2)
REFERENCES Doherty, M.F. and G., Buzad. 1992. ICHEME Symposium series. 128:A51-A68. Hauan, S., T., Hertzberg and K.M., Lien. 1995. Computers chem. Engng. 19: $327-$332. Higler, A.P., R., Taylor and R. Krishna. 1999. Chem. Eng. Sci. 54: 1389-1395. Estrada-Villagrana A.D., D. Bogle, E. P6rez-Cisneros and R. Gani. 1999. Computers chem. Engng. 23:$339-$342 Kaczorek, T. 1992. Linear control systems. Research studies press. England. 379 p. P6rez-Cisneros, E.S., R., Gani, M.L., Michelsen. 1997. Chem. Eng. Sci. 52:527-543 Schrans, S.T., S., de Wolf and R., Baur. 1996. Computers chem. Engng. 20: S 1619-S 1624 Skogestad, S. and I., Postlethwaite. 1996. Multivariable Feedback Control. John Wiley & Sons. England. 559p Sneesby, M.O., Tad6 and T.N. Smith. 1998. Trans IChemE. 76(A): 525-531
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
HYBRID SIMULATION
OF CONTINUOUS-DISCRETE
163
SYSTEMS
Vishal Bald and Andreas A. Linninger
Laboratory.forProduct and ProcessDesign Department of Chemical Engineering,Universityof Illinois at Chicago, Chicago, IL 60607, U.S.A, email: {vbahl,linninge}@uic.edu
Abstract Many process models in the chemical industry involve discrete phenomena superimposed on continuous system dynamics. While the continuous behavior is described by mass, energy and momentum balances, discrete behavior may occur due to physico-chemical discontinuities or discrete logical controller actions. Such systems with interaction between the continuous and discrete system dynamics are classified as Hybrid Systems. For these systems, standard numerical treatment via "continuous" integration methods break down. Equally, discrete simulation e.g. finite state machine cannot handle the continuous dynamics. Hence, a new mathematical framework capable of handling both continuous and discrete aspects of hybrid systems is needed. This paper presents a new environment for modeling and simulation of hybrid systems. It offers a high-level design language for the automatic or semi-automatic generation of the process models. The efficacy of the simulation language is demonstrated by means of continuous-discrete implementation of a Dynamic Matrix Control strategy.
Keywords Modeling, simulation, discrete events, modeling language I. INTRODUCTION Many of the operations in the process industries exhibit continuous-discrete dynamic behavior. Examples include the dynamics of the reacting system together with its safety. device. The discontinuities could also stem from physical behavior such as phase changes, changing reaction order or logical process input via fault diagnosis or alarms, In all the above processes there are control tasks or logical control functions interacting with the continuous process dynamics. In this literature these mathematical models are classified as Hybrid Systems. The need for discontinuities handling in a physical system has been addressed by Gear [Gear, 1980]. His work revealed that gross inefficiencies may result from the use of multi-step integration procedures without special consideration for discontinuities. In the worst case this may even lead to a numerical crash of the solver algorithm. In less severe cases, the solver will simply produce an incorrect system response. Several simulation environments have addressed physical systems with discontinuities. [Fahrland, 1970], in his earlier work identifies the fundamentals of continuous-discrete systems and describes how this class of problems may be solved as a sequence of initial value problems. This methodology requires a combined system to be decomposed into continuous and discrete subsystems that are allowed to interact during the course of simulation. Batches [Joglekar
164
and Reklaitis, 1984], Omola [Andersson, 1990], gProms [Barton and Pantelides, 1994] and Stateflow [Simulink, 1998] offer models for discontinuities modeling. Despite these efforts some of the earlier approaches are specialized to specific classes of discontinuities. More complex situations such as simultaneous events have been addressed insufficiently. In this paper we will demonstrate a new mathematical approach for event detection and handling. Subsequently, we will show how the mathematical theory can be used advantageously in conjunction with a high level language. Section 2 will review mathematical foundation for event handling and some of the existing algorithms. Section 3 presents an event detection algorithm for use with an ODE system and discusses synchronous events briefly. A new highlevel language for hybrid systems will be presented and its use demonstrated by a case study in section 4. 2. EVENTS C A T E G O R I Z A T I O N IN A PHYSICAL SYSTEM A combined continuous-discrete simulation is advanced by a solution of a sequence of initial value problems. Mathematically, the "continuous" dynamics of process models can be described by a set of explicit ordinary differential equations as in Eq. (1). y'(t) = f ( y ) (1) The continuous evolution of the state variables may be interrupted by a discrete event. The occurrence of such an event is usually tied to a conditional function. In consequence to this event, the system transits to a new discrete state. To detect and locate the events, a discontinuity function (z-function) can be used as expressed by Eq. (2). z ( y , t ) <_o (2) The event function depends on the state variables y and is normalized to zero. It crosses the value of zero when a state event takes place. Integration of the initial value problem proceeds until one or more discontinuity functions change sign. As an example consider the liquid level on a tray of a distillation column. The change of the filling on the tray can be described by the ODE given by Eq. (3). dh (3) A d---t= qin - qoverflow
An event occurs if the liquid level reaches the overflow weir and the liquid overflows the tray for the first time. The conditional h_>h,~ is of the form as given by Eq. (2) and after normalization the z-function can be represented as given by Eq. (4). z = (hweir - h(t)) <_0 (4) 2.1 E a r l i e r E v e n t H a n d l i n g A l g o r i t h m s
Several algorithms have been developed for event detection and location. The major differences between these approaches lie in the manner in which the event fimction is used to locate the state event. Joglekar and Reklaitis [1984] established a linear discontinuity function and solve the constraint z(t) = 0 using a Newton iteration scheme. However, their approach is restricted to systems with linear discontinuities. Preston and Berzins [1991] developed an event location algorithm for specific class of problems, e.g. valve operations. Pantelides [1988] uses the state variable conditionals directly to detect a state event. An event is located by a bisection method and successive interpolations in time interval [tk, tk+l]. Park and Barton [1994] developed a polynomial root-finding procedure to detect a state event and use a Newton's method with recursive interval bisection to locate the events.
165 Despite previous work pertaining to the mathematical aspects of hybrid systems, logical consistency and the semantic content of the continuous-discrete process models needs further discussion. In the following section, the mathematical foundation of a new algorithm to address the deficiencies including simultaneous events will be described. Depending upon the semantics of the physical process and the event conditionals, the events can be categorized as compiled in table 1. ' Event Categorization ,,
Time explicit Time implicit .
.
.
.
.
.
.
.
Consecutive Simultaneous
Type of Event
Example
Sampling using digital PID controller Event location known in advance , Reversible time implicit ' ' ,Operation o'fPressure,'re!iefvalve Operation of a pressure burst disc Irreversibletime implicit Secondary event is triggered due to a Consecutive event in digital Cascade....... control scheme principal event. SynchronousEvents . . . . . Controllers samplingat same time Table 1" Events categorization
3. A L G O R I T H M To detect whether an event has taken place in a physical system, an event function, z is defined as given by Eq. (2).The derivatives of the event function, z, are added to an augmented system given by Eq. (5) and Eq. (6). = f(y)
(5)
dz =G(y)
(6)
dt dt
The derivatives of the z-function are obtained by symbolic differentiation of the Eq. (2). In our modeling system this is done by automatic differentiation [Linninger et. al., 1999]. The event handling algorithm proceeds through three phases: (i) event detection, (ii) event location and (iii) step-completion. A detailed examination of the three phases follows. 3.1 Event Detection
The initial value problem is integrated using a fifth order Runge-Kutta method with adaptive step size control. During the integration the system equations are "locked" into the current state ignoring any discontinuities [Park and Barton, 1994]. After each successful integration step, the discontinuity function is checked for zero-crossings. This can be achieved by checking for the signs of the event function at the beginning and end of the time interval. If no event has been detected the integration is continued. If there is a sign change of the z-function an event has been detected. For each event function with a zero-crossing, the exact time location t* has to be found next. 3.2 Event Location
For each event function, zi, the intermediate estimator values kz, k3, k4, k~ are examined. These are computed and stored by the Runge-Kutta method in each integration step. For each interval spanned by the supporting points ki and kj, the condition for the zero-crossing is
166 checked again. There must be exactly one pair ofki, kj at which this condition holds. Fig. 1(a) depicts the zero-crossing of one z function. Fig l(b) shows the detailed view between the supporting points ki and kj which embrace the zero-crossing. The trajectory of the z-function in this interval is interpolated by a second order polynomial p(t) given by Eq. (7). p(t) = At 2 + Bt + C = 0 (7) The fitting parameters A, B and C are found through the three constraints expressed by Eq. (8), Eq. (9) and Eq. (10). p(ti) = k~ (8) p(tj) = kj (9) dp/dt(t0 = dkdt, (10) where, trtj are the time instances associate with the support points ki and kj and dkdti is the gradient at point k~. In the next step, the exact zero-crossing of the interpolation polynomial p(t) is found by solving for p(ti ~ = 0 using Eq. (8)-(10), to identify the exact time for event o c c u r r e n c e ti*.
l'd dti
Z ~
i k,
z
7
i
k,
ii~)i~iiiIiIi (Iii
h~ V
!t~J
t ti
t ~
z
_ _ _ ~
Ti me kj
Fig. 1 Event location using quadratic approximation
The same procedure is repeated for all the z-functions that had a zero-crossing. The actual event time t* is then set as the minimum t* as described in Eq. (11). t * = min(t i * )
(11)
3.3 Step Completion The interval in which the event has been detected is re-integrated with a reduced step size t-t ~ while leaving all equations in their settings before the event i.e. locking the equations. After the integration to the location of event at time t* we swap to the new state. This may entail change in the assignment of the system state variables or even exchange of model equations. The initial value problem is then further integrated.
3. 4 Synchronous Events A situation which is often omitted in hybrid simulation refers to synchronous events. In synchronous events two events occur at same time or the integration step size between the two events is close to zero. In a practical example this occurs when two digital controllers sample at same time. Such types of events are efficiently handled by our algorithm by checking for simultaneity or near simultaneity via the condition of Eq.(12).Without this
167
treatment, closeness of the two events could lead to singularity in the Runge-Kutta method and an event not being detected at all [Bahl and Linninger, 1999]. if ti*-tj*l<-e "both events are fired" (12)
4 A NEW HYBRID S I M U L A T I O N LANGUAGE As can be seen, the algorithm discussed is quite complex for a process modeler to implement for continuous-discrete simulations. We have developed a high-level language that eliminates the need to directly deal with the event handling algorithm. The language constructs can be used to model systems with the discontinuities which have been discussed in this paper. The high-level language has been integrated into TechTool which is a phenomena-oriented environment for the generation of the steady and dynamic process models [Linninger and Krendl, 1999]. Table 2 shows the vocabulary of the declarative modeling language for event handling in physical systems usi~gchTool.
Event Categorization Every x seconds do
Purpose
Whenever (condition) {expression A} else {expression B } When(condition ) {expression A}else(expression B}. . . . . . While (condition) do {expression A }. At time x seconds do{expression A} else {break;) Schedule
Reversible implicit events Irreversible implicit events//Pre-condition Irreversible implicit events//Post condition Explicit time events
Periodic explicit events
Example Sampling using digital PID controller Flow of liquid through a vessel with a weir Bursting of a pressure disc Irreversible reaction in a react0r " Feeding of fuel in a furnace /
Explicit time events
Charging in a vessel at pre t / deternlined time
Table 2: Language constructs for Hybrid simulations in TechTool. 4.1 A p p l i c a t i o n s : S a m p l i n g u s i n g M P C s t r a t e g y
The Hybrid simulation methodology has been applied to model and tune a Model Predictive control strategy for a physical process model. The process under consideration is a simple first order system representing the level control of liquid in vessel as in Fig. (2). The change in the level of the tank is described by Eq.(13). dy = flow _ it, - flow _ out dt area
(13)
The system process and the DMC were simulated on a PC. Alternatively these studies would require expensive Distributed Control Systems Network (DCS). Fig. 3 shows the response of a DMC to a load change from 4000dhto 4100cn~. As can be seen from the figure a DMC takes about 22 seconds for the system to reach steady-state again.
168
,1 H ,~,ge5
o
2
4
e
s
lo
12
14
le
la
20
22
z,l rm
Fig. 2 Tank with an overflow weir.
2e
2s
30
32
:~
38
:~
4o
42
44
=
~
5o
(~=l
Fig. 3 Response of MPC to a load change
CONCLUSIONS A N D S I G N I F I C A N C E An a l g o r i t h m that offers precision on sign and a c c u r a c y o f the event condition is introduced. A high level m o d e l i n g l a n g u a g e has also been introduced which allows engineers to construct hybrid simulations without presenting the m a t h e m a t i c a l details. The v o c a b u l a r y o f the declarative m o d e l i n g l a n g u a g e for hybrid simulation is presented. An application has also be s h o w n to d e m o n s t r a t e the m o d e l i n g and simulation o f practical systems. REFERENCES Andersson, M.; "Omola - An Object-Oriented Language for Model Representation, Ph.D. thesis, Lund Institute of Technology, 1990. Bahl, V; "Applications of the Hybrid Simulation to Model Predictive Control Strategy"; UIC-LPPD Lab Report 2 (07/05/1999). Bahl, V; "Hybrid Simulation in TechTool"; UIC-LPPDLab Report 3 (09/23/1999). Barton, P.I; " The Modeling and Simulation of Combined Discrete/Continuous process", Ph.D. Thesis, University of London, 1992. Fahrland, D.A;" Combined discrete event continuous system simulation", Simulation, 14.61-72, 1970. Joglekar, S., and Reklaitis, G.V.;"A simulator for batch and semi-continuous processes", Computers and Chemical Engineering, 8:315-327,1984. Linninger, A., S. Chowdhry, V. Bahl and H. Krendl; "TechTool a Process Model Generation Environment", lASTED International Conference on Modeling, Identification and Control, Innsbruck, Austria, February 1999. Linninget', A.A, Kxendl, H and Pinger, H;" An initiative for integrated Computer-Aided Process Engineer, FOCAPO '98 Conference, Snowbird, Utah, July 5-July 10. Linninger, A.A and Krendl, H; "TechTool- Computer-Aided Generation of Process Models (Part 1- A Generic Mathematical Language ), Comp. Chem. Eng., 23, $703-$706, 1999. Pantelides, C.C;" Symbolic and Numerical Techniques for the Solution of Large System of Nonlinear Algebraic Equations", Ph.D. thesis, University of London, 1988. Park, T and Barton, P.I;" State Event Location in Differential Algebraic Models"; ACM Transactions on Modeling and Computer Simulation, Vol. 6, No. 2, April 1996, Pages 137-165. Preston, A.J and Berzins, M.; "Algorithms for the location of discontinuities in dynamic simulation processes"; Computers and Chemical Engineering, 15, pp. 701-713, 1991. Simulink User Guide; Mathworks, lnc, Version 11.1, January 1998.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
169
Interaction between Design and Control of Heat-Integrated PFR C.S. Bildea, A.C. Dimian and P.D. Iedema Department of Chemical Engineering, University of Amsterdam Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands E-mail: [email protected]; [email protected]; piet @chemeng.chem.uva.nl. This article analyzes the interaction between design and control of a heat-integrated plugflow reactor. The space of the design parameters, feed-effluent heat-exchanger (FEHE) efficiency and steam-generator duty, is divided by bifurcation varieties into regions with different controllability properties. Close to the cusp variety, the system is sensitive to disturbances and control is as difficult. The same conclusion is valid for designs with large FEHE. Disturbances can be rejected, but fast control can not be achieved. The range of design parameters for which the system cannot be stabilized is computed. 1. INTRODUCTION In processes where an exothermic reaction takes place, heat exchange between the effluent of adiabatic reactor and the feed stream may be used for energy saving. Startup, control, or plant energy balance may require additional units: furnace, steam-generator and quench. The resulting structure (Figure 1) will be called heat-integrated PFR. Although attractive from the viewpoint of energy saving, it may exhibit state multiplicity, isolated solution branches and oscillatory behavior. Hence, control problems are expected. Bildea and Dimian (1998) proposed a design procedure to ensure a desired multiplicity pattern and a stable point of operation, and to avoid high sensitivity. Similar systems have been analyzed using linear models (Silverstein and Shinnar, 1982, Tyreus and Luyben, 1993) or nonlinear dynamic simulation (Luyben et al., 1998). 2. DESIGN CONSIDERATIONS In this article, we consider a case where moderate conversion is required. The kinetic, thermodynamic and design parameters correspond to the toluene hydrodealkylation. STEAM For this reaction, the boundary between ~ GENERATOR unicity and multiplicity regions is located at QUENCH a rather low value of FEHE efficiency
(e*=0.32). Feed flow rate (F), reactor inlet temperature (T2) and conversion (X) are typical initial data. During conceptual design, the reactor volume, FEHE efficiency, furnace and steam-generator duties are of interest, but detailed equipment sizing is not necessary.
tqh * T6
REACIOR
FURNACE
Fig. 1. Heat-integrated PFR
170 The reactor can be designed if reaction kinetics and thermodynamics (adiabatic temperature rise, ATaa) are known. First design decision regards the FEHE efficiency (e). The size of FEHE determines the multiplicity pattern, the stability of the operating point and the sensitivity to energy disturbances. Bildea and Dimian (1998) discuss this issue in detail. The net energy requirement (q) is given by the heat-balance equation: q-O-E)'FcpTo +(qh-e~'qc)-(1-e)'FCpT2-C'FcpATadX (1) However, it must be split between feed enthalpy (To), furnace and steam-generator duties (qh, qc). Considering fixed feed temperature, the second design decision concerns the steamgenerator duty. According to the above decisions, four design alternatives are considered. Design 1 has a small FEHE (e=0.28). The operating point is in the unicity region, but close to the unicitymultiplicity boundary. Design 2 has a large FEHE (e=0.745). Hence, the operating point is in the multiplicity region. In Designs 1A and 2A, a large quantity of energy is used to generate steam (qc=10575 kW), and a large amount of energy is provided in the furnace (qh=16436 kW and 12665 kW, respectively). Conversely, less steam is generated in Designs 1B and 2B (qc= 1048 kW), a smaller furnace being necessary (qh= 13969 kW and 6011 kW, respectively). A dynamic model was developed, based on the following assumptions: 1. Reactor dynamics is described by a plug-flow, pseudo-homogeneous model. 2. An instantaneous model is used for the steam-generator, because its duty is regarded as a disturbance. 3. For FEHE, an efficiency model is adequate because of fast dynamics. 4. A fast fuel flow r a t e - flue-gas temperature control loop is used to set the furnace duty at the prescribed value. Consequently, the heat transfer from burner to tubes (hot-side) is not included in the model. The temperature variation along the tubes is neglected and the coldside heat-transfer coefficient is assumed to be constant. These assumptions allow an approximate furnace design (number of tubes, diameter, length, and thickness). The model describing the essential dynamic behavior is: fluid:
tubes wall:
3Th aTh 4 O---7-:--Wh. Oz + d~w
Kwf
~.(T p.Cp
w --Th)
-Th)dZ / ~---~ ~Tw = Pw "cpwl( . V . qh-/r" dw "Kwf "I(Tw L
(2) (3)
0
3. LINEAR CONTROLLABILITY ANALYSIS The main control objective is to keep the reaction conversion and selectivity within acceptable limits. Because both variables depend on the reaction temperature, three different control structures are considered, which include the reactor inlet temperature as the main controlled variable. In the first structure (CS 1), furnace duty is used to control the reactor inlet temperature. In the second one (CS 2), FEHE bypass and furnace inlet temperature are the manipulated and controlled variables, respectively. The setpoint of this loop can be used, in a cascade manner, to control reactor inlet temperature. In the third control structure (CS3), both furnace duty - reactor inlet temperature and FEHE bypass - furnace inlet temperature loops are used. Feed flow rate, steam-generator duty and FEHE heat transfer coefficient are considered as disturbances. The axial coordinate of model equations was discretized by finite differences and the model was linearized around steady-state. In order to obtain meaningful controllability results, the inputs, disturbances and outputs were scaled such that the control objective is to keep the
171 control
error
manipulated
inputs
[d(t]
le(tl < 1,
lu(t] < 1,
using when
._~
0.04
K iy
X disturbances < 1 affect the process. < 0.02 Linear model was used only for ~. ~ "~,~_~+ tcontrollability analysis (Skogestad and "~ o Postletwaite, 1996). The full nonlinear E model was used to evaluate the -0.02 performance of the control system. The operating points are stable for -0.04 Designs 1A and 1B, but unstable for -0.04 -0.02 0 0.01 Designs 2A and 2B. Figure 2 presents a Real Axis part of the root-locus plot for Design 2A and CS1 (similar results are Fig. 2. Root locus plot for Design 2A and CS 1 obtained with CS2). Two eigenvalues located near the imaginary axis are of interest. They are denoted by/~+ and ,L according to their sign when the controller gain is zero. When the controller gain is increased,/1,+ moves towards the left half plane and crosses the imaginary axes. Hence, there is a minimum controller gain for stability. In the same time, L moves to the right. When &+ and &_ meet each other, a pair of complex conjugate eigenvalues emerges and moves to the right half plane. Hence, there is also a maximum controller gain for stability. For Design 2B, &+ and ~_ meet each other in the right half plane, near the imaginary axis. Hence, for all controller gains, there is at least one eigenvalue located in the right half plane. Consequently, linear analysis predicts that Design 2B can not be stabilized by a P-controller. For the system analyzed, we have found that integral controller action has the usual, destabilizing 1000 effect. This is in contrast to results of the study of a catalytic reactor / preheater process (Tyreus and 100 Luyben, 1993), where integral action improved closed loop stability. 10 For CS3, RGA is used to evaluate the inputoutput pairing under decentralized control. The 1 diagonal RGA elements for designs 1A, 1B, 2A and 0.0001 0.01 rad/s 2B are 4.274, 4.289, -0.954 and -0.964, respectively. Decentralized integral controllability requires positive values for stable systems, but ~00 negative values for unstable systems with one RHP pole. This condition is fulfilled in all cases. Closeto-one diagonal elements denote little interaction. 10 Small values in the RGA matrix also indicate that model uncertainty is not a problem. The RGA_number has small value and falls to zero for 1 0.0001 0.01 rad/s 1 high frequency, showing that good control performance is possible. Fig 3. CS 1" Loop (~)and To analyze the disturbance rejection properties, disturbance gains. I feed flow rate" the Closed Loop Disturbance Gain (CLDG) is A steam-generator duty; X FEHE calculated (Figure 3). Designs with the same steamefficiency. generator duty, as 1A-2A and 1B-2B, gave similar ._
-
_.
('~;'__+_
.........
!'\
m
i
~
~
i
i
172 results. At low frequency, Ig.I >
Vk, for Design 1A and 2A. Hence, disturbances can
be rejected. For Design 1B and 2B, it is necessary to increase slightly the maximum allowed control action, in order to avoid input saturation. It all cases, fast disturbances can not be rejected. Feed flow rate is the worst disturbance. Large steam-generator duty (Design 1A and 2A) is also a demanding disturbance. Note that we are only concerned with low frequency response for FEHE disturbance, because fast change of FEHE heat transfer coefficient is not expected. When bypass around FEHE is the manipulated variable (CS2), changes of the feed flow rate can not be handled. Disturbances in steam-generator duty can be rejected only if the steam-generator duty is small (Design 1B, 2B). Only Designs 2A and 2B can cope with FEHE fouling. In CS3, a second controller is added to CS1. It regulates the furnace inlet temperature using a bypass around FEHE. RGA elements show that, compared to CS 1, the gain of main loop (furnace duty - reactor inlet temperature) is lower for Design 1A and 1B, and is practically unchanged for Designs 2A and 2B. The Relative Disturbance Gain indicates that the interactions increase the gain from disturbance to reactor inlet temperature. Hence, there is little incentive to add the second control loop. Moreover, except Design 2B, bypass around FEHE is not powerful enough to control furnace inlet temperature. 4. NONLINEAR ANALYSIS The control of the heat-integrated reactor in the HDA plant has been studied by Luyben et al. (1998). They pointed out that is difficult to stabilize systems with large FEHE. However, they did not quantify the meaning of "large" and "small" units. In this section, we bring a quantitative approach of this problem by using elements of the bifurcation theory (Guckenheimer and Holmes, 1983). Hence, we identify the region in the design parameter space leading to designs that can not be stabilized. Figure 4 presents, qualitatively, the conversion vs. controller gain (Kc) and Damkohler number (Da) for two sets (FEHE efficiency, steam-generator duty). The first diagram corresponds to design 2A (large FEHE, Design 2A large steam-generator). Model equations are used to compute the controller bias (the value of the manipulated variable when the controlled variable equals its set point). This is denoted as the nominal case and is represented by the dark lines. The system exhibits three steady-states for small controller gain. The middle state is of interest, but is unstable. Increasing the controller gain, the branches become closer. They coalesce for a critical value of the Design 2B controller gain. This is a pitchfork bifurcation point. For higher values of the controller gain, only one Fig. 4. Conversion vs. controller steady-state is possible. When the controller gain is gain and Damkohler number. further increased, the system loses stability due to a Hopf bifurcation. Hence, the pitchfork and Hopf points bound the range of controller gains that ensure state unicity and stability.
/---Hopf
Nominal/
173 The extent of the unicity and stability region decreases when the FEHE efficiency increases or the steam-generator duty decreases. For a critical value of FEHE efficiency, pitchfork and Hopf bifurcations occur at the same value of the controller gain. Incidentally, the Design 2B is very close to this limit. This situation is presented in the second diagram of Figure 4. If FEHE efficiency is further increased, the design can no longer be stabilized. The correct value for the controller bias can not be calculated when design parameters are uncertain. This is denoted as the perturbed case and is presented in Figure 4 by the light lines. The Hopf point is one limit of the stability region. Depending on the parameter uncertainty, the other limit is the fold point located on the lower or the upper solution branch. Note that the region of state unicity and stability is larger. However, zero control error can not by achieved by proportional control. In order to classify the steady-state and dynamic behavior of the controlled system, we choose the controller gain as bifurcation parameter. The Hopf and fold points are 1codimensional, which means that the value of one parameter is fixed. In the general case, a pitchfork point has codimension 3. However, the assumption that the parameters used for bias calculation are exactly known introduces Z2-symmetry. Consequently, the pitchfork point becomes also 1-codimensional (Golubitsky and Schaeffer, 1985). Consider a fixed, large FEHE efficiency. Then, there is a particular value of the steamgenerator duty for which the pitchfork and Hopf bifurcation occur at the same value of the controller gain, like in the second diagram of Figure 4. This point has codimension 2. We call it "Hopf-and-pitchfork". Computing its locus in the space of the design parameters (FEHE efficiency, steam-generator duty), we trace the boundary between the designs that can be or can not be stabilized. Similarly, computing the locus of the cusp variety, we trace the boundary between state unicity and multiplicity. The results are presented in Figure 5. The cusp and Hopf-and-pitchfork varieties divide the design parameter space into three regions. In region I, the unique steady-state is stable for low controller gain, 1A 9 2A 9 >, "5 0.8 but unstable for high controller ~, -~ oO~ S 06 o** gain. State multiplicity appears -~ $~, when the cusp variety is Controllergain ~k, , ~ ''~ crossed to region II, but it is = 04 N-X o ,,** Controllergain E possible to find a controller ~ 02 2B ~ ............... 1B 9 gain that gives a unique stable (i) (II) ~ ~ , "~j~iiiil, , Oil) ~g steady-state. When the Hopf 0 0.4 0.6 0.8 0.2 locus is crossed to region III, it FEHE effieciency is impossible to get both state unicity and stability. Fig. 5. Phase diagram in the space of design parameters. o
5. D Y N A M I C S I M U L A T I O N We verified first that design 2B can not be stabilised. Figure 6a presents the results of dynamic simulation, when CS 1 is used. Initially, the system is at steady-state. After 2000 s., the feed flow rate is changed by 0.1%, for 100 s. In the first run, the controller gain is lower than the gain at the pitchfork point (Kc=31.3). For this reason, there are multiple steady-states and the operating point is unstable. Hence, the disturbance drives the system to the extinguished state. In the second run, when the controller gain is slightly increased past the pitchfork point (Kc=31.3), only one steady-state exists. However, it is unstable and surrounded by a limit cycle.
174 A question that naturally arises is: "The results of linear controllability analysis are applicable to such nonlinear systems?" To answer this question, we apply feed flow rate and steam-generator duty disturbances to Designs 1A, 1B and 2A. CS1 with Tyreus - Luyben controller settings is used. The results are presented in the Figures 6b and 6c. In all cases, the manipulated variable is strong enough to reject disturbances. However, disturbance rejection is slow and the control error is initially large. For Designs 1A and 1B, feed flow rate is the worst disturbance. For design 2A, the disturbances considered are almost equally difficult. These results agree with linear controllability analysis.
650
Kc=31.5
(a)
i i
640
._
rr
630
620 5000
10000 Time
760
1500(
(s)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(b)
o, v 710
o_
+25% feed flow rate
F: 660
610
-
-
U
113 -25% feed flow rate
CC
.
560
3600
.
. 7200 Time
. 10800
14400
(s)
6. CONCLUSIONS (c) 1. The controllability of the heat-integrated PFR is 710 +25%steam generator duty determined by two design decisions: FEHE la efficiency and steam-generator duty. -25%steam generator 2. Systems with small steam-generator and large ~, 61o duty u FEHE can not be stabilized. The Hopf-andpitchfork variety bounds the range of design (s) parameters for which the system can be stabilized. Fig. 6. Dynamic simulation results 3. Fast changes in feed flow rate and steamgenerator duty lead to high temperature excursions. For this reason, they have to be avoided in operation. However, large disturbances can be handled, if they are slow. 4. Disturbance rejection is also difficult for designs with small FEHE, when the operating point is close to the cusp variety. 5. The results of linear and nonlinear analysis are confirmed by dynamic simulation. The approach presented here may be extended to similar problems, where state multiplicity and instability limits the range of controllable designs. E 660 _=
__ .E
rr
56o
o
3600
7200
10800
14400
Time
REFERENCES Bildea, C.S. and A.C. Dimian, AIChE. J., 44(1998), 2703. Golubitsky, M. and D. Schaeffer, Singularities and Groups in Bifurcation Theory, SpringerVerlag, New York, 1985. Guckenheimer, J. and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983. Luyben W.L., B.D. Tyreus and M.L. Luyben, Plantwide Process Control, McGraw-Hill, New York, 1998. Silverstein, J.L. and R. Shinnar, Ind. Eng. Chem. P.D.D., 21(1982), 241. Skogestad, S. and I. Postlethwaite, Multivariable feedback Control, John Wiley, Chichester, 1996. Tyreus, B.D. and W.L. Luyben, J. Proc. Cont., 3 (1993), 241.
EuropeanSymposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V.All rightsreserved.
175
Optimal control of batch reactors using Generic Model Control (GMC) and Neural Network N. Aziz a, M.A.Hussain b and I.M. Mujtaba c* a'CComputational Process Engineering Group, Department of Chemical Engineering, University of Bradford, West Yorkshire BD7 1DP, UK email: [email protected], [email protected] bDepartment of Chemical Engineering, University of Malaya, 59100 Kuala Lumpur, Malaysia email: [email protected] On-line implementation of the optimal reactor temperature profiles in batch reactors is considered here. The optimal reactor temperature profiles are obtained by solving dynamic optimisation problems off-line to achieve maximum conversion to the desired product. Generic Model Control (GMC) algorithm is used to design the controller to track the optimal temperature profiles (dynamic set points). Neural Network technique is used as the on-line estimator to estimate the amount of heat released by the chemical reaction. A complex reaction scheme is considered in this work to illustrate the ideas. The results clearly show that the GMC controller coupled with a Neural Network based estimator tracks the optimal temperature profiles very well to obtain the desired products on target. 1. INTRODUCTION Batch reactor is an essential unit operation in almost all batch-processing industries. The control of a batch reactor in a simple case consists of charging the reactor, controlling the reactor temperature to meet some processing criterion, and shutting down and emptying the reactor. Batch reactor is an inherently unsteady-state process and therefore modelling of such reactor results in a system of Differential Algebraic Equations (DAEs). Operating batch reactors efficiently and economically is very important as far as overall profitability is concerned. The dynamic optimisation (optimal control) of batch reactors has received major attention in the past. The objective was to determine the optimum reactor temperature profiles for cases where there are competing side reactions, so that an increase in the yield (productivity, profit etc.) may be obtained using the optimal profiles [1,2]. However, all these researchers considered only the off-line optimisation problems. None of them have implemented these results on-line. Designing controllers to implement the optimal control profiles or tracking the dynamic set points is an important area of research for inherently dynamic batch processes.
* Correspondence should be addressed to I.M.Mujtaba. Email: [email protected].
176 In their study, Cott and Macchietto [3] have used Generic Model Control (GMC) algorithm of Lee and Sullivan [4] as the controller in order to track the reactor temperature set point (Trsp). They used a three-term difference equation and exponential filters as the on-line estimator for heat-release. Later, Kershenbaum and Kittisupakorn [5] considered the same reaction scheme of Cott and Macchietto and also used the GMC algorithm for the controller. They however used the extended Kalman filter for the on-line heat-release estimator. In this work, we also consider the GMC controller but use Neural Network techniques for on-line estimator of the heat-release. We demonstrate the idea using the reaction scheme considered by Cott and Macchietto [3]. First, an off-line dynamic optimisation problem is solved with fixed batch time to find the optimum temperature profile that will maximise the conversion of the desired product (details in Aziz and Mujtaba [6,7]). The optimum temperature profile thus obtained (off-line) is used as the set point to be tracked (on-line) by the GMC controller. 2. GMC F O R M U L A T I O N FOR BATCH REACTORS
Generic Model Control (GMC), a model based control strategy developed by Lee and Sullivan [4] is one of several advanced process control algorithms implemented recently. The GMC uses nonlinear models of a process to determine the control action. The desired response can be obtained by incorporating two tuning parameters. The main advantage of the GMC is that the nonlinear process models do not need to be linearised because it directly inserts nonlinear process models into the controller itself. In addition, the GMC algorithm is relatively easy to implement. The GMC control algorithm can be written as: dx/dt = K1 (X~p- x) + K2 .[ (Xsp - X) dt
(1)
where x is the current value of control variable and Xsp is the desired value of control variable. The first expression in the algorithm (K1 (Xsp- x)) is to bring the process back to steady state due to change in dx/dt. In order to make the process have a zero offset, the second expression (K2 ~ (Xsp - x) dt) is introduced. Details of this GMC method can be seen in Lee and Sullivan [4]. The batch reactor system of interest in our control strategy is shown in Figure 1. For temperature control of batch reactor, a process model relating the reactor temperature, Tr, to the manipulated variable i.e. the jacket temperature, Tj is required. Assuming that the amount of heat retained in the walls of the reactor is small in comparison to the heat transferred in the rest of the system, an energy balance around the reactor contents gives the following model: dTr/dt = (Qr+UA(Ti - Tr))/WrCp
(2)
Replacing T~ for x and Trsp for Xsp in equation (1), combining equation (1) and (2) and finally solving for the manipulated variable, Tj, the control formulation under GMC is given by: T i = Tr + (WrCp/UA) {KI [Trsp - Tr] + K2 I [Trsp - Tr] dt} - QJUA
(3)
where Tj gives the jacket temperature trajectory required so that the reactor temperature, Tr, follows the desired trajectory incorporating the values of GMC tuning parameters, K~ and K2.
177
Vent condenser ~
Coolant/ Steam inlet
[ I
lr [Vr ~
r
Tr (k-2)
Feed
Stir
I
Coolant Steam Outlet
Tr(k-1)
T~ (k)
Tj(k-1)
Tj(k)
Qr(k-1)
[-~
Input Output
~Product Figure 2: Input/Output Map of Neural Network
Figure 1: Schematic diagram of a jacketed batch reactor
The discrete form of equation (3) for the k th time interval is implemented for the on-line control which is given by: k
Tj (k) = Tr (k) + (WrCp/UA) {Kl [Trsp - T~ (k)] + K2 Y~[T~p- Tr (k)] At} - Qr/UA
(4)
where At is the sampling time. However, equation (4) gives the actual jacket temperature, Tj(k) which is not the jacket temperature set point, Tjsp(k), needed to control the reactor temperature at its set point Trsp. It is reasonable to assume that the dynamics of the jacket temperature control are approximately first order [8] with time constant ~.i and hence, the Tjsp can be further calculated using the following equation: Tjsp(k) = Tj(k-1) + ~:j [Tj(k)- Tj(k-1)]/At
(5)
3. ON-LINE HEAT-RELEASE ESTIMATION USING NEURAL N E T W O R K METHOD
The success of the GMC controller as formulated in equation (4) is largely dependent on the ability to measure, estimate, or predict the heat-release, Qr at any given period of time. As Neural Networks have been proven to be an accurate and fast on-line dynamic estimator, it is used to carry out the task in this work [9]. A multilayered feedforward network (3 layers with hidden (middle) layer consisting of 20 nodes) is used which is trained using the backpropagation method. Since the process being studied is a dynamic system, it is necessary to feed the Neural Network with past historical data. Here the input layers consists of the present and past value of Tr (Tr(k-2), Tr(k- 1), Tr(k)), Tj (Tj(k- 1), Tj(k)) and the past value of Qr (Qr ( k - 1)) and the output layer estimates the present value of the heat-release, Q~.
178 With the 6 inputs, the Neural Network is trained through forward modelling methodology to get the value of the output i.e. present value of Qr. The input-output map for the Neural Network training can be seen in Figure 2. Here the Neural Network is placed in parallel with the estimator for Qr and the error between them and network output (i.e. prediction error) is used as the training signal for the Neural Network [10]. The estimated, Qr is then used in equation (4) to estimate the value of Tj. This strategy outlined above is then applied to the reaction scheme as described below. 4. E X A M P L E
Here the reaction scheme is same as that used by Cott and Macchietto [3], which is: A + B --) C; A + C --) D; where A and B are raw materials, C is the desired product and D is the waste product. The model equations for the reaction can be written as: dMA/dt = - klMAMB - k2MAMc
(A1)
dMB/dt = - klMAMB
(A2)
dMc/dt = + klMAMB
-k2MaMc
(A3)
dMD/dt = + kzMAMc
(A4)
dYr/dt = (Qr + UA(Tj - Yr))/(CPAMA + CpBMB + CpcMc + CpDMD)
(AS)
dTj/dt = (Tj sp - Tj)/'cj - U A ( T i - Tr)/VipjCpj
(A6)
kl = exp (kl I - kl 2 / (Tr + 273.15))
(A7)
k2 = exp (k21 - k22 / (Tr + 273.15))
(AS)
Qr =-AH1 klMAMB- AH2 k2MAMc
(A9)
where MA, MB, Mc and Mo are number of moles of component A, B, C and D respectively; kl and k2 are the rate constant for reaction 1 and 2 respectively. All the parameter and constant values used in the model and in equations (4-5) are: CpA=18.0kcal/kmol~ CpB=40.0kcal/kmol~ Cpc=52.0kcal/kmol~ CpD=80.0kcal/kmol~ AH1 = -10000.0kcal/kmol, AH2= -6000.0kcal/kmol, kl 1 = 20.9057, kl 2 = 10000, k21=38.9057, k22= 17000, Cp-0.45 kcal/kg~ U=9.76kcal/min.m2~ A=6.24m 2, Vj=0.6912m 3, pj=1000.0kg/m 3, Cpj=0.45kcal/kg~ Wr=1560.0 kg, At=0.2 min, zF3.0 min 4.1. Results First, an off-line dynamic optimisation problem is solved to find the optimum temperature profile that will maximise the product "C" and minimise the by-product "D". Two runs were carried out; RUN1 uses one control interval (time) and RUN2 uses three fixed control intervals. The batch time is 120 minutes and the initial values of [MA, MB, Mc, MD, Tr, Tj] are [12.0, 12.0, 0.0, 0.0, 20.0, 20.0] respectively. The reactor temperature is used as the controlled variable and is bounded between 20 to 100~ The manipulated variable, Tj is bounded between 20 and 120~ The model in the dynamic optimisation problem does not require Equation A5, A6 and A9 to be used.
179 Table 2 Summary of the results Off-line Optimum Temperature Profile
Run
Temperature,~
GMC Tuning Parameters Kl(min -1) K2(min -2)
Off-line Product Mc
On-line Product Mc
6.5126
6.3294
0.21
1.11E-4
6.5171
6.3330
0.21
1.11E-4
92.46
I
I
Switching time, min t = 0 120.0 Temperature,~ 92.83 91.17 93.41
120.0
t=0 40.0 80.0 Switching time, min
140 ]
1800 1600 1400 '] 200
120 100
0"80
~
60 40
Tr Trsp
...............
20 0
I
20
40
I
60
I
1
I
80 100 120
800 400 200 -_,j 0
1
I
I
I
I
I
20 40 60 80 100120 t(min)
t(min)
Figure 3: GMC response with 1 step
-
1000
600
....... TTisp I
-
Trsp
Figure 4: Performance of heat-released estimator with 1 step Trsp
The results (optimal temperature profiles) are shown in Table 2 for both runs. It can be seen that by using three control intervals, the product achieved can be slightly improved. GMC controller (equation (4)) is now used in order to track the optimum temperature profiles achieved in the optimisation stage. Figures (3-4) show the response of the GMC controller in tracking the set points (Trsp) and the performance of the Neural Network in estimating heat-released for Run 1 and Figures (56) show those for Run 2. It can be seen that the GMC controller was able to accommodate very well both constant and dynamic set points although was a little sluggish for the latter. This is due to the use of the same tuning parameters for both runs. Again Figures (4) and (6) show that Neural Network gives good estimation for the heat released by the reaction. Also Table 2, shows that for both runs the amount of desired product obtained on-line (after implementing GMC controller) were within only 4% of that obtained by off-line dynamic optimisation. This clearly shows the effectiveness of implementing the GMC controller combined with the Neural Network estimator.
180
140 ]
1800 1600 -: 1400 9 200 000 800 600 400 200 0
120 100
~'8o60 -
'"'r r
40-
....
r
20-
. . . . . . . Tj
~Tjsp
0
I
0
I
20 40
I
60
I
I
I
80 100 120
/~
--
Actual
I
9
t(min)
Figure 5: GMC response with 3 steps
Trsp
Figure 6: Performance of heat-released estimator with 3 steps Trsp
5. C O N C L U S I O N S Off-line dynamic optimisation has been carried out to obtain the optimum temperature profiles to maximise the desired product in batch reactors using a complex reaction scheme. The GMC controller was then implemented to track these optimal reactor temperature profiles. Neural Networks have been used as the heat-release estimator within the GMC formulation. In the example presented, the GMC coupled with the Neural Network based estimator has been found to be effective and robust in tracking the optimal reactor temperature profiles to keep the desired product close to the target values. ACKNOWLEDGEMENTS
The Fellowship support to N.Aziz from the Universiti Sains Malaysia and the UK Royal Society support to M.A. Hussain are gratefully acknowledged. REFERENCES
1 J.S. Logsdon and L.T. Biegler, Comput. Chem. Engng., 17 (1993) 367. 2 R. Luus, J.Proc. Cont, 4 (1994), 218. 3 B.J. Cott and S. Macchietto, IEC Res, 28 (1989), 1177. 4 P.L. Lee and G.R. Sullivan, Comput. Chem. Engng., 12 (1988), 573 5 L.S. Kershenbaum and P. Kittisupakorn, Trans IChemE, 72 (A) (1994), 55. 6 N. Aziz and I.M. Mujtaba, IChemE Advance in Process Control Conference V, 2-3 September, 1998. 7. N. Aziz and I.M. Mujtaba, submitted to AIChEJ, 2000. 8. G. Liptak, Chem. Engng., May (1986), 69. 9. M.A. Hussain, Artificial Intelligence Eng., 13 (1999), 55 10. I.M. Mujtaba and M.A. Hussain, Comput. Chem. Engng., 22 (1998), $621.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
181
Stability analysis of delayed chemical systems L. Pellegrini a, M. Ratto b and M. Schanz c aDept of Industrial Chemistry and Chemical Engineering "G. Natta" - Politecnico P.za Leonardo da Vinci, 32 - 20133 Milano- Italy bDept of Environmental Engineering- Universith di Genova Via Opera Pia, 15 - 16145 Genova - Italy cIPVR- University of Stuttgart Breitwiesenstrasse, 20-22 - D-70565 Stuttgart- Germany A CSTR with a time delay XD in the control action is studied. The stability analysis of the infinite dimensional system is performed by means of an analytical approach that allows describing the Hopf locus. The presence of a critical XDvalue is emphasised. 1. INTRODUCTION Time delay plays an important role in simulating the dynamical behaviour of nonlinear feedback systems. The few papers in chemical engineering literature on systems, which are influenced by the so-called history or memory effects, agree on the destabilising effects of the time delay. Pellegrini et al. (1995) pointed out the role of the time delay for two examples by comparing the dynamics of the delayed and undelayed system. The simulation of a delayed dynamical system means a remarkable numerical and computational effort; it requires the approximation of the continuous evolution of the infinite dimensional system by a finite number of elements whose value is considered at discrete times. In this way a continuous infinite dimensional dynamical system is replaced by finite dimensional iterated mapping. Pellegrini et al. (1999) performed a bifurcation analysis for a delayed feedback controlled CSTR following the technique first utilised by Soliman and Ray (1972) and applied again by Boe and Chang (1989): the infinite dimensional delayed system is reduced to an infinite dimensional undelayed system by adding a hyperbolic Partial Differential Equation. In the present paper the stability analysis is tackled by directly considering the original delayed equations. This leads to a trascendental characteristic equation for the eigenvalues according to the method described by Wischert et al. (1994).
2. THEORY Considering a n-dimensional dynamical system with a constant time delay rD and assuming that, in the problem under study, only one time delay is physically relevant, the following generic system has to be analysed:
182 dx(t)
= f(x(t),u(t-
%))
x ~ ffln,u e fil m m < n
dt
(1)
where x is the vector of phase variables and u is the vector of the components ofx at time t-rD. The function u(t) is given for the interval [to - % , t o ]. In particular, if the functionfis linear in u ( t - z-D),it can be expressed as:
dx(t) dt
= A x ( t ) + B x ( t - rz~) + g ( x ( t ) )
A , B e fit n•
(2)
where g(x(t)) contains only non-linear terms and the matrix B can be singular, if m
= J t 2 ( t ) + J~Y,(t - r D) J , J ~ e 9t n•
dt
(3)
where Jt and Jr contain the components of the Jacobian matrix linear in x(t) and x(t-rD) respectively: Jt=A+
Og, Oxj
Jr=B
(4)
By doing the following ansatz (5)
Y(t) = Meht2o,
where M is a non-singular square matrix and A is the eigenvalue matrix, and considering the linear Eqs (3), the following linear system in the unknowns 2 0 is directly obtained, A - M - l J t M - M - 1 J ~ M e -A~o )2 o = 0
6)
The system has to be verified for any value of the initial conditions 20 ~0; hence the characteristic equation is obtained as det(A - M -1Jt M - M -1J~Me -A~ )= O.
(7)
Since M'IJtM and M'IjrM simply describe the transformation of the matrices from one basis in the phase space to another and being the determinant invariant with respect to the basis, the eigenvalues are obtained by solving d e t ( M - J *)= 0
with
J * = (Jt + J~ e-a~~ ) .
(8)
183 3. A P P L I C A T I O N TO C O N T R O L L E D CSTRS
The equations describing the CSTR with a delayed PI control device can be written as"
d~
- - = -~: + Da(~9)(1 - ~') dt d3
~=
dt dcp
(9)
~9os - kp ( ~ 9 ( t - z"D ) - t g s ) -
k, Tqg(t- r D ) + NL9 e
-(1 +N ) O + D a ( 3 ) ( 1 - ~ )
~=~9-3,
(10) (11)
dt
where the notation and the operating conditions are those adopted by Pellegrini and Biardi (1990): 4 - CAo - cA
-
CAo
v = - -V
t=- i
Q
r
N = ~UA
ATa=
Q,o%
- Z~Hr
cA~ pCp
, 9 T= -- ~Tre f AT~
(12)
and (13)
Da(~9) = k ~ exp - R(Eel + A T . 3 )
The PI controller acts on the inlet temperature in order to keep the reactor temperature at the set point: ~90(t) = ~9os-k0(61(t- rD)-61s)-k.r~o(t- rD),
(,o(t -- rD) = ~-~ (~9(T) -- 3,)d7
(14)
An in depth bifurcation analysis has been performed on the above system without time delay. Pellegrini and Biardi (1990) studied the undelayed system showing the destabilising effect of the integral control mode, while Giona and Paladino (1994) obtained the analytical expression of the Hopf locus for the case of a genetic reaction of order n. Pellegrini et al. (1997) performed a high codimension bifurcation analysis for the ideal CSTR and for the Lo-Cholette model of a real agitated reactor (Pellegrini and Tablino Possio (1996)). Ratto (1998) studied the effects of noise in the control system on the steady state behaviour, while Paladino and Ratto (2000) considered a sensitivity analysis of the Hopf locus with respect to model and parameter uncertainty. The system with time delay in the control action was then analysed by Pellegrini et al. (1999) from a numerical point of view. This delayed system of differential equations has a unique singular point, represented by the set temperature of the controller. The local stability analysis in the neighbourhood of the unique steady state (~s,~gs,q)s) can be performed as described in the preceding section, defining:
a12 -a l 0
- (kee -~
+ N +
1
0 1) + al2
- kIV. e -~~
0
/ (15)
184 The coefficients of the characteristic equation are not constant but depend exponentially on the eigenvalues and the characteristic equation can be expressed as: 2,3 - tr(J*)2, 2 + m 2 (J*)2 - det(J*) = 0
(16)
In the control gains plane, the stability region is delimited by the k i t = 0 axis, where det(J*) changes sign. Another boundary is given by the Hopf bifurcation locus; for a 3D system without delay, the Hopfbifurcation condition is given by: det(J) = m 2( J ) , with the constraints tr(J) < O, m2(J) > O, det(J) < 0 (17) tr(J)
from which the analytical expression of the Hopf locus can be obtained (Giona and Paladino, 1994). In the case of a delayed system, such a kind of condition cannot be obtained, since a delayed system is infinite dimensional, with an infinite number of eigenvalues. Hence, the general definition has to be considered. Starting from a stable equilibrium point, a Hopf bifurcation occurs when a conjugate pair of eigenvalues crosses the imaginary axis, while for every other eigenvalue the constraint 9~(2)<0 holds. The relevant locus can be obtained by simultaneously solving the following system of equations in the two unknowns .~(2) and p (p is a bifurcation parameter, i.e. kp, kit, or zo):
~.~2(~,). [a12
_
a21 _ 2 - N - kp cos(rz).~(2)) ]
+ .~(2)[kp (1 + a2, ) + k,r]sen(%.~(2)) + (1 + az,)k,r, cos(%.~(2)) = 0 -- 3 3 (/%) "l- .~j2 (/],) . [kp
sen(ro..~(~,)) ]
+ .~(2)[(N + 1)(1 + a21 ) - al2 + (kp (1 + a21 ) + kit) cos(%.~(2))] -
(18)
(19)
(1 + a21 )k~r-sen(%.~(2)) = 0
3.1. D e t e r m i n a t i o n o f t h e c r i t i c a l rD at f i x e d
ke, k t r
To compute the critical rz) value, it is useful to apply the following change of variables: x = cos(%.~(,t)) y = sen(%.~(2)) (20) So, starting from Eqs (18) and (19), the following equation in the unknown .~(2) is obtained: (AzC 1 - A, C2) 2 + ( A 1 B 2 - AzB,) 2 - ( B , C
2 - BzC1) 2 =
f(.~(2)) = 0
(21)
The equation (21) is a polynomial of 10th order, with only even indices, i.e. the function f(.~(2)) is even. If kit>0, Eq (21) has only one pair of real roots +.~c(2). Hence, x and y can be computed as functions of .~c(2) and the critical value of the time delay derives from Eqs (20). Among the infinite periodic solutions of Eqs (20), the minimum z'D value has to be taken, i.e. the solution to Eqs (21) in the interval 0< rD.~c(2)<2n has to be considered. In Figure 1, the Hopf loci in the rD-ke plane are shown for different values of the dimensionless integral control gain kit. The Hopf loci delimit from above the stability region.
185 0.15
. . . . . .
I
,
/(,' /
0.1
.-
. . . . .
i
k r=0.01 / - ; ~ .......\ I , ; ~1r=0.7 ~, i/
/
/ / // //' // /! :i
-
,
'\\
k I r=l
//'
.,
,/
~,'\,
iI /
0.05
//
I
0 10 -1
!
/i i
. . . . . . .
/k
/' /
I
fi
.,i
i
,
i
'
~'
/
/
/
i
<>
t r=lO
~, "%.
/"
/i
/ I
10 o
10 ~
10 2
km Fig 1. Hopf loci in the plane kp -
7~D
at constant ksr.
A high kit value tends to destabilise the system, as well as a too high kp value. Moreover if kpo~ the critical value ro tends to zero, i.e. a too high kp destabilises the system, whatever small the time delay in the control device. Furthermore a critical value of rD exists above which the steady state is always unstable. For the present case study, such a critical ro is about 0.15. This means that the delay in the control system cannot be larger than about 15% of the mean residence time in the reactor r.
3.2. The Hopf locus in the ke-klrplane at fixed ro To compute the Hopf loci at fixed z'D, it is useful to define the variable z=rD,~(2). Since the integral control gain can be explicitly expressed as a function of z, the equation
G(z, kp,%)=O
(22)
can be solved for the unknown z for any kp at fixed values of z-D. The equation (22) has infinite solutions, due to the presence of the trigonometric terms. The critical zc value at the Hopf bifurcation is found by considering the lowest root of the equation. The results of such computations are shown in Figure 2, where the Hopf loci in the kzz-kp plane at fixed values of rD are shown. The stability region with a null time delay, corresponding to the ideal controlled CSTR, is an open set above the Hopf locus, which is represented by a quadratic function; therefore, in the ideal case, if kp is sufficiently large, the steady state is stable for any value of kit. On the other hand if a non-null time delay is considered, whatever small the rD value is, the stability region becomes a closed set, which is delimited by the kp axis and by the Hopf locus; such a behaviour couldn't be detected with the numerical approach (Pellegrini et al. (1999)). This means that, unlike the undelayed system, sufficiently high kit or kp values always exist, which are able to destabilise the system. The present results reveal the upper
186 10
3
10
2
.... z-o =0.01
......
10
1
0
10
~ 10
" , "i" ,
/
/ ,., //
............................................................................................................................................................................................ -~ i . - f ";.i--:.::/--~...." -'k --'iCT-iii-;-7'L~:;S:j
r o =0.13 10
"
-~
10
o
kl~"
r o =0
10
5
Fig 2. Hopf loci in the plane kiT- kp at constant rD limits for the control gains which are not put in evidence by the analysis of the ideal model, while encountered when any non-ideality is considered: time delay, non-perfect mixing, etc. REFERENCES
-Boe, E. and Chang C.H., Chem. Eng. Sci, 44, 1281 (1989). -Giona, M. and O. Paladino, Bifurcation analysis and stability of controlled CSTR, Comput. Chem. Engng 18, 877-887 (1994). -Paladino, O. and M. Ratto, Robust stability and sensitivity of real controlled CSTRs, Chem. Engng Sci. 55, 321-330 (2000). -Pellegrini, L. and G. Biardi, Chaotic behaviour of a controlled CSTR, Computers Chem. Engng 14, 1237-1247 (1990). -Pellegrini, L., Biardi G., Tablino Possio C., Albertoni G., The role of time delay on the stability of chemical system, Proceeding of the Conference Chaos and Fractals in Chemical Engineering, 227, World Scientific, Singapore (1995). -Pellegrini, L. and C. Tablino Possio, A non-ideal CSTR: a high codimension bifurcation analysis, Chem. Engng Sci. 11, 3151 (1996). -Pellegrini L., C. Tablino Possio, G. Biardi, An Example of how Nonlinear Dynamics Tools can be Succesfully Applied to a Chemical System, Fractals, 5, 531-547 (1997). -Pellegrini, L., M. Ratto, O. Paladino, Bifurcation diagram of delayed chemical systems, Proceedings of the II International Conference on Tools for Mathematical Modelling, St Petersburg, june 14-19, 1999. -Ratto, M., A theoretical approach to the analysis of PI controlled CSTRs with noise, Computers Chem. Engng 22, 1581-1593 (1998). -Soliman, M. A. and Ray W. H., Int. J. Contr., 1972, 15, 609. -Wischert, W., Wunderlin A., Pelster A., Olivier M., Groslambert J., Physical Review B, 49,203 (1994).
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
187
N o n l i n e a r m o d e l b a s e d control o f optimal temperature profiles in polystyrene p o l y m e r i z a t i o n reactor G. Ozkan a, S. Ozen b , S. Erdogan b, H. Hapoglu a and M. Alpbaz a Department of Chemical Engineering, Ankara University, Tandogan 06100, Ankara, Turkey bDepartment of Chemical Engineering, Gazi University, Maltepe, 06570, Ankara, Turkey E-mail: [email protected] a
In this work, nonlinear model based control was applied to the free radical solution polymerization of styrene in a jacketted batch reactor and its performance was examined to reach the required monomer conversion and molecular weight. Optimal temperature profiles for the properties of polymer quality were evaluated using the Hamiltonian optimization method. Total simulation program having mass and energy balances of the jacketed polymerization reactor was used to calculate the optimal trajectories. For control purposes, several experimental and theoretical dynamic studies have been made to observe the validity of simulation program. Expefimemal and theoretical nonlinear model based control have been investigated to track the temperature at the optimal trajectory Two types of parametric and nonparametric models were evaluated to achieve the temperature control. For this purpose, reaction curve was obtained to calculate the system dynamic matrix as a nonparametric model. In all control work, heat input to the reactor was chosen as a manipulated variable. NARMAX (Nonlinear Auto Regressive Moving Average eXogenous) giving a relation between heat input and reactor temperature was chosen to represent the system dynamic and this model was used to desing the related control system as a parametric model. NARMAX model parameters were determined by using Levenberg Marquard aigorittun. A Pseudo Random Binary Sequence (P.R.B.S.) signal was employed to disturb the system. Total simulation program was used to calculate the system and control parameters. Several types and orders were used to construct the NARMAX models. The efficiency and the performance of the nonlinear model based control with the NARMAX model and dynamix matrix were tested to calculate the best model. Nonlinear model based control system was used to control the reactor temperature at the desired temperature trajectory experimentally and theoretically. Theoretical simulation results were compared with experimental control data. It was concluded that the control simulation program represents the behavior of the controlled reactor temperature well. In addition, nonlinear model based control keeps the reactor temperature of optimal trajectory satisfactorily. 1. INTRODUCTION Significant improvements on polymer plant operation and economics can be obtained by the development and application of state estimation, process optimization and model-based
188
predictive control. In fact, polymerization reactor control is difficult due to nonlinear timevarying dynamic behaviour and the lack of the on-line process sensors to measure polymer properties. In recent years, there is considerable literature on the computer control of reactors and control algorithms [1,2]. For industrial application and academic studies DMC algorithm was used [3]. Most of the model based control techniques are based on linear model and they are not able to control effectively and precisely the nonlinear systems within a very wide range. For this reason, recently the studies to extend model based control techniques to nonlinear systems have increased [4]. In this study, nonlinear model based control method is implemented to control the temperature of a jacketed batch reactor in which styrene radical solution polymerization takes place. The optimal conditions for the case of one-charge of initiator addition to achieve a predetermined final monomer conversion and desired number average chain length were computed by using mathematical model equations derived for the styrene polymerization reactor system. The polymerization of styrene proceeds a free-radical mechanism under normal conditions. Then, nonlinear model based control of this system under optimal conditions found for different initial initiator concentrations was investigated experimentally and by simulation. Heat given to the reactor was taken as manipulatedvariable. Different models were tested. For this aim, P.R.B.S. signal was applied to the manipulated variables, NARMAX model was developed and model parameters have been obtained using Levenberg Marquardt method. In theoretical work, simulation program has been used for dynamic analysis and to obtain reaction curve. Nonlinear model based control algorithm was applied to keep the reactor temperature at optimal trajectory. Experimental conversion and number average chain legnth were compared with optimum values. 2. NONLINEAR MODEL BASED CONTROL ALGORITHM The Extended Dynamic Matrix Control algorithm (EXDMC) described by Peterson et al. [4]is used for predictive control purposes. This extension of the linear DMC algorithm [5] was developed for the use of nonlinear model based control. The linear estimates of the system s future outputs are based upon the known past inputs J"~' and the sytems s dynamics as lin yp~t y = + A A u +d (1) Where A is the dynamic matrix which consists of the step response coefficients, A u is future control moves and d is the disturbance effects due to modelling and measurement errors on the output. In linear model based control, the vector d is assumed to remain constant over the prediction horizon. To calculate the control inputs an objective function which minimizes the square of the output error over the time horizon is taken. The nonlinear model based control algorithm can be expressed by the following equations, x =x"~'+ [b,+,_~-b,~](AU)k +~d"'(k+i), x =x +~ia.k(AU):" (2) - N~+ i
"
k=O
old
-
cL,'~
oL,~?
=
t
The optimal control increments are conaputed as Au = (ATA + f2I)-~AT( x - d ' ) For nonlinear model based control, NARMAX model is given as y(k)= a~ka(k-1)+ a2ku(k- 2)+...+ a,,c~(k- N - 1)
(3) (4)
189
NA ~RMAX
Ill , I
,
ii
i,
i ,i
i
loa~
I I
il
~,---~O-~lMod~
,
t-i
I ul
II
'.: : .
.
.
'.
,
.
Fig 1. Nonlinear Model Based Control System In nonlinear model based control algorithm, an extended linear model based control is defined. There is a new disturbance vector d which contains contributions from the nonlinearities of the system as d 'a and external disturbances as ar x. So, the estimates of the future outputs over the prediction horizon can be written as y,,lDp,,,t + A A u + d '~t+d "1 (5) Where arxt is assumed constant and ar is varying from one operating point to another over the prediction horizon. The purpose here is to obtain the vector dz. One method of solving nonlinear equations of d "t is the fixed -point algorithm [5,6] as d I i+~ = d l i + fl(Y i "t - y i et) (6) Here, l is the iteration number and fl is a factor used to enlarge the region of convergence. The flow chart for the nonlinear model based control algorithm which uses the fixed-point algorithm is given in Figure 1. The control action a u ( k ) at the increment of time in which convergence is viewed is implemented to the plant and then iterations on nonlinear model based control calculations are repeated at the next increment time. 3. EXPERIMENTAL SYSTEM A schematic diagram of the batch reactor system is shown in Figure 2. Experiments were carded out according to the following main steps: firstly, initial mixture was preheated from ambient surrounding temperature to the desired reaction initial temperature precalculated. Then, the mixture temperature was maintained at the predefined temperature profile by the implementation of the control methods to the reactor control system. 4: RESULTS AND DISCUSSION This work provides theoretical and experimental studies to obtain the dynamic behaviour of the jacketed batch reactor. Also, model and control parameters for the nonlinear model based control method were evaluated. In the firstpart of the study, experimental reaction curve was
190
86
G %
85
84 _
83
.... ' 82
,,
~}oop NARMAX
~',,
~',,
~',,
Time
Fig. 2. Exp~'nental system
Table 1. The experimental conditions Run Mo(mol?) m~ (ml/min) i 6.092 0.5 2 6.092 0.5
F i g 3. O p e n - l o o p
4',,
Model
~',,
~,,
(rain)
tern p e r a t u r e
response
L (mol/lt) Tf (min) T R(o C) 0.0125 127 97 0.0185 222 89
obtained using the open-loop step test. For this aim, styrene-toluene mixture in the reactor was heated with an immersed heater. At the same time, the cooling water was passed through the reactor cooling jacket. In this case, the polymer reactor which is a mixing chamber have been considered to be continuous in terms of energy, because the heat given by the heater was absorbed continuously by the cooling water. Therefore the system can reach the steady state condition with defined values of heat input and cooling flowrate (Table 1). The heat released during the reaction was taken as a disturbance for this system and nonlinear model based control change in the heat input (AQ = 11.39W) was applied after the system reached to the steady state conditions. The temperature response to this effect obtained from theoretical simulation and experimental work are given in Figure 3. The dynamic matrix A a n d other parameters for model based control were obtained from this curve. In all control works, the amount of heat from the immersed heater was taken as a manipulated variable. In the second part of this study, the calculation of the dynamic behaviour of the system and the evaluation of the NARMAX model parameters was carried out using total simulation program having mass and energy model equations. The estimated equivalent equation for this system was found as y(t) = -c0Y(t - 2) - clY(t - 1) + c2u3(t - 1) (7) The coefficients of this equation were obtained in open-loop using a pseudo-random binary sequence (P,R.B.S) as the input function and also Levenberg Marquard algorithm. In this work, the P.R.B.S generation procedure with fourtythree was used. The magnitude of the P.R.B.S. and the rate of sampling were determined by a knowledge of the process dynamicsand the behaviour of the control element. The coefficients were founds as co = - 0.9024, cI = - 0.7025 xl0 -l c2 = 0.7102 xl0 11 . The open-loop step responses of the actual and simulated plant and estimated
191 NARMAX model are presented in Figure 3. The agreemem among the temperature responses obtained tu the experiment data, total simulation and estimated model is sufficiently good. The third part of this study deals with the theoretical and experimental application of nonlinear model based controller to the free radical solution polymerization of styrene in a jacketed batch reactor. The best values of the control parameters were determined from the total simulation program, and then used in the experimental control studies. The parameters of model based control system for the implementation of this algorithm were found as NP--4, NC=l and f=0. The experimental operating conditions are summarized in Table 1. Because of complex dynamic behaviour or gel formation at high conversion levels, the control objective of reaching 50 % monomer conversion and 500 average number of chain length in a minimum time was selected in this work. Nonlinear model based control results of the polymerization reactor at optimal temperature profiles are plotted in Figures 4 and 5. These figures represent the temperature and heat variations with time for two different initiator concentrations and different desired properties. Also, the calculated temperature profiles under control-free conditions are given to see the performance of nonlinear model based controller. The results obtained showed that temperature remains very close to its desired trajectory with some fluctuations at the beginning of the reactions. These fluctuations become less significant as optimal temperature increases. Experimental, theoretical and desired monomer conversions and average number of chain lenghts for both I0= 0.0125mol/1 and I0= 0.0185mol/1 are given in Table 2. As it is seen, the experimental conversion data agree well with the desired optimal values. For the number average chain length, the agreement between experimental and desired values is very well for I0=0.0125 mol/l. But, the experimental value for the number average chain length at the end time for I0= 0.0185mol/1 is higher than the desired. This result can be attributed to a longer polymerization time and more solvent evaporation then previous one during this time.
~3o ~ 120 ] ] ~" 115 ~_ i ~ | I ~ 110 ~- ,
" ..... .......... ~
f',,
L i ] !,1
..........."
T heoretical Control ------ Dvnamic Simulation ........ O ptim al Profile Experimental Control
120 o~ / 4 ~ 1 1 0 [-- ] ~100 ~!
Theoretical Control Dynamic Simulation Optimal Profile Experimental Control
~' 105 1oo 95 90
80 0
20
40
60
80
I00
120
Time (rain)
F i g . 4. Dynamic, experimental, simulation nonlinear model based control of the reactor temperature profile (10= 0.0125mol/1)
0
40
80
120
160
200
Time (rain)
Fig. 5. Dynamic, experimental, simulation nonlinear model based control of the reactor temperature profile (I0= 0.0185mol/1)
192 Table 2. Experimental results at the optimal temperature profiles Io .... (mo!/1) ... 0.0125 0.01.85
Optimal Values m . xa* 50 500 50 500
Experimental Results m x, 44.83 501.2 54.25 745.9
5. CONCLUSION A Nonlinear model based control algorithm was applied to a jacketed batch reactor for tracking a precalculated optimal temperature profile. The efficiency and the performance at different conditions were observed well with the NARMAX model constructed. The NARMAX model parameters were determinedby using Levenberg Marquard algorithm. The experimental study shows that the temperature can follow satisfactorily optimal trajectories and nonlinear model based controller is acceptable with respect to changing system parameters, Computer simulation results also were found in good agreement with the experimental results. Because most chemical processes in industry have nonlinear natures, the implementation of the nonlinear model based controller at the industrial scale is promising when compared to linear controllers. 6. NOMENCLATURE A : dynamic matrix vector Mo : initial monomer concentration, mole/1 b/. a1 9element of dynamix matrix re,m* : conversion, desired conversion c~ : parameters of NARMAX model m~ : coolant flow rate, kg/sec d : disturbance effect T : reactor temperature, ~ /0 : initial initiator concentration mole/1 Y :output variable AU : vector of manipulated variable Au : control output Xn,Xn*: number average chain lenght, desired number average chain length w
REFERENCES 1. S.R. Ponnuswamy, S.L. Shah, C.A. Kiparissides, Compmer optimal control of batch polymerization reactors, Ind. Eng. Chem. Res. 26 (1987) 2229-2236 2. G. Ozkan, H. Hapoglu, M. Alpbaz, Generalized predictive control of optimal temperature profiles in a polystrene polymerization reactor. Chemical Eng. and Processing, 37 (1998) 125 3. S.Ytice, A. Hasaltun, S. Erdogan, M. Alpbaz, Temperature control of a batch polymerization reactor, Chem. Eng. Res. Des. 77(1999) 413-420 4. T. Peterson, E. Hemandez, Y. Arkun, F.J.Schork, Nonlinear predictive control of a semi batch polymerization reactor by an extended DMC. Proc. American Control Conference, (1989) 1534 5. E. Hemandez and Y. Arkun, Control of nonlinear systems using polynomial ARMA models. AIChE, 39 (1993),446-451 6. A. Draeger, S. Engell, Nonlinear model predictive comrol using neural net plant model, Proc. the NATO advanced study, Turkiye (1994)627
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
193
Experimental Verification and Optimisation of a Detailed Dynamic High Performance Liquid Chromatographic Column Model H.K. Teoh,1)M. Turner,~)N.Titchener-Hooker 2) and E.Sorensen 1) 9 1) Dept. of Chemical Engineering 2) Dept. of Biochemical Engineering University College London, Torrington Place, London, WC1E 7JE, United Kingdom A detailed mathematical model of chromatographic separation processes is of crucial importance due to the increased application of this separation technique in the downstream processes of the biotechnological, pharmaceutical and fine chemical industries. In this paper, the development of a detailed dynamic model of an HPLC unit based on the equilibrium-dispersive approach and its verification by comparison with real experimental data, are described. Good agreement between the simulated elution profiles and the experimental data was obtained. Using this model, optimisation of chromatographic separation was conducted on a closed-loop recycling preparative chromatographic unit. It was found that the effective column length can be increased by recycling the feed sample back to the column for further purification which may improved purity and yield of the process.
1. Introduction Chromatographic techniques, or analytical chromatography, have been used routinely for chemical analysis since the 1950s and for automated analysis of process streams in process control since 1961. Large scale, or process chromatography, is a relatively new entrant to the range of separation unit operations available to the process engineer. Though to the best of our knowledge, detailed dynamic models of process scale high performance liquid chromatographic (HPLC) units, verified against real scale data, have so far not been presented in the open literature. The main objectives of this paper are therefore: 1) to develop a detailed dynamic model of a HPLC unit based on the equilibrium-dispersive modelling approach, 2) to verify the model by comparison with real experimental data and 3) to examine the use of the model for the optimisation of the performance of a HPLC unit by operating using recycling.
2. Equilibrium-Dispersive Model In this section, the equilibrium-dispersive model on which this work is based will be described [4]. The concentration in the mobile and stationary phases are considered to be in equilibrium in the whole column. This assumption is valid as long as the column efficiency is greater than several hundreds of theoretical plates, which is the case in high resolution process chromatography. The contributions of axial dispersion and possible additional mass resistance that cause band broadening can be described through an apparent dispersion coefficient, Dap,i. There are no radial concentration gradients and no temperature gradients in the column. *Author to whom correspondence should be addressed: Fax: +44 20 7383 2348; Phone: +44 20 7419 3802;
email: [email protected]
194
2.1 Model Equations Equations 1 is the differential mass balance for component i of a volume element, 5x, in the column, qi and Ci are related through the adsorption isotherm, in which qi and Ci are the solute concentration in the mobile and stationary phases accordingly. The phase ratio, F, is defined as the ratio of the volumes of the solid and liquid phases in the column expressed as a function of the total porosity, CT, i.e. F = 1-~T ~T OCi
~t
OC~
O---t + F
-k- u " ~ x
0 2C~
i - 1 N
(1)
- - D ap ,i O x 2 ,
The apparent dispersion coefficient, Dap,i can be estimated from the number of theoretical plates of the column, Np ,i (Dap,i -- 2u~,~ uc ) [4]. The initial and boundary conditions are given in Equations 2, 3 and 4. CE(t) is the injection profile which is usually rectangular in shape. The total separation time is equal to t y = tin j q- t e.
@(t0,~)
-
0
qi(to, x)
-
0
c~(t, zo)
-
{ c~(t) t < t~j o
oc~ Ox
(2)
~=L --
0
t~ <_ t
(Injection) (Elution)
(3) (4)
2.2 Isotherm and UV Adsorbance's Correlations The simple Langmuir isotherm was employed to describe the relationship of the solutes between the stationary and mobile phases (Equation 5). ai and bi are the Langmuir isotherm's numerical coefficients for component i. The Beer-Lambert Law was used to relate the outlet concentration of component i, Cout,i, to the UV adsorbance in Equation 6 [2]. In this work, the response of the UV detector was determined to be linear in the range of experiments undertaken. UVi and Ei are the UV adsorbance and the Beer-Lambert Law's coefficient for component i, respectively.
qi
=
UVi
-
aiCi 1 + b~Ci EiCo~t,i
(5) (6)
2.3 Number and Height of Theoretical Plates The number and height of theoretical plates of the chromatographic system can be determined either by experimental results or parameter estimation. The experimental number of theoretical plates can be obtained from the chromatogram according to Equation 7 [9]. tn,i is the retention time for component i and Ai is the peak width of component i at half height. The experimental HETP or Hp,i(exp) can then be calculated from Equation 8 where L is the column length. A modified van Deemter equation (Equation 9) was used to calculate the HETP or Hp(moad) for the chromatographic system for comparison with the experimental values. The modified
195 van-Deemter equation parameters, A and C~ can be estimated from experimental data to yield a theoretical plate height, Hp(~od~t).
Np,i(exp)
=
5.54(~)2
(7)
=
Hp,i(~;)
(8)
A + C~u
(9)
L
Np,i(~p)
Hp(.~odet) =
3. Experimental Verification and Parameter Estimation In order to verify the dynamic model, an aromatic mixture of four components was considered: nitrobenzene, naphthalene, fluorene and fluoranthene. The mixture was separated using a HPLC column (4.6 X 250 mm) at different flowrates (1.0, 1.2 and 1.5 mL/min). The stationary phase consisted of 5 #m diameter silica particles. The mobile phase used was 90 % acetonitrile and 10 % water. The injected feed concentration, City,i, was 0.0004 g/L for each of the aromatic compounds. 3.1 Determination of the Experimental HETP Experimental data at different flowrates were used to calculate Hp,i(exp) for the aromatic components as a function of the linear velocity. The variation in theoretical plate height at different linear velocities is given in Figure 1. From Figure 1, we can conclude that the column used is very efficient. At high linear velocities (> 1.4e - 3 m/s), the experimental HETP or Hp,i(exp), only increases very slowly. This is particularly true for the small particle size used in this study. These results agree with those reported in the literature [9]. This is of great importance as it enables the column to be operated at a wider range of velocities. For linear velocities greater than 1.4e-3 m/s, at which the colunm is normally operated, all the reduced plate heights for the aromatic components satisfied the common rule of thumb, 2 < hi < 4 (hi - Hv,~(~v) where @ is the particle diameter). dp 3.2 Parameter Estimation Model parameters were estimated using gEST within the gPROMS package [3]. In this case, an overall HETP for all components, Hp(modei), was employed instead of individual values for Hp,i(exp). The experimental data when the flowrate was 1 mL/min was used for the parameter estimation. An orthogonal collocation finite element method was applied. Table 1 lists the estimated parameters obtained for Equations 5, 6 and 9 in which subscripts 1, 2, 3 and 4 correspond to nitrobenzene, naphthalene, fluorene and fluoranthene respectively. Figure 2 shows the simulated and experimental elution profiles for a flowrate of I mL/min. It can be seen that good agreement in terms of peak position and height was obtained. The estimated parameters were then used to predict the elution profiles for flowrates of 1.2 and 1.5 mL/min. These simulated elution profiles were then compared to the real elution profiles as shown in Figures 3 and 4. This was done to test the goodness of fit for the estimated parameters. From Figures 3 and 4, it can be shown that good agreement in terms of peak positions and height was obtained between the simulated and experimental elution profiles. Some oscillations along the time axis were observed due to the numerical method employed to discretise the model. Attempts to use a finite difference method for solution showed that
196 the numerical error produced was unacceptably large and required longer computational times compared to the orthogonal collocation finite element method (results not shown). From these results, we can conclude that the dynamic model with the estimated parameters accurately describes the column behaviour under the experimental circumstances investigated. 4. Optimisation The model was used for optimisation of a closed-loop recycling preparative chromatographic system where the aims were to investigate how improvements to the performance and yield of an HPLC system could be achieved. A binary mixture of 1-,3-,5-,Tri-tert-butyl-benzene (TTB) and 1-,3-,5-,Tri-hydroxybenzene (PHL) was considered. The detailed experimental conditions and parameters were presented by Heuer et al. [5]. Feed samples of different composition mixtures (1:1, 1:3 and 3:1) and different samples volumes (10 and 50 #L) were fed into the column. The main objective was to maximise the amount of purified product produced. To simplify the optimisation problem, only the amount of pure TFB (the least retained component) produced was optimised subject to a purity constraint. In order to investigate the effect of column length on purity and yield of the product, the feed sample was recycled back to the column for further purification. By so doing, the effective column length can be increased while keeping the matrix volume, which is a large cost item, constant. Optimal operating conditions were found for different numbers of operating cycles. Purities of 98 to 99 % was used as the cut-off points for the separation requirement. The optimum number of cycles and hence the effective column length can then be determined. Figures 5 and 6 show the yield of TI'B against number of cycles for the 50 and 10 #L samples volumes. The yields increase with the number of cycles as can be expected since the effective column length increases. Both the purity and yield increase proportionally with the amount of TI~B in the feed sample (1:3, 1:1 and 3:1 mixtures). The yields increase as the sample volumes increase from 10 #L to 50 #L. For a sample size of 50 #L, the optimum number of cycles is 4 for both the 1:1 and 3:1 mixture. For the 1:3 mixture, the optimum number of cycles is 3 as there is no feasible solution for 4 cycles (purity below 98 % constraint). It is found that it is more difficult to purify the minority component in the binary mixture. When the sample volume is 10 #L, the optimum number of cycles is 4 for all the different feed compositions. Also, increasing the sample volume while keeping the feed composition constant, will prolong the diffusive tail effect of the first component [4]. The band broadening effect increases with recycling time and, as a result, the purity of TTB decreases. As the number of cycles increases, the band broadening effect is increased but at the same time, the overlapping area between the components is reduced (not shown). This will have a counter-effect on the purity and the recovery yield of the purified products as shown in Figures 5 and 6. 5. Conclusion An equilibrium-dispersive chromatography model has been verified against experimental data based on the separation of a mixture of aromatic standards. Good agreement between the experimental data and the simulated elution profiles was obtained under different operating conditions.
197 The effective column length can be increased by recycling the feed sample back to the column which will improve the yield of the purified products. As found in this paper, careful consideration is required to decide the optimum number of cycles to be adopted as this greatly affected the yield of a recycling chromatographic separation. Future work will consider the verification of the recycle predictions at a process scale of operation.
References [1] Bellot J.C. and J.S. Condoret, Process Biochemistry 26 (1991), 363. [2] Boyer R.F. Modern Experimental Biochemistry, The Benjamin/ Cummings Publishing Company Inc., California, 1993. [3] gPROMS Advanced User Guide: Release 1.7, Process System Enterprise Ltd., UK, May 1999. [4] Guiochon G.,S. Golshan-Shirazi and A.M. Katti, Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, Boston, 1994. [5] Heuer, C., A. Seidel-Morgenstern and R Hugo. Chem. Eng. Sci. 50 (7),(1995) 1115. [6] Heuer C., R Hugo and A. Seidel-Morgenstern. Separation Sci. & Tech. 34 (2),(1999) 173. [7] Heuer C., R Hugo, G. Mann and A. Seidel-Morgenstern. J. ofChrom. A 752 (1996), 19. [8] Katti A., J. Huang and G. Guichon, Biotech. & Bioeng. 36 (1990), 288. [9] Snyder L.R. and J.J. Kirkland, Introduction to Modern Liquid Chromatography, John Wiley & Sons, New York, 1974.
2.8
x 10 -5 ,
2.6
E
2.4 2.2
-'E'. 9 I:L --r-
2
..\-\
@
1
ot--
"
\
"
b
.(3 O
1.8
c0
"o
<: >
1.6 1.4
,
cI.l') El O4
E
D_ X
,
1.5
0
'
0.5
'
'
1 1.5 Linear Velocity, m/s
'
2
'
2.5 x 10 -3
Figure 1: Height Equivalent to a Theoretical Plate against Linear Velocity for Aromatic Standard ('x'= Nitrobenzene, ' * ' - Naphthalene, 'o'= Fluorene and '+'= Fluoranthene
0.5
0
0
I
2
III III
4 Time, min
6
Figure 2: Simulated and Real Elution Profiles for the Aromatic Standard with flowrate = lmL/min (Continuous l i n e - Simulated Elution Profile and 'x' = Exp. Elution Profile)
198
IE c.-
1.5
E 1.5
I.Q r 04
@ lD O t-t~ .sL o
t43 04
@
1
co
0.5
,,~ 0.5 > Z3
> 0
III
l
. . . . . . . . . . . 2 4
III
0
0
6
1
2 3 Time, min
Time, min Figure 3" Same as Figure 2 but flowrate = 1.2 mL/min
9
II
0
__ , ~ A
^,
4
Figure 4: Same as Figure 2 but flowrate = 1.5 mL/min
Table 1" Estimated Parameters for Aromatic Standard Parameters
Estimated Values
Parameters
Estimated Values
E1 E2 Ea E4 al a2 a3
1E+4 5.9E+3 2.6E+4 2.52E+4 1.285 2.3 1.9
a4 bl b2 ba b4 A Cs
2.617 1.449E+5 2.7E+4 8.462E+3 4.75E+3 1E-5 1E-12
m
I-.I--
"5 >-
100
100
95
90:
.....O. ......
,~
O.......
0
. O ' ' "
90
9
80
m
.
-9O
85 (
O
I
I
/
/
75
/
70
/ /
60
/ /
50
/ /
40
/
1
o
->-
t
/
70
.0""
'
2
'
'
3 4 Number of Cycles
Figure 5: Yield against Number of Cycle for T r B of 50 #L sample at 98 to 99 % Purity for different feed composition ( * = 1:3 mixture, o = 1:1 mixture, x= 3:1 mixture)
'
5
30
1
|
2
a
|
3 4 Number of Cycles
Figure 6: Same as Figure 5 but for 10 #L TrB
5
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
199
Expert Control of DO in the Aerobic Reactor of an Activated Sludge Process M. Galluzzo, R. Ducato, V. Bartolozzi and A. Picciotto Dipartimento di Ingegneria Chimica dei Processi e dei Materiali, Universit?~ di Palermo, Viale delle Scienze, 90128 Palermo, Italy An expert control structure is proposed for the control of DO in a NDBEPR plant to account for the several processes that are influenced by the dissolved oxygen concentration in the aerator. In the scheme a supervisory fuzzy controller determines the set point of an inner DO control loop where an Adaptive Robust Generic Model Control (ARGMC) controller is used. The fuzzy supervisory control has a hierarchical structure. Off-line measurements of biological parameters of influent and effluent streams can be used to periodically update the set points of the fuzzy controllers. The complete control scheme has been designed for an actual plant and tested by a simulation programme that uses for the plant the n. 2 IAWQ model. Simulation and experimental results show that good operation conditions can be obtained. 1. INTRODUCTION The control of dissolved oxygen concentration (DO) in the aerobic reactor of an activated sludge process is traditionally carried out using a simple feedback loop starting from the DO measurement and using the air flow rate as manipulative variable. The set point of the control loop is normally fixed at a constant value on the basis of theoretical and heuristic considerations concerning the different biological processes that take place in the aerobic reactor. Obviously the set point value is chosen as a compromise between the various values that would be more suitable in different operational conditions. The biological nitrification, denitrification and phosphorus removal processes are strongly dependent on the concentration of DO in the aerobic reactor. In particular while the nitrification and phosphorus removal processes are aided by higher concentrations, for the denitrification process an opposite influence is found. It is therefore clear that well timed strategies in the choice of the dissolved oxygen set point can help in assuring suitable conditions for the development of the above processes. The variation of the set point of the dissolved oxygen control loop has been proposed by Olsson (1992) and applied successfully in ND (Nitrification Denitrification) plants (Nielsen and Onnerth, 1994). This strategy, if applied in a suitable way, could allow to have a more settleable biomass together with energy savings and a higher standard effluent. In fact if a lower aeration rate allows the utilization of nitrates by the heterotrophic biomass for the denitrification on the other side the lack of oxygen determines a reduced denitrification with an increase of ammonia in the effluent. In addition in NDBEPR (Nitrification Denitrification Biological Excess Phosphorus Removal) plants there is the need of assuring the presence of an electron acceptor in the final sedimentation tank in order to avoid that the phosphorus removing biomass releases soluble phosphorus in the effluent.
200 A possible control strategy for the dissolved oxygen set point should: i. lower the set point when high concentration values of nitric nitrogen in the aerator or high concentration values of carbon substrate in the aerator influent are found; ii. raise the set point when the soluble phosphorus or the NH4-N in the aerator effluent overcome limit values. In any case one should assure that in the effluent of the final sedimentation tank the soluble phosphorus and the NH4-N remain within acceptable limits since the starting of biological reactions in the sedimentation tank could prevent the success of the whole depuration process. For the implementation of a such control strategy the use of a rule-based fuzzy multilevel controller appears a good solution since it allows to express the control strategy in a linguistic way and at the same time to introduce the operator experience by heuristic rules. A fuzzy control scheme for an NDBEPR plant has been recently reported (Cassinera et al., 1998) in which the DO set point is changed using a fuzzy algorithm that receives as input variables several process variables while the DO control loop itself uses a fuzzy control algorithm. In this paper a similar scheme is proposed in which a supervisory fuzzy controller determines the set point of the DO control loop where an Adaptive Robust Generic Model Control (ARGMC) controller is used. The complete scheme, as shown in Fig. 1 can be considered as a cascade control structure in which the DO control loop corresponds to the slave controller while the task of the supervisor fuzzy controller is to keep the whole depuration process at the best operating condition, with a compromise among the several biological processes involved.
~
NH3, NO3 controller
1st level
~
~ controller 2nd level
"1 ~
Fuzzycontroller 3ra level
Adaptation L rrechanism ]"
I
-
controller [flow-rat~
Process DO
Fig. 1. The complete control structure
2. T H E S U P E R V I S O R Y C O N T R O L S C H E M E
The scheme is designed for an NDBEPR plant in which an experimental monitoring system has been installed for control research activities. The availability of several measured variables allows the application of both the supervisory fuzzy control and the ARGMC techniques. The supervisory fuzzy controller has a hierarchical structure with three different levels. It generates the set point of the internal DO controller on the basis of the most important variables that characterize the process of nitrification, denitrification and phosphorus removal.
201 The first level receives as input variables the NH3-N and NO3-N concentrations in the effluent of the aerator and gives the first approximation of the DO set point. The elements that characterize the internal structure of the controller are: a. Singleton Fuzzifier b. Minimum Intersection c. Fuzzy Rule System d. Minimum Implication e. Maximum Aggregation f. COG Defuzzification The second level controller has the same internal structure. It uses as input variables the output of the previous controller and the PO4-P concentration in the effluent of the aerator. The task of the controller is to correct the DO set point to account for the phosphorus removal process. The third level controller introduces another correction factor of the DO set point in order to adapt the aeration process to the changing influent flow rate. In the actual plant for which the controller is conceived the influent flow rate is often changed by activating a different number of pumps, so determining very different operating conditions. The latter controller can be considered as a feedforward controller that allows to eliminate in a very fast way the consequences of what can be considered the main disturbance of the process. Also the third controller has been designed with the same internal structure. The use of a hierarchical structure requires a number of rules lower than what is necessary in an equivalent controller with only one level. The design of all fuzzy controllers has been carried out by "trial and error", on the basis of the knowledge of the system dynamics and the experience used.
3. THE ADAPTIVE ROBUST GENERIC MODEL CONTROLLER (ARGMC) The choice of a controller based on Generic Model Control (GMC) for the inner control loop derives from the consideration that the oxygen dissolution process is highly non linear and the GMC allows to include a non linear model in the control algorithm. Lundberg e Bezanson (1990) highlighted the limited robustness of GMC controllers when a critically damped or overdamped closed loop response is imposed. They suggested a more robust version of the control named Robust Generic Model Control (RGMC). However the latter algorithm, that includes a derivative feedback action, although succeeds in compensating parameter changes of the process model, cannot account for model structure errors. Rani e Gangiah (1991) proposed an adaptive control strategy, the Adaptive Robust Generic Model Control (ARGMC), that has been used in the application reported in this paper in order to take into account the non stationary characteristic of the dissolved oxygen dynamics. The initial assumption of a constant OUR (Oxygen Uptake Rate) value or of an oxygen mass transfer coefficient value proportional to the air flow rate (Holmberg et al., 1989) could be assumed. However the oxygen uptake rate has usually daily variations and the oxygen mass transfer coefficient has a non linear behaviour so that an adaptation mechanism can improve the controller performance. For the dissolved oxygen dynamics the discrete process model suggested by Lindberg e Carlsson (1996) was assumed. On the basis of on-line measurements provided in the aerator (air flow rate, DO, influent flow rate) a "software sensor" was designed in order to make available both themass transfer
202 coefficient and the OUR at the same time. The sensor is constituted by a recursive state estimator that uses the air flow rate and DO measurements and allows the parameter estimation by a Kalman filter. The estimation is carried out in two phases (Carlsson,1993). In the first phase, a few hours long, the air flow rate is subjected to large variations in consideration of the exponential behaviour of the mass transfer coefficient. After that it is possible, with a good approximation, to consider as a constant, for a large time interval, the oxygen mass transfer coefficient and to provide to the estimation of the only OUR, that does not require frequent and large variations of the dissolved oxygen. 4. EXPERIMENTAL AND SIMULATION RESULTS The control scheme previously described has been tested by simulation and partially by some experiments. The simulations were carried out using a programme developed by Ducato and Galluzzo (1995). The programme is based on the n.2 IAWQ model for the biological reactors while a complex model that considers separately thickening and clarification has been used for the sedimentation tank.
2,6
-
2,5 2,4 2,3 2,2 2,1 1,9 1,8 1,7
. 1
3
. 5
. 7
.
. 9
. 11
. 13
.
. 15
.
. 17
. 19
.
. 21
23
N u m b e r of sampling intervals (T=5 min) Fig. 2. Step response of the ARGMC loop The ARGMC scheme was firstly tested by simulation and then implemented in the actual plant. Very good control results were obtained in several different plant conditions with the air flow rate ranging from the minimum to the maximum operating values The response of the control loop to set point changes was particularly studied in consideration of the fact that a changing set point is expected. In fig. 2 the response to a step change of the set point is reported. Data obtained from monitoring the actual plant and the experiencegained in the implementation of the ARGMC controller have been of fundamental importance in chosing all the elements of the fuzzy controllers. After several simulations five fuzzy subsets have been defined for all input and output variables.
203
The feedforward fuzzy controller for the influent flow rate was implemented in the plant control system. In fig. 3 the response of the plant controlled by the ARGMC and the feedforward fuzzy controllers to step variations of the influent flow rate is shown. In fig. 4 the response to a set point change and to a contemporary step disturbance in the influent flow rate is shown.
DO-SP 57
~9
_
~
~
-" D O
4 I
~350
o__ Qin
~300 250
3r
. _
21%~_~r
0
~j._.
1
_
.
.
"-"
....
150 .~
50
I i ......
,
1
6
,
,
,
11
16
21
'
'
,
" ,
26
,
31
0
36
Number of sampling intervals (T=5 min)
Fig. 3 - Response of the ARGMC and feedforward fuzzy controller to step variations of the influent flow rate
-- 400
_
3,5-
- 350
32,5 -
300
2-
250 200 ~
9 1,5-
15o ~
1-
l
..
0,50
I
1
100
o Qin I
6
I
11
- 50 1
16
0
21
Number of sampling intervals (T= 5 min)
Fig. 4 - Response of the ARGMC and feedforward fuzzy controller to a step variation of the influent flow rate and to a step change of the set point
204 The simulation tests of the complete control scheme show that while the internal DO control loop is always able to manage the set point changes required by the first and second level controllers, the monitored NH3-N, NO3-N and POn-N are always maintained between acceptable limit values, not higher than 10 % of their normal value. The simulation results were used to change off-line the set point of the DO control loop. Experimental results did not confirm the simulation results. In particular the control tended to be slower than estimated by simulation with higher deviations of the controlled variables. 5. CONCLUSIONS Simulation and experimental results show that good operation conditions can be obtained by the proposed control scheme. The outlined control strategy requires the on line measurements of NH3-N, NO3-N and PO4-N. This certainly limits its application both for reliability and cost reasons. Another important limitation is the availability of only a manipulative variable: the air flow rate. The possibility of an additional manipulative variable like the recycle flow rate to the anoxic reactor could make the control more effective. AKNOWLEDGEMENTS
This work has been partially financed by the Regional Government of Sicily and the European Union with an ERDF grant. The authors thank the staff of the treatment plant of AGIP Petroli- Gela (IT) for all the help in the experimental work. REFERENCES
Carlsson B., "On-line Estimation of the Respiration Rate in an Activated Sludge Process", Wat. Sci. Tech., 28 (11-12), 427-434, 1993. Cassinera S., R. Ducato, I. La Barbera, A. Runci and M. Galluzzo, "Controllo fuzzy sperimentale dell'ossigeno disciolto nel reattore aerobico di un impianto NDBEPR", Proceedings of GRICU Conference, pp. 315-318, Ferrara, 1998. Ducato R. and M. Galluzzo, "Dynamic Simulation of a NDBEPR Activated Sludge Process" Computers Chem. Engng., Vol. 19, Suppl., pp $441 - $446, 1995. Holmberg U., G. Olsson and B. Anderson, "Simultaneous DO Control and Respiration Estimation", Wat. Sci. Tech.,21, 1185-1195, 1989. Lindberg C.F. and B. Carlsson, "Estimation of the Respiration Rate and Oxygen Transfer Function Utilizing a Slow DO Sensor, Wat. Sci. Tech., Vol. 33, No. 1,325-333, 1996. Lundberg B.A. and L. W. Bezanson, "Enhanced Robustness Generic Model Control Using Derivative Feedback", AIChE J. ,36, 2, 283-290, 1990. Nielsen M.K. and T. B. Onnerth, "State of the Art Control of Activated Sludge Plants", Proceedings of the Conference on Modelling and Control of Activated Sludge Processes, Copenaghen, 1994. Olsson G., "Process Control", Dynamic and Control of the Activated Sludge Process, Andrews J.F. (ed.), 67-104, Technomic, Lancaster (USA), 1992. Rani K.Y. and K. Gangiah, "Adaptive Generic Model Control: Dual Composition Control of Distillation, AIChE J., Vol. 37, No. 11, 1634-1644, 1991.
EuropeanSymposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rights reserved.
205
Dynamic Behavior of a Counter-Current Fixed-Bed Reactor with Sustained Oscillations M. Mangold*
E Klose
E.D. Gilles
Max-Planck-Institut ftir Dynamik komplexer technischer Systeme, Leipziger Strage 44, 39120 Magdeburg, Germany E-mail: {mangold, klose, gilles} @mpi-magdeburg.mpg.de
Abstract Travelling reaction zones have proven to be energetically advantageous for weakly exothermic reactions. In this contribution, a new coupled system of catalytic fixed-bed reactors is presented, which utilizes travelling reaction waves created by autonomous periodic oscillations. The system consists of two parallel fixed beds with inlets for gaseous reactants at opposite ends. Thermal recoupling is established by heat exchangers which connect the ends of the two beds. So far, several variants of the system with co-current and counter-current heat exchange have been studied in dynamic simulation as well as by nonlinear analysis. The investigations reveal a complex dynamic behavior. In a far range of operation conditions, two types of autonomous oscillations are found to coexist.
Keywords: catalytic fixed bed reactor; creeping reaction fronts; travelling waves; autonomous periodic solutions; nonlinear dynamics
1
Introduction
Under appropriate conditions, exothermic gas phase reactions in catalytic fixed beds can take place in creeping reaction fronts (Wicke and Vortmeyer 1959). The creeping reaction fronts are characterized by a propagation velocity far below the velocity of the gas flow and by a maximum temperature over the adiabatic steady state value, caused by heat accumulation inside the front (Kn6pp 1989). In technical applications, that over-adiabatic temperature rise can be used for carrying out weakly exothermic reactions autothermally in cases, where conventional steady state operation would require supply of external heating energy. One example is the circulation loop reactor proposed by Lauschke and Gilles (1994) which requires no external forcing but makes use of sustained autonomous periodic oscillations caused by internal recuperative heat recovery. The benefits and the technical feasibility of the principle of the circulation loop reactor have been demonstrated in theoretical as well as experimental investigations for the catalytic combustion of hydrocarbons as well as chlorinated hydrocarbons in waste air streams (Lauschke *Corresponding author, phone: +49 391 6117-519, fax: +49 391 6117-545
206 and Gilles 1994; Kienle et al. 1995; Mangold et al. 1999). In this contribution, a generalization of the principle is discussed. The generalized reactor scheme consists of two reactor tubes connected by heat exchangers at both ends, as shown in Fig. 1. Both tubes have separate inlets and out~\\\\\\\\\~ I I ,, ,, lets. The arrangement allows heat exchange ,, ,,' ,, / ,, between the tubes, but no mass exchange ", ,'" 9 or mass recycle. Starting from that gen'ii! ...'....~ eral scheme, a variety of different reactors -. . . . . . . . . . . . . . . . can be realized. Degrees of freedom in the scheme are the type of heat exchange (e.g. 9 O co-current, counter-current, or cross-flow), / ..... , 'l ...... A the reactor geometry (heat exchange over the ,,', whole reactor length or a part thereof), and i' / 0 0 0 _(~)_ / ~ , 41 ...... h the positioning of catalytic material in the arol ~ io . . . . . . . . . . . . . . ='~,\\\\\'~~ rangement. The topic of this paper is to characterize the dynamic behavior of the described reactor " ~169 ......... OO ......... @o {) scheme for an exothermic gas phase reaction of first order and to identify variants and opFigure 1: Proposed scheme of two catalytic eration modes promising for technical applifixed beds coupled by heat exchangers; variants cations. In the first step, a highly idealized of the scheme with respect to the type of heat model is used for the heat exchanger sections exchange and to the sections filled with catalyst to identify the principal dynamic behavior. In (hatched areas) the second step, the influence of the type of heat exchange is investigated by more detailed models. In the last section, possible technical applications for the found dynamic phenomena are briefly discussed.
Dynamic Behavior under Simplifying Assumptions for Heat Exchanger Sections To get a first overview of the dynamic behavior of the coupled system, a strongly simplified model is used for the heat exchanger sections. It is assumed that ideal co-current heat exchange takes place, that the heat exchange is quasi-stationary, and that there is no reaction in the heat exchanger sections. In this case, the inlet and outlet temperatures of the heat exchanger are related by simple algebraic equations. The reactor tubes between the heat exchangers are described by a standard one-dimensional pseudo-homogeneous model. The kinetic parameters used in the simulations were obtained from experiments by (Richter and Roschka 1998) for the total oxidation of ethene. The model equations and parameter values are given in (Mangold
2000). In the following, identical inlet temperatures, inlet compositions, and flow velocities in both beds will be considered. The question arises if the symmetric operation conditions in the reactor permit different temperature and concentration profiles in both tubes. It can be shown that such non-symmetric solutions can be excluded in the steady state case, if the steady state solutions of the single reactor tubes (without heat exchange and re-coupling) are unique functions
207 of the conditions at their inlets (Mangold 2000). This can be guaranteed for high Peclet numbers (Hlavacek and Hofmann 1970) and hence for most technical fixed bed reactors. Therefore, under conditions typical for technical applications, the described reactor system can only show symmetric steady states, where both tubes are extinguished or ignited. It should be noted that the uniqueness in the steady state solutions in the uncoupled reactor tubes still allows multiplicities in the solution of the coupled system. The situation becomes more complicated when the dynamic behavior of the system is considered. It is found that two different types of stable autonomous periodic solutube #1 tube #2 tions can exist under the same inlet , ~ 1600 conditions, and that only one of them t5oog is symmetric. Temperature and conP 4 0 0 ~ L2 q 400 centration profiles of the two solution 300~" 0 : ~ 0.5 types are depicted in Fig. 2. Fig. 2 0.3 (a) shows an asymmetric, partly ignited ~ ~ t4 periodic solution, characterized by a Z 0.2 t5 t6[t5i t4/talt2~ 1tI:I~ travelling reaction zone in only one of the two tubes. In Fig. 2 (a), the trav00 0.5 elling reaction zone is in tube #2, mov(b) 600 ing in the direction of the gas flow from 600t t ~ 500 ~" right to left, whereas temperature and Z5 o o ~ conversion in tube #1 are low. The travelling reaction zone gradually increases 0 0.5 1 0.5 0 the temperature at the outlet of tube #2 and due to the thermal coupling also the temperature at the inlet of tube #1. This causes the formation of a new re' action zone at the inlet of tube # 1. After 0 0.5 0.5 0u time t6, which marks about half the pez1 [m] z2 [m] riod of the periodic solution, the travelling reaction zone leaves tube #2. This Figure 2: Temperature and concentration profiles for causes a temperature drop at the end of periodic solutions at TR,i,, = 300K and XR,in = tube #2 as well as at the inlet of tube 0.0025" (a) partly ignited periodic solution; (b) fully #1, which drives the newly formed re- ignited periodic solution. action front towards the outlet of tube #1. The second type of oscillations is a symmetric fully ignited periodic solution, shown in Fig. 2 (b). Now, travelling reaction fronts exist in both beds. When the reaction zone in one bed reaches the end of the bed, it ignites a new zone at the inlet of the other bed and vice versa. In this operation mode, high conversion is achieved in both beds at all times. In the region of co-existence of the two types of oscillations, it depends solely on the initial conditions in the reactor system, whether the fully or the partly ignited solutions will develop. The regions of existence of the two types of periodic solutions differ only slightly. As an example, steady state and periodic solutions are depicted as functions of the temperature at both reactor inlets in the
6oof
t
1
f
~
-~
i'l'
'if'
J:
208 bifurcation diagram in Fig. 3. (a)
5000 4500 4000 ~_ 3500 x
.t-- 3000
x x x
x x x x xx x
x
xA~
HP2
~ 1 7 6~ ~ ~ ~ ~ ~ ~ 1 7 6 1 7 6 1 7 6 1 7 6 1 7 6 107 6oo 1 7 6o<111 1 7 6 1HP4 76176
2500 20O0
"*o*
1500
.,.'''*'****g
HPl
ooooooooo
a~o
10% 0
(b)
""
9
HP3
.;o
a;o
450
650~ 600 550 t
I
500 E
450
-stable steady state . . . . o .... unstable steady state * - s t a b l e periodic oscillations o o unstable periodic oscillations 350 9 Hopf bifurcation
400
3%0
A
t o m s bifurcation300
350
Tin[K]
~ HP4
400
450
Figure 3" Periodic and steady state solutions for different inlet temperatures at X R , i n - - 0.0025; (a) period of oscillations, (b) maximum temperature at z=0.5 m; *,o - symmetric solutions" x, ~ = asymmetric solutions
3
The bifurcation diagram was obtained by using the continuation methods in the simulation environment DIVA (Mangold et al. 2000). For high inlet temperatures, the reactor always reaches an ignited steady state. For very low inlet temperatures, it is always in an extinguished steady state. In between, the two branches of periodic solutions and a multiplicity of steady states are found. For both periodic branches, the lower boundary of existence is given by cyclic turning points at nearly the same inlet temperature. For high inlet temperatures, the branch of stable fully ignited oscillations ends in a Hopf bifurcation. The branch of partly ignited oscillations loses its stability in a toms bifurcation before meeting the branch of steady states at Hopf bifurcation point HP2. For temperatures above the toms bifurcation, a transition occurs from the partly ignited to the fully ignited periodic solutions. Further investigations (Mangold 2000) show that the described bistability of the two types of periodic solutions is not a result of the idealizing symmetric conditions in the simulations, but persists when there are considerable differences in the inlet conditions of the two tubes.
Dynamic Behavior for Co-current and Counter-Current Heat Exchange
The simple heat exchanger model used in the previous section gives a rough picture of the qualitative behavior to be expected in the coupled reactor system. However, the model cannot answer important questions concerning the reactor design. It is neither able to describe the influence of the type of heat exchange, nor can it make predictions about the optimal reactor length of the heat exchanger sections in relation to the total reactor length. In order to answer those questions, a slightly more detailed model is studied in the following. Now the adiabatic reactor tubes as well as the heat exchanger sections are modeled as one-dimensional pseudohomogeneous systems. It is assumed that also in the heat exchangers reaction can take place. The cases of co-current and of counter-current heat exchange are considered. The influence of the reactor geometry is studied by varying the lengths of the heat exchanger sections. Changing the heat exchanger lengths may have an additional effect: It also changes the
209 intensity of the coupling between the two reactor tubes for a constant heat transfer coefficient. To suppress that effect in the simulations as far as possible, the heat transfer coefficient o~ is adapted to the heat exchanger length luE in such a way that the product (o~luE) is always constant. The results are summarized in the bifurcation diagrams in Fig. 4. The diagrams show regions of existence for periodic and steady state solutions in a parameter space spanned by the inlet temperature and the heat exchangers' length. It is found that in the co-current case (Fig. 4 a) the qualitative behavior of the reactor system depends only weakly on 1HE. The partly ignited periodic solutions as well as the fully ignited periodic solutions even exist when the heat exchanger sections stretch over the whole reactor length. This means that to some extent, increasing the area of heat exchange can compensate for a poor heat transfer and leaves the qualitative behavior unchanged while changing the quantitative behavior only slightly. The influence of the heat exchanger lengths is much stronger in the counter-current case (Fig. 4 b). Counter-current heat exchange is less supportive to the formation of travelling reaction zones since the thermal recoupling counteracts the movement of the fronts. Nevertheless, both types of periodic solutions are found, if an intensive heat transfer takes place in a short section of the reactor. But the region of periodic solutions diminishes rapidly when the length of the heat exchanger is in- Figure 4: Periodic and steady state solutions for different inlet temperatures and lengths of the creased. The autonomous oscillations vanish heat exchanger sections; (a) co-current heat excompletely for the chosen set of parameters if the heat exchanger sections cover more than change; (b) counter-current heat exchange. two thirds of the total reactor length.
4
Conclusions and Possible Applications
The system of two catalytic fixed beds coupled by heat exchangers shows a bistability in its autonomous periodic oscillations. Under symmetric operation conditions in both beds, the reactor possesses a symmetric, fully ignited solution and an asymmetric partly ignited solution, whereas only symmetric solutions exist in the steady state case for large Peclet numbers. The actual mode of periodic operation is determined by the initial conditions. This gives the reactor a memory for past asymmetries in the reactor state or the operation conditions.
210 The temperature and concentration profiles of the symmetric solutions are identical to those found in the circulation loop reactor by Lauschke and Gilles (1994). This opens the same area of application as for the circulation loop reactor, that is when high conversion is desired for a weakly exothermic reaction and energy costs are an issue. The asymmetric solution clearly offers no advantages for that type of reaction, since the region of existence is approximately the same as in the symmetric case, while the time averaged conversion is much lower due to the half periods of extinguished states. However, technical benefits may be drawn from the asymmetric periodic operation mode for more complex reaction schemes where a regeneration of the catalyst is required. A possible application could be reactions which seriously coke the catalyst. Reactions of hydrocarbons at metal catalysts, e.g. hydrogenations, partial oxidations, and cracking reactions, often belong to that class (Wittcoff and Reuben 1996). Here, a travelling reaction front in one of the beds, e.g. bed #1, can be used to combust the coking deposit at high temperature and return the catalyst to an active state. Simultaneously, the desired reaction is carried out in the other bed under conventional conditions at a lower temperature level. When the travelling reaction front reaches the end of bed #1, it ignites a new zone in bed #2. In the second half period, bed #2 is regenerated, and bed #1 can be used for production. In most cases, the regeneration phase and the production phase will require different inlet compositions. Therefore, this mode of operation will be autonomous only with respect to the temperature oscillations. A synchronization will be necessary between the production phase and a subsequent cleansing of the bed (e.g. with nitrogen) on one hand and the velocity of the moving front on the other hand. This should be easily accomplished, since many parameters are available for changing the velocity of the front, such as inlet temperature, inlet compositions and flow velocity. Therefore, this mode of operation might be applicable to a bigger class of reactions. Experimental verification of the theoretical results obtained so far is a prerequisite for further investigations. For that purpose, a laboratory set up is currently under construction.
References Hlavacek, V. and H. Hofmann (1970). Chem. Engng. Sci. 25, 173-185. Kienle, A., G. Lauschke, V. Gehrke, and E. Gilles (1995). Chem. Engng. Sci. 50, 2361-2375. Kn6pp, U. (1989). Mathematical Methods in Applied Sciences 11,599-625. Lauschke, G. and E. Gilles (1994). Chem. Engng. Sci. 49, 5359-5375. Mangold, M. (2000). Ph.D. thesis (in preparation), Universit~it Stuttgart. Mangold, M., A. Kienle, E. Gilles, M. Richter, and R. Roschka (1999). Chem. Engng. Sci. 54, 2597-2607. Mangold, M., A. Kienle, K. Mohl, and E. Gilles (2000). Chem. Engng. Sci. 55, 441-454. Richter, M. and E. Roschka (1998). Research report BMBF 03D0034B. German Ministry of Education and Research. Wicke, E. and D. Vortmeyer (1959). Zeitschrififiir Elektrochemie 63, 145-152. Wittcoff, H. and B. Reuben (1996). Industrial Organic Chemicals. New York: Wiley.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
211
Use of Gap Metric for Model Selection in Multi-Model Based Control Design: An Experimental Case Study of pH Control Omar Gal~a t, Jos6 Romagnoli*, Yaman Arkun + and Ahmet Palazoglu $ I" Laboratory of Process Systems Engineering, Department of Chemical Eng., University of Sydney, NSW 2006, Australia +College of Engineering, Koc University, Istanbul, 80860 Turkey Department of Chem. Eng. and Mat. Sci., University of Califomia, Davis, CA 95616, USA The gap metric concept is extended to multi-linear model-based control framework. The concept of distance between systems is used as a criterion to select a set of models that can explain the nonlinear plant behavior. Gap metric is used to analyze the relationships among candidate models, resulting in a reduced model set which provides enough information to design a H~-robust co,roller. 1. INTRODUCTION Most control design strategies are often based on a mathematical model of the actual system to be controlled, hence incorporating relevant process information in design. Most physical systems are inherently nonlinear, and this behaVior exacerbates the control problem [1 ]. However, if the operating range of a control system is small, then the control system may be designed based on a linear model. The limitation of the linear control design is that a single nominal model may not be able to represent the nonlinear plant in the whole operating range. This deficiency has been addressed using gain scheduling [2,3] or adaptive control [4]. Despite the fact that classical linear design tools may not be able to deal with nonlinear processes, they provide a diverse set of methods for controller design, which incorporate robustness and performance requirements in a natural manner. Considering such powerful design features, another research area receiving significant attention is the multi-linear model approach for controller design [5]. The key is to represent the nonlinear system as a linear combination of linear systems where classical control design techniques can be applicable. The controller design based on multiple linear models requires either simultaneous stabilization using a single controller, subject to performance and stability constraints [6,7], or interpolation using validity functions, where local controllers are scheduled as a function of the current state of the process [8,9]. Despite the fact that, in most cases, it may be possible to account for the nonlinear behavior with a finite number of linear models, the question of how many models are necessary remains largely unanswered. We shall formulate the problem in hand by assuming a set of local plants as well as local controllers that stabilize these plants, and by asking the question, "Is there a reduced set of controllers, which are 'close' in some sense, and that can stabilize the system?" To determine when two systems are close to each other is a nontrivial task. If we have a controller that internally stabilizes one system, then it should also be internally stabilizing for the other
212 system. Also, if such a stabilizing controller is applied to each one of the systems, then the closed-loop systems should be close to each other. The first requirement considers robustness of internal stability. The second requirement prevents other performance criteria from being highly sensitive to perturbations even though stability is preserved. The challenge in this formulation is how to define the concept of distance between two systems. Since systems are input-output operators, a natural distance concept would be an induced operator norm. Yet, the norm cannot be generalized as a distance measure [10]. The aim of this paper is to introduce the application of a distance measure between systems, the so-called Gap Metric, to select a reduced set of models that contain non-redundant process information for robust stabilization of feedback systems based on multi-linear controller design. 2. THE DISTANCE BETWEEN LINEAR MODELS The gap metric has been introduced into the literature [10,11,12], as being appropriate for the study of uncertainty in feedback systems. The metric defines a notion of distance in the space of (possibly unstable) systems, which do not assume that plants have the same number of poles in the RHP. The gap between two systems Pl and P2 is defined by 5 (P~, P2) where O<5(P1,P2)
(1)
_o~11 Here, (i,j)
equals to (1,2) and (2,1), and (Ni,Di)
is the normalized fight coprime
factorization of P~(s),i = 1,2 [10]. If the gap metric is close to zero, it indicates that the "distance" between two systems is close. If, on the other hand, it is close to 1, then, two systems show distinct dynamic behavior. The impetus behind the derivation of this metric has been to quantify the perturbations of a (possibly unstable) open-loop system that would still maintain stability under constant feedback. In other words, if two systems are "close" in open-loop, they would be expected to be also "close" in closed-loop. Our use of the gap metric, while does not exploit such closed-loop implications, focus only on the quantification of the distance between models in a set. 3. pH-NEUTRALIZATION PROCESS We shall consider a mixed tank reactor with a constant volume, where an acidic solution with a time-varying volumetric flow is neutralized with an alkaline solution made up of a base and a buffer agent. For a model of this system and simulation details, please refer to [7]. Transfer functions are derived based on five distinct operating regions in the steady-state map (Figure 1),
~,,(S) =
Kp,, p~S + 1
(2)
213 Table 1 displays the model parameters. As mentioned above, the gap metric values 5~j = 0 and 5~ - 1 stand for "identical" and "different" systems respectively. At this stage, it is important to define a threshold value to define the demarkation between them. Let us assume 0.5 heuristically to be this limit value.
Region ..... uss 1 0.14 2 o.26 3 0.45 4 0.60 5 0.82
y,~
Kp,,
r~,,
3.3
3.89
131.25
5.0 7.4 8.6 10.4
101.62 5.24 25.95 1.76
118.55 103.52 94.23 82.58
. . . .
8~j .....
1 2 3 .... 4 .....
Table 1. Information for five local linear models.
1
2
3
4
0.000
0.933
0.256
0.8'05
0.262
5
0.933
0.000
'0.889
0.514
0.952
0.256
0.889
0.000
0.689
0.418
0.805
0.514
0.689
0.000
0.857 i
0.262
0.952
0.418
0.857
0.000
Table 2. Distance between linear models.
Table 2 shows the gap metric among the pairs of five linear models representing the whole operation range for the nonlinear system (Figure 1). The numbers suggest that model subsets ~ / s = {f22, f24 } and f2Ls = {f21, f23, f25 } are closer to "different" while the member in each subset 'are closer to "similar." Model 2 is considered different than models 1,3 and 5 as the gap metric is around 1, but close to model 4. This similarity between models 2 and 4 can be explained physically by the fact that those models represent high sensitivity regions, ff2ns. One can contrast this with models 1,3 and 5, which represent the low sensitivity regions, ~LS.
Io9
[24~
I1
8 7
:/6
f2
5
3
20
O. I
0.2
0.3
0.4
U
0.5
0.6
]
0.7
0.8
0.9
Fig. 1. Model identification along the steady-state map. In principle, it is possible to select a minimal subset of models to represent this system, choosing only one element from each subset. In the next section, a reduced set of models is used to perform numerical simulations and real-time experiments to validate the gap metric as a reliable selection tool. 4. CONTROLLER IMPLEMENTATION Classical techniques such as linear H~o control theory have gained popularity in engineering applications [14,15]. The main reason is the possibility of including robustness considerations explicitly in the design in addition to the fact that meaningful physical
214
performance objectives can be expressed as H~o design specifications. We will use the methodology outlined in [7] to construct the local controllers. The controllers for different operating regions are combined to form a complete control system. Membership functions are introduced to create a transition region in the measured variable "y" [ 16].
~p = ~P(Y) ~_d~p(y)
(3)
p=l
with ~p "y ~ [ 0 , 1 ] , where, we have
1 Y- .Vp
d~p(y) = exp --~
(4)
Crp
Here ,~p and Op represent the mean and the standard deviation of each model "p".
The
desired contribution of combined controllers can be represented as a function of the N
membership functions, resulting in the control signal,
u(t)= E~pUp(t).
This is a single
p=l
control action generated by the "weighted" contributions of local controllers. In the /-/~
~(s),
approach, for a SISO p'lant
a controller
ffi(s)is
designed, such that the closed-loop
system satisfies the basic requirements of robust stability, and robust performance [17]. In our design, the penalty weights are specified as [7]:
r
lO
4.4s + 1 440.8s + 1
.r
3.6s + 1
28.6s + 1
0.1 ls + 1
0. 04s + 1
(5)
Then, the local H~o controllers are obtained as:
fq(s) =
0"0019s3 + 0"06916s2 + 0.4867s +0.0037
(6)
s 4 + 9.82s 3 + 2.86s 2 + 0.309s + 0.00069
fc:(s) =
- 0.001 ls 3 + 0.04372s 2 + 0.5327s +0.00611 S4 +
10. 93s 3 + 11.78s 2 + 4. 05 ls + 0. 009 - 0.0022s 3 + 0.0308s 2 + 0.5239s +0.0051
fq(s) = s4 +10.27s3 +3"72s 2 +0.4846s +0.0011 fq(s) = - 0"0023s3 +0.031 ls 2 + 0.5293s +0.0057 s 4 + 10.54s 3 + 6.91s 2 + 1.62s +0.0036 0. 0018s 3 + 0. 069 ls 2 + 0. 4985s + 0.006
lOs(S)= s 4 + 9.83s 3 + 2.61s 2 + 0.254s +0.00056
(7)
(8) (9)
(lO)
Simulations are performed to assess the closed-loop performance when a complete set and a subset of models are chosen in the controller. Table 3 contains the parameters for the membership functions.
215 ......
Region ~p
1 3.0
Crp
0.25
,,,,,
2 5.0 , ,,
0.25 ,,,,,,,,,
,,
3 7.0
0.25 ,,, ,,,,,,
4 9.0
5 10.5
0.25
0.25
,
Table 3. Parameters for the membership functions. The closed-loop profile for the complete set of models (models 1,2, 3, 4 and 5) is depicted in Figure 2. However, satisfactory performance can also be achieved using a subset of models (models 1 and 2) as shown in Figure 3. Note that in the application of the reduced set of models, we use model 1 to explain regions 3 and 5 and model 2 to explain region 4. If we only use the subset f2 Ls , Figure 4 shows degradation in the performance due to the lack of information from the high sensitivity regions in the selected subset as expected. Here, we use model 1 to explain region 2, and model 3 to explain region 4. As depicted in Table 2, these models show distinctly "different" behavior, hence the lack of success of the controller in these regions, especially in region 2. 10
t
9 8
pH 7!.~, J
'ItL,
6
5t
500
1000 1500 2000 250 fim~,scxxl~ Fig. 2. Closed-loop profile using models 1, 2, 3,4 and 5.
40 '-
t~e~~
Fig. 3. Closed-loop profile using models 1, 2. 9i"
t'i
J~ 6
pH ~l
1
1000
!~
1500
pH 54[t
2000
time, seconds Fig. 4. Closed-loop profile using models 1, 3 and 5.
2500
":'
' i,! :. 'i.. ,~.~,.j F
"
J 30 200 400 600 800 iooo 1200 ~400 ~600 ~800 time, seconds
Fig. 5. Closed-loop profile using models 1, 2, 3, 4 and 5.
5. EXPERIMENTAL CASE STUDY The experimental study is conducted at UC Davis using a bench-scale pH neutralization experiment. An acid stream (HC1 solution) and an alkaline stream (NaOH and NaHCO3
216 solution) are fed to a 2.5-1iter constant volume, well-mixed tank, where the pH is measured through a sensor located directly in the tank. The control objective is to drive the system to different pH conditions (tracking control) and also to maintain the tank pH at a specified value despite variations in acid stream flowrate (disturbance rejection) by manipulating the alkaline stream flowrate. The acid flowrate is considered a measured disturbance. As expected, the closed-loop pH profiles (Figures 5, 6 and 7) show a very similar trend, despite the fact that there is the inevitable model-plant mismatch. In fact, the experimental results show more clearly the loss of performance when the model set is varied, hence providing evidence for the interpretation of the gap metric as a measure for model similarity.
101, I[t
i
~0~-. . . .
""/"~'+'+'~'~"
]
h
"1+"
~
+
:
!
~
iv i i
!
pHI;,:
\
9
6f
I
+I
',4 i
.
+[ 30
|
200 400 600
t
800 1000 1200 1400 1600 1800 tin/e, seconds
Fig. 6. Closed-loop profile using models 1, and 2.
3;
200 4J30 600 800 I(X)O I~0 time, seconds
I~0
1
1600 180
Fig. 7. Closed-loop profile using models 1, 3, and 5.
REFERENCES 1. J.J.E. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, NY, 1991. 2. F.G. Shinskey, Process Control Systems: Application, Design, and Tuning, 4th ed., McGraw-Hill, NY, 1996. 3. W. Rugh, IEEE Control Syst. Magazine, 11, (1991) 79. 4. K.J. Astr6m and B. Wittenmark, Computer Controlled Systems, Prentice-Hall, NJ, 1984. 5. R. Murray-Smith and T.A. Johansen (eds.), Multiple Model Approaches to Modeling and Control, Taylor & Francis, London, England, 1997. 6. E. Schrming, M. Sznaier and U. Ly, J. Guidance, Cont. and Dynamics, 18, (1995) 525. 7. O. Galhn, A. Palazoglu and J.A. Romagnoli, Chem. Eng Sci., in press (2000). 8. B.A. Foss, T.A. Johansen and A.V. Sorensen, Control Eng. Practice, 3, (1995) 389. 9. A. Banerjee, Y. Arkun, B. Ogunnaike and R. Pearson, AIChE J., 43, (1997) 1204. 10. M. Vidyasagar, Control System Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985. 11. G. Zames and A.K. E1-Sakkary, Proceedings of the Allerton Conf., (1980) 380. 12. A.K. EI-Sakkary, IEEE Trans. on Automatic Control, AC-30, 3, (1985) 240. 13. T.T. Georgiou, Systems & Control Letters, 11, (1988) 253. 14. U. Christen, H.E. Mush and M. Steiner, J. Process Control, 7, (1997) 19. 15. S. Skogestad, M. Morari and J.C. Doyle, IEEE Trans. on Automatic Control, AC-33 (1988) 1092. 16. C.W. de Silva and A.G.J. MacFarlane, Knowledge-Based Control with Application to Robots, Springer-Verlag, NY, 1989.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
217
DYNAMIC AND CONTROL OF HIGH PURITY HETEROGENEOUS AZEOTROPIC DISTILLATION PROCESS Vasconcelos, C.J.G and Wolf-Maciel, M.R. Separation Process Development Laboratory (LDPS), Faculty of Chemical Engineering, State University of Campinas (UNICAMP) P.O Box 6066, Zip Code 13081-970, Campinas - SP, Brazil, E-mail: [email protected]; [email protected] The three phase azeotropic distillation is a common separation process in chemical and petrochemical industries. This is a complex process, presenting high instabilities. In this work, it was used HYSYS.Plant commercial simulator for calculating the ethanol-water separation using different kiinds of entrainers. The following studies were carried out: steady state and dynamic simulations, which can lead to different process performance according to the liquid phase distribution in the decanter, optimization of the operating conditions in terms of energy consumption and entrainer selection taking into account economical aspects, environmental restrictions and column control. 1. ~ T R O D U C T I O N The chemical and petrochemical industries have often the task of separating close boiling point and azeotropic mixtures. Three processes normally are used for this kind of separation: heterogeneous or homogeneous azeotropic distillation and liquid-liquid extraction. Heterogeneous azeotropic distillation operations represent the most complex behavior of these processes and this reflects in difficulties of convergence mainly when the complete process is considered. The azeotropic distillation using benzene as entrainer has been used to alchool-water separation for many decades. But due to the toxity of benzene, it must be substituted by other entrainers or alternative processes have to be considered. There are many studies in this area looking for the best alternative to the alchool purification. In this work, it was studied the complete process for heterogeneous azeotropic distillation. Simulations were performed using HYSYS Plant commercial simulator (by Hyprotech Ltd., 1999). It was studied the actual process with benzene as solvent and compared with azeotropic distillation using cyclohexane. There are three important steps to be considered when comparing different processes: the steady state behaviors, optimization and unsteady state. Steady state modeling allows to establish the flowsheet topology, specifications and all process conditions necessary to achieve the desired product. Before comparing different alternatives, it is fundamental to assure that each one is on its best configuration. Therefore, the processes were optimized in terms of energy consumption using factorial design. Besides the economical aspect, it will be also considered the environmental restrictions. Finally, the dynamic modeling is considered to evaluate the process stability and a control scheme is proposed to maintain the desired specifications when disturbances are introduced.
218
Organlc I reflux J
I L,~
i -J-"
_t~r 7~
Recycle
Pure Ethanol Solvent recovery
er
Fig. 1. Flowsheeting for the complete azeotropic distillation process. 2. PROCESS DESCRIPTION It was used the ethanol-water separation to illustrate this study. This system is important as an illustrative example of nonideal system and it is a renewable source of energy. For this reason, the ethanol production by fermentation has been studied to decrease the energy consumption. Figure 1 shows the complete process for the azeotropic distillation. It was considered a flesh feed in the composition of the ethanol-water azeotrope (ethanol mole fraction 0.8900). The specifications for the products are: mole fraction of ethanol at the bottom of the first tower equal to 0.9999, and mole fraction of water at the solvent recovery tower about 1.0000. 3. PROCESS OPTIMIZATION The second step is the optimization of the process. Optimizations for structural and operational variables (number of trays, feed position, reflux temperature, position of recycles and split of the reflux stream from the decanter) were performed. The objective of the optimization is to minimize the energy consumption in the reboiler of the two columns, maintaining the specifications of ethanol purity and recovery. Moreover, the intention is to improve the convergence facilities and to avoid the formation of two liquid phases in internal trays. It was used the Factorial Design and Response Surface Methodology to optimize the process, under rigorous calculation methodology. Factorial design is used to study process where two or more variables affect the response. In this study, the response is the energy consumption and the independent variables are: number of stages (NS1) and feed position (FP1) in the azeotropic column, the decanter temperature (TD), the number of stages(NS2), feed position (FP2) and reflux ratio (RR) in the solvent recovery column. Firstly, it was proposed a fractional factorial design to verify which variables have the main effects on the response. The low level (-1) and the high level (+ 1) are in Table 1. The number of simulations necessary to the fractional factorial design with six variables is 32 (26"1). To the process using benzene, the mean value of the runs was 1.459 x 107 KJ/h and the variables main effects are shown in Table 2.
219 Table 1 Levels for the independent variables to the factorial design Variable NS1 FP 1 TD NS2 FP2 RR
-1 30 4 30 10 4 0.8
Table 2 Main effect for the independent variables
+1 50 10 60 20 8 2.0
Variable NS1 FP 1 TD NS2 FP2 RR
Main effect 0.030 -0.010 -0.188 -0.016 -0.016 0.148
It can be observed that the decanter temperature presents the larger effect on the energy consumption. It was verified that increasing decanter temperature, the energy consumption decreases. It was made an analysis changing the decanter temperature to verify the component distribution in the organic and aqueous phases. The results are in Table 3. The increase of the temperature is favorable to the ethanol in the organic phase. Table 3 Liquid-liquid equilibrium in the decanter- mole flows in kmole/h Organic phase
TC 1 = 30~
TC 1 = 60~
Aqueous phase
TC 1 = 30~
TC 1 = 60~
Ethanol
76.3
82.3
Ethanol
27.8
20.6
Water
7.6
11.9
Water
19.7
20.3
160.7
161.9
Benzene
2.6
1.6
Benzene
The same procedure was made to the process with cyclohexane. The main difference between the equipment is on the solvent recovery column. For the process using benzene, it was used 10 stages and, when using cyclohexane, it was necessary 16 stages. In both cases, the azeotropic column has 30 theoretical stages. The results are shown in Table 4. The advantage of the process using cyclohexane is about the environmental restrictions. The main difference is the composition of the two phases in the decanter. In Table 5 it is observed that, when using cyclohexane, the aqueous phase is richer in ethanol instead of in water. This influences greatly in the solvent recovery column performance. Table 4 Comparing energy consumption (106 kJ/h) for benzene and cyclohexane as entrainer.
Energy consumption in the azeotropic column Energy consumption in the solvent recovery column Total energy consumption
Benzene 7.18 1.81
Cyclohexane 8.69 4.70
8.99
13.39
220 Table 5 Aqueous phase composition. Component Ethanol Water Entrainer
Benzene 0.4834 0.4773 0.0394
Table 6 Organic phase composition. Cyclohexane 0.6766 0.2122 0.1112
Component Ethanol Water Entrainer
Benzene 0.3358 0.0517 0.6126
Cyclohexane 0.0768 0.0015 0.9217
4. DYNAMIC SIMULATION AND CONTROL STRATEGY FOR AZEOTROPIC DISTILLATION The dynamic simulation and the control of the azeotropic distillation process are not simple tasks and have been discussed for many authors, for example, Rovaglio et al (1992 and 1995) and recently by Chien and Wang (1999). In this work, it was proposed a different strategy and it was considered the multiple steady state phenomena. It was evaluated the dynamic behavior of the process and it was implemented PI controllers on the azeotropic column, decanter, and recovery column, aiming high purity in all product streams and level control. The proposed scheme includes the following controllers: 9 Level control for the aqueous phase by manipulating the aqueous phase flow; 9 Level control for the decanter by manipulating the organic phase flow; 9 Level control for the reboilers of the two columns by manipulating the bottom product stream flows (ethanol and water); 9 Temperature control for the bottom of the columns by manipulating the reboiler duties. The temperature control is used to obtain the desired purities of ethanol and water. 9 An on/off controller to maintain the ratio of organic reflux/recycle upper to the value obtained in the steady state. The azeotropic distillation control becomes complex when the existence of multiple steady states is considered. There are recent works that prove experimentally the existence of multiple steady states in azeotropic distillation (Guttinger et al, 1997 and Muller et al, 1997). In this work, the rigorous simulation of the process, performed using HYSYS, allowed to observe that a variation in the organic phase to aqueous phase flow ratio can induce the process to a steady state where no separation occurs: the bottom product has the composition of the binary azeotrope ethanol/water. So, it was proposed an on/off controller that open a escape valve when the reflux/recycle ratio decreases. It is a new improvement in azeotropic distillation control. Also at this point, the equilibrium in the decanter is of fundamental relevance. In the steady state simulations the reflux/recycle ratio, considering mass flows, was 18.0 to benzene and 2.2 to cyclohexane. Due to the lower value of the reflux/recycle ratio, the process using cyclohexane presents more instabilities: a small variation in the feed flow causes a variation of the equilibrium in the decanter and the process goes to an undesirable steady state. These are other disadvantages of the process using cyclohexane: operating instability and, consequently, the process becomes difficult to control.
221 1.0
1.0
.:than01 ~
o.go,
( I
0.80. c 0.70 .o 0.00, L
Water ~
~hanol
Water " ~ ' - I
..-x-
,yolohex me
0.80
=J
C 0
u_ 0 . 5 0
0.60
a)
S 0.40
---g 0.40 0
f
0.30, 0.20'
'
\
0.10,
0.00
---B-
C yclohexJIne
0
0.20 0.00
5
10
15
20
25
30
35
0
5
10
15
Fig. 2. Liquid phase composition profile for azeotropic distillation using cyclohexane as entrainer with high purity ethanol production
20
25
30
Fig. 3. Liquid phase composition profile for azeotropic distillation using cyclohexane as entrainer with low purity ethanol production
Figure 2 shows the liquid composition profile for the azeotropic distillation using cyclohexane with high purity ethanol production and, Figure 3 shows the same process and operationg conditions atter a disturbance introduced in the reflux to recycle ratio (observe that the composition remains constant in the binary azeotrope). In both situations, Figure 2 and 3, the reboiler duty in the columns and the product flow rates are the same, however the temperature and concentration profiles are quite different showing two different steady states for the same operating conditions. 5. ALTERNATIVE PROCESS
An alternative process to the ethanol production is the homogeneous azeotropic distillation using ethyleneglycol as solvent. This process has the following advantages: non toxic solvent, operational stability, simple control scheme. The complete process is depicted on Figure 4.
Bhanol
,,
35
Stage
Stage
OR1 EG+lAhater
EG
Fig. 4. Homogeneous azeotropic distillation for the ethanol-water system
222 Table 7 Tuning parameters for the controllers
Reflux l~hamol
I:i'hyleneglu
Controller CL RL TC RC
Feed
TC
QR1
kp 1.50 0.50 1.00 0.80
xi 20 15 10 15
,| ~ e r
Fig. 5. Control scheme for the extractive column It was proposed the level controller to the condenser and reboiler (LC and LR), a temperature controller (TC) to avoid the ethanol loss at the bottom and, to maintain the top product purity, it was used a reflux control that adjust the ratio of reflux mass flow to the feed flow equal to 1.75. Figure 5 shows the PI controllers and Table 7 shows the tuning parameters. Steady state simulations were carried out with feed flow rate of 100 kmol/h. To test the control scheme it was proposed disturbances of 80 kmol/h and of 120 kmol/h. The scheme proposed was efficient to maintain the product purities and ethanol recovery. This process was the best alternative studied to substitute the process using benzene and cyclohexane in all of the items considered: energy consumption, operational stability and environmental restrictions. 6. CONCLUDING REMARKS The results of the complete studies of simulation, optimization and control reveal that with the structural and operational modifications proposed in this work, it is possible to use more effectively (in terms of improved stability) for the azeotropic distillation. The main modification proposed were about the recycle position (returning to the azeotropic column) and the temperature in the decanter. Industrial experience confirm the results comparing benzene and cyclohexane: the ones that have made the substitution observed that the volume in the decanter and vapor consumption in the reboilers have increased and it was necessary to increase the number of stages in the recovery column. It was shown that the process using cyclohexane becomes more instable, due to the multiple steady states formation at the same operating conditions. It was proposed the extractive distillation for the same separation and it was shown that a simple control scheme at the optimal point. This process was better than azeotropic distillation in terms of energy consumption, operational stability and environmental restrictions. REFERENCES 1. Chien, I-L., Wang, C. J., Ind. Eng. Chem. Res., 38, (1999) 468 2. Guttinger, T. E., Dorn, C., Morari, M., Ind. Eng. Chem. Res., 36, (1997) 794. 3. HYSYS.Plant version 2.10 by Hyprotech Ltd., 1999 4. M011er, D., Marquardt, W., Ind. Eng. Chem. Res., 36, (1997) 5410. 5. Rovaglio, M., Faravelli, T., Gaffuri, P., Di Palo, C., Dorigo, A., Computers Chem. Engng, 19, (1995) $525.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
223
TRAINING A RECURRENT NEURAL NETWORK BY THE E X T E N D E D K A L M A N F I L T E R AS A N I D E N T I F I C A T I O N T O O L R. Scheffer and R. Maciel Filho LOPCA/DPQ, Faculty of Chemical Engineering, State University of Campinas (UNICAMP), Cidade Universit~iria Zeferino Vaz, CP 6066, , Campinas SP, Brazil, CEP 13081-970, e-mail: [email protected] ABSTRACT In this work we evaluate the behaviour of the extended Kalman filter as a recursive training algorithm for neural networks in real-time. It shows that the extended Kalman filter is a very potential candidate for on-line training of neural networks and as fast as the sequential backpropagation algorithm with momentum, but with better global convergence properties. It could not be shown that a recurrent neural network with external feedback is sufficient to give a dynamic representation of the process. 1. INTRODUCTION Control of non-linear chemical processes relies on a good dynamical model, which usually is linear and has to be continuously updated to follow the process behaviour. In the past decade, this has lead to an increased interest for non-linear process control, which can be divided in optimisation and transformation methods. Only recently, artificial neural networks (ANN) attracted a great deal of attention, providing a simple way to describe non-linear processes at low computational cost. If trained by appropriate input/output data, an ANN is able to approximate any function mapping of the process input/output behaviour insides its training area. So, an ANN can be viewed as a black-box model of the process. The classical approach to identify a non-linear dynamical process by an ANN, is to augment the number of inputs of the ANN by past values of the input data. For simple systems, which can be described by a one-dimensional state space system, this will result in a small and workable ANN. But this approach can result in too large ANN's suitable for training more complicated systems, such as fixed-bed reactors, fluidised beds and bubble columns, and for systems with slow dynamics. Especially, when the process is subjected to environmental or internal changes, such as catalyst deactivation, the ANN should be dynamical and be trained in real time to be able to represent the process. Therefore, the main objective of this work is the development of a dynamical ANN trained by a recursive algorithm as the Kalman filter, which can be used to identify a non-linear process on-line and afterwards to be used in a adaptive control scheme. 2. THEORY
The most popular ANN is the feedforwarded network trained by backpropagation of Rumelhart et al. (1986), which consists of an input layer, a hidden layer and an output layer.
224 The output of a neuron j, yj in the hidden or output layer k is calculated as a function of the outputs of the former layer as follows:
Yk,j-- f(Vk,j)-- f~N2olWk,jiYk-l,i)
(1)
where wji is the weight for the input Xi of neuron j and Nk-1 is the number of inputs and the number of neurons in the former layer. The bias or treshold is defined by putting input Yk-l,0 equal to -1. k ranges from 1 to the number of chosen layers, where layer 0 is the input layer. The function f(.) is typically a sigmoidal or tangent hyperbolic function for a neuron in the hidden layer and linear for the output layer in case of function approximation. If the data is appropriately scaled, the latter can be a non-linear function also. The ANN's output and the desired output define an error, e = dj - yj, which can be propagated back through the system. The error for intermediate or hidden neurons is calculated by:
%, = ~'~j~l Wk+l,jiak+l,J
(2)
8kj is the local error gradient for neuron j in layer k and is calculated by:
6 k,j = df(vk'j ) e(k, j) dvk,j
(3)
In case of the backpropagation algorithm, this leads to a parameter adjustment based on the steepest descent by:
Wk,ji(n + 1)= Wk.ji(n)'+ ~Takjy(k - 1, i)
(4)
A faster convergence can be obtained by adding a momentum term (Rumerlhart et al., 1986), which makes the backpropagation algorithm more stable. Optimization methods, as the conjugate gradients (Fletcher et al., 1964) and the method of Levenberg-Marquardt can be used to obtain a much faster convergence using second-order gradient information. These methods need a good estimate of the gradient, and therefore can only be used in a batch training mode, where an average gradient to the weights is calculated over the whole training set. A dynamical ANN is known as a recurrent neural network (RNN), where some of the neurons in the layer k have a feedback connection with the neurons in layer 1, where 1 < k. In this work was chosen only external feedback connections, which lead the outputs from the output layer back to the input layer. The advantage of this type of RNN is that during the training phase the target values instead of the RNN's outputs, can be fed to the input layer, which leads to a faster convergence. When the error of the output is small enough, the network outputs are fed to the input layer. In this way non-linear state-space approximations are trained by the network.
225 As for the ANN an error can be declared comparing the output of the RNN with the desired output (d), normally, a quadratic cost function of the error is declared, which can be minimised by the gradient methods mentioned above. For on-line applications, the optimisation methods usually do not have a recursive calculation scheme and cannot be used, but the backpropagation algorithm is typically slowly and forgets the past data. A system identification method such as the Kalman filter could be used to update the networks parameters, which has the advantage to take into account the past data when it calculates a new optimal estimate with the new arrived data. The actualisation of the weight parameters by the Kalman filter is by definition of the following dynamical system:
wk,ji(n + 1)= wk,ji(n)+ qk,ji(n) d,j(n), = f(~yrk,
i=O
k-l,i
i= O...Nk_1
(n)wk,ji(n)l+ r,,j (n)
(5)
where qkji and rkj are stochastic variables with a Normal random Gaussian distribution, N(0, Q) and N(0,R) respectively. According to Shah e Palmieri (1990), the use of multiple extended Kalman filters (MEKA) can be used to train a feedforwarded neural network to obtain a much faster convergence than with the gradient methods. It has to be found out if this is valid for a recurrent network also. Puskorius et al. (1994) successfully implemented a decoupled Kalman filter algorithm training recurrent networks and were able to use these networks in various control problems. An additional advantage is that MEKA can be used in a recursive scheme for process identification in a advanced control loop, without the disadvantages of the backpropagation algorithm. 2.1.
Extended Kalman filter training algorithm
The training algorithm with the extended Kalman filter falls down into two parts, the dynamic actualisation and the actualisation of the observation. The dynamic actualisation is done by (5), where the estimates of q and r are 0. In fact, it is the same as the feedforwarded pass (1) of the backpropagation algorithm. The only difference is that the Kalman filter algorithm needs to update the covariance matrix Pkji:
Pk,jii(n+l)=ek.jii(n)+a
(6)
Since the states of the dynamical system (5) are parameters, there would be in fact no process noise qkji, but the addition of a very small process noise (in the order of 0.001) makes the filter more stable. A too large a process noise leads to oscillatory and stochastic behaviour. The errors of the intermediate outputs are calculated by propagating back the error (eq 2 and 3). With the errors known for every neuron the multiple extended Kalman filters update the weight parameters by:
226
Wk,ji(l/l "Jr"1)= Wk.ji(M)-]-Kk,ji(n)ek,j(n) e*'(')=k
" ~
Pk,jiicTS x ,,i =
i =O...Nk_ , Yk-l,N,_l(n)l [Wk,j,O
%j,o ~ = CW (7)
/tc ,, ic + e)
- (I- xc)P(I- xc) + The advantage of using a Kalman filter per neuron is that the product (cPCT+R) becomes a scalar and therefore no matrix inversion is needed. The only disadvantage of the Kalman filter is the memory requirement for the process covariance matrix. To ensure that the Kalman filter still reacts after some time, the matrixes Q or R have to be made a function of the error or the derivative of the error. This makes the system more sensible to new data. The observation covariance matrix, R, has a large effect on the stability of the Kalman filter. With a too small value the filter diverges. R has to be increased when the number of neurons and or layers are increased, otherwise the filter becomes unstable. The matrix R was fixed and only made variable when the total summed square error of the outputs became lower than 5.0. If so, R was made equal to the total number of neurons multiplied with the total summed error. Networks were trained with data obtained from simulation of a fed-batch penicillin process. The process model used was validated with experimental and pilot plant data. The RNN's were trained by the multiple extended Kalman filter, and by the different minimisation algorithms as conjugate gradients and Levenberg-Marquardt and the standard backpropagation algorithm with momentum in sequential and batch mode. 3. R E S U L T S The ANN's trained with the different training algorithms were all able to give an accurate representation of the penicillin production process, when the target values of the outputs were not re-fed to the inputs. The convergence of the conjugate gradient method was much faster than the Levenberg-Marquardt algorithm, because the latter is much more dependent on the local gradient behaviour and the initial shoot (Figure 2). Both were much faster than the standard backpropagation algorithm with momentum. The ANN trained with the multiple extended Kalman filters was in the initial training phase faster than the sequential backpropagation algorithm with momentum, but became slower at the end phase (Figure 1).
227 700
- - Sequential Backpropation (eta =
,
2 600 w 500
0.00125 e mom= 0.9) MultipleExtendedKalmanFilters
" I.U
i
[ ' ' " Batch
Backpropagation,eta
~
; 0.0125 and mom=
~ 1800000~
I
= 400
1400 ~.....~ ~ 1200 ~
m C'onjugateGradients
| |
300
~'J 600
i i
E 200
E
= 100
400
~
-- - Levenberg-Marquardt t ~ ,
200
0
.
0
10
.
.
.
.
20
.
.
Iterations
.
.
.
.
30
0
.
40
0
50
Iterations
1O0
150
Figure 1" Summed square error for the Figure 2: Summed square error for the batch sequential training algorithms for an ANN training algorithms for an ANN
This can mean that the chosen dependence of the Kalman filter to the observation covariance matrix should be changed to make the Kalman more sensible in the mid and end phases. But from a simple feedforwarded problem with the identification of a bioreactor, it showed that the generalisation performance of the multiple extended Kalman filters was two times better than with the backpropation algorithm (Table 1). Table 1 Summed square errors of the test set for a simple feedforwarded network Algorithm Sequential Backpropagation (rl= 0.00125 e ~=0.9 ) Multiple extended Kalman filters Batch Backpropagation (1]= 0.0125 e Gt=0.9 ) Gradientes Conjugados Levenberg-Marquardt
Summed square error of the test set 10.705 5.1281 26.082 2.0119 8.4759
In Figure 3, the error during the training of the RNN for the penicillin production process are shown, for when the target values were re-fed. It can be seen that the multiple extended Kalman filters converge in a low number of iteration, but stays on a higher error plane than the sequential Backpropagation algorithm. This does not mean automatically that the training by the sequential backpropagation algorithm is better that the extended Kalman filter algorithm, as could be seen from the identification of the feedforwarded network.
228 1200
Backpropagation sequential
i
1000
,
.
.
m
--
.
="= -.
800
,Extended Kalman filter "-- k e v e n b e r g - M a r q u a r d t
~=-=
k, I~1
l
..... Backpropagation batch
,
- - Conjugate Gradients "'il
600
, 9 il
M . "~.,~
400
IL 9
"'""----.
" 0
0
,= m
............ .
=..,
999
"'ii.i
li m
-~-~
!
I
10
20
I
30
. . . . . . . . . . I
40
50
Itorations
Figure 3: Summed square error for the different algorithms for identification of the penicillin production process by the RNN When the outputs of the RNN were connected to the inputs this resulted in a high final error for the training set for all the training algorithms. This indicates that the RNN with only an external feedback is not enough to learn the dynamical behaviour of the penicillin production process and probably full recurrent network should be used. 4. CONCLUSIONS It was shown that the multiple extended Kalman filters can be used as a training algorithm for neural networks in real-time. The extended Kalman filter converges faster and maintains better generalisation properties. It could not be shown that the RNN with external feedback can be used as an dynamical identification tool, and probably the RNN has to become fully recurrent to better describe non-linear chemical processes. LITERATURE Fletcher, R. and C. Reeves, Function Minimization by Conjugate Gradients, Computer Journal, 7, 149-154 (1964) Puskorius, G.V. and L.A. Feldkamp, Neurocontrol of Nonlinear Dynamical Systems with Kalman Filter Trained Recurrent Networks, IEEE Transactions on Neural Networks, vol. 5, no. 2, 279-297 (1994) Rumelhart, D.E. and J.L. McClelland, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Volume 1: Foundations, The MIT Press, Cambridge (1986) Shah, S. e F. Palmieri, MEKA - a fast, local algorihm for training feedforward neural networks, San Francisco, CA, vol I, 226-231 (1990)
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
229
An Algorithm for Efficient Specification Analysis in Large-Scale Dynamic Process Simulation Jorge R. Paloschi AspenTech UK Ltd - j o r g e . p a l o s c h i @ a s p e n t e c h , corn Castle P a r k - Sheraton House - C A M B R I D G E - CB3 0AX - UK Abstract: An efficient algorithm is presented whose input is the Jacobian occurrence matrix for a dynamic (or steady state) problem, and the subset of columns (variables) which are free (the others are considered to be fixed). Using very little CPU time, the algorithm recommends a new set of specifications for the columns in such a way that the resulting Jacobian becomes structurally nonsingular. A problem with 138000 equations can be solved in less than 17 CPU seconds on a Pentium600 PC. This specification analysis algorithm is available in the commercial dynamic simulation package Aspen Custom Modeler TM
1
Introduction
The dynamic simulation of a chemical process can be represented by a system of differential algebraic equations (DAEs):
/(r(t),x'(t), y(t),t)=o,
(1)
where x are the state variables, x' are their derivatives with respect to t, y are the algebraic variables and t the independent time variable.. The values of x(t),x'(t) and y(t) indicate how the process changes from values given at a time to towards a time tl: We will denote with n; and ;;,, the dimensions of the spaces of state and algebraic variables respectively, and with ;; the dimension of the range of the function f~ If we define the space F =Rn~ x Rns x Rna, then f: F x R --> R n. The Jacobian matrix for this problem consists of n rows and 2ns+n, columns. For this problem to be well defined we need n < 2n,.+n,. In order to solve this problem it must be well posed to ensure that a unique solution exists. A necessary condition is to define a subset of the state, derivatives and algebraic variables to be fixed so that the Jacobian matrix is structurally nonsingular (which implies being square). The exact form of the DAE system (1) depends on the selection of specifications. For instance, the steady state case consists of setting x'(t)=O while the initialisation case will fix the values of some variables at time t=to solving for the remaining ones. In any case, we will need to have the problem well posed (i.e., at least structurally nonsingular).
230
The problem of finding the subsets of variables to fix in each case is not trivial for complex simulations, especially when interactions between different units because of recycles are not always evident. This can be a difficult problem, even for experienced process design engineers. In this paper, we propose a simple algorithm that enables engineers to solve this problem very efficiently, even when dealing with very large industrial simulation cases. The problem can be expressed mathematically as follows: Problem 1 Given the space F and a subspace ~1 ~ ~, find a subspace 1:2 c F by adding and/or deleting variables from Fl, and such that the Jacobian matrix related to 1:2 is structurally nonsingular (i.e., the matrix F where the columns corresponding to the complement of 1:2 in F were removed). That is, we start with an initial selection of variables to be considered free (or to be solved lor). Then, we find out which variables need to be fixed and/or freed, in order to have a well posed problem from a structural point of view. Notice than we could just start with F1 as an empty subset, but in general there will be unit variables that will be known to be either free or fixed according to the simulation problem posed. In general, since the problem needs to be square, an indication of the state of the specification problem is given by the degrees of freedom (dof) of the simulation problem. Its departure from n gives an indication of whether we need to fix or free variables in order to define a well posed problem, but it is not as simple as that since we also need to ensure structural nonsingularity, therefore not every variable is a candidate. Notice that solving index problems in DAE's is a particular case, and is covered as well by our discussion. In this case, all state variables are set to initialised for the analysis. Morton and Collingwood(1998) considered this problem only for the steady state case ( although they indicate that their analysis can be extended for dynamic problems). They considered well-specified problems not only in a structural sense but also from a numerical point of view. We will discuss this problem not only for the steady state case but also for dynainic simulations, and our algorithm will handle specific requirements arising from ll]em, not occurring in the steady state case. Our emphasis will be based only on the structural point of view, with the aim of obtaining a useful fast algorithm to help users of simulation packages with the specification analysis problem. Therefore, we do not consider the numerical point of view. We present in Section 2 a brief description of the simulation cases considered while the proposed Specification Analysis algorithm is explained in Section 3. We give an overview of the algorithm and a brief description of each step. Finally, we present in Section 4 an exanlple of use plus some timings obtained for some large problems. 2
Simulation cases considered
Two simulation cases will be considered. The first is the steady state case. Here,
_v'(t)=O and we need to determine a subset of the algebraic and state variables which
231 will be fixed, hence solving the system of equations (1) for the remaining free variables. The second is the dynamic case, which itself results in two subcases, initialisation and integration. In the initialisation subcase, a subset of variables is fixed for a given time to and the system f(x(to),x'(to),y(to),to)=O is solved for the remaining ones. In the integration subcase, the derivative variables are replaced by an approximation formula (which depends on the integration method used), hence they are replaced by implicit substitution by the corresponding state variables. For x ( t ) - x ( t - At) instance x'(t) = ,t - t o > At is one of the most commonly used. At Both subcases need to be solved in a coordinated way. Any variables that are fixed in the integration stage would need to be fixed as well in the initialisation stage for consistency. Some variables will be fixed only for the initialisation stage, thus they are called initial variables. Furthermore, the state and derivative variables are tightly related in pairs. Each state variable will have its corresponding derivative and both can not be initialised at the same time. This is a feature that the specification analysis algorithm must take into account. ~ In addition, some variables are known beforehand to be either free or fixed. Hence, there are some constraints regarding which variables can be considered candidates to change status. The algorithm must also be able to take this into consideration. For this, we use the concept of preferred variables; that is, we specify those variables that we prefer not to change specification.
3
Specification Analysis algorithm
The input to the algorithm consists of an occurrence matrix Jo and vectors V,, and V,.. The matrix Jo is such that its entries eiTjoej=l if equation i contains variable j, otherwise they equal 0. The entries in the vector V~ indicate the variables status according to the following: I. 2. 3. 4. 5. 6.
Fixed, input preferred Free, output preferred Fixed Free Fixed, to be Free, to be
Fixed means that the variable has a fixed value, therefore its corresponding column in the Jacobian is not active. Free means the variable is solved for (active Jacobian column). Input or output preferred means that the status of the variable should be preserved, if possible. "To be" (cases 5 and 6) means that the variable originally was free and the algorithm is marking it to be fixed, or conversely it was fixed and the algorithm is marking it to be fi-eed. On input V~ will contain only entries between 1 ~nd 4. The values 5 and 6 are set internally and can only be present on output. The
232
vector V~. has as many entries as variables and each entry is either 0 or a variable number. If eirV~-j, it means variablesj and i collide with each other in the sense that if one is fixed, the other must be free. This is useful for handling pairs of state variables and their derivatives. The output of the algorithm consists of V,., where the eventually modified entries indicate with values 5 or 6 whether a variable needs to change its specification. After making the changes indicated in V~ the resulting status, is given in the variable do.[ containing the degrees of freedom. A zero value means a square system, a negative value means the problem is underspecified, and a positive value means overspecification. A non-zero value indicates that the algorithm was not able to find a solution due to the problem not being well posed. For instance, there could be more rows than columns, which is an unsolvable problem. Finally, the output variable n,, indicates the number of equations that remained unassigned and EL, contains a list of them. If there are equations unassigned, it means the problem is not well posed.
3.1
Algorithm overview
Fi ndSpecs(Jo, V,., V,..dofn,,E,) 9 If Jo has fewer rows than columns, add zero rows to Jo to make it square. 9
stctttts=l
While (status>O) OrderVariables(V,., V~,,P,.) BuiidReducedProblem(Jo, P,,,Jj,J2) Fi ndAs si gnment (~, P l ) J.~=PI r Jl PI r , FindAssignment(J3,P2) J4=P2 r,13 P2 r , FindNewSettings(J4,Pi,P2,status, n.,E.) Evaluate resulting degrees of freedom in d o f
9
9 9 9 9 9 9
3.2
Algorithm details
3.2.1
OrderVariables(V,.,V,.,P,,)
A permutation matrix P,, is calculated such that the columns of Jo are ordered according to: I. 2. 3. 4. 5. 6. 7.
Preferred free variables which collide with others Remaining preferred free variables Remaining free variables Variables which are marked to be freed (in the first pass there will be none). Variables currently fixed which were free on input Remaining fixed not preferred to remain as such Remaining fixed variables preferred to remain fixed
233
BuildReducedProblem(Jo, P,,,Js,~)
3.2.2
Define the matrix G as G e j ~- O~'j gj, where ~j.= 1 if variable j is marked as free or to be freed, 0 otherwise. Define the matrix @, as G but with o4=1 if variable j is marked free or to be freed and is preferred to remain so. Obtain J#=Jo G P,, and ~=Jo @, P,,.
FindAssignment(J,P)
3.2.3
This uses Duff's assignment algorithm [2] to find a permutation P which causes pT j pr to have the maximum possible nonzero elements in its diagonal (see [3]). 3.2.4
FindNewSettings(J,P, Q, status, n,,,E,,)
This first finds which rows remain unassigned. For each unassigned row, if it was originally one of those added to make the problem square (i.e., a zero row added to the end), the corresponding variable (the one in the diagonal of the unassigned row) is marked to be fixed. If it was one of those originally in the problem, it is marked as unassigned. 4
An example of use
T a b l e 1 : Results obtained using the proposed algorithm Figure 1 shows a screenshot of Aspen Custom Modeler while using the algorithm proposed in this paper. The chosen example is a simple flowsheet consisting of a network of heat exchangers. The basic case is a pure simulation, where all feeds and unit parameters are known. From the basic case, we pose different problems by fixing/freeing different variables and applying the algorithm. The results are shown in Table 1. The last column indicates variables that were fed into the algorithm with an indication to be preferred to remain unchanged. The first case is one where although the modified problem is square, it is structurally singular. In the other two cases, the modified problems are respectively over and under specified. In all cases the algorithm proposes to put the variables back to the simulation base case. The algorithm has proven to be very efficient. A problem with 15000 equations is solved in less than 1 second of CPU time on a Pentium600 machine, hence the user hardly notices any time lapsed between requesting the specification analysis and obtaining back the results. The largest problem we have tested in Aspen Custom Modeler TM has 138000 equations and a typical specification analysis takes 17 seconds of cpu time on a Pentium 600 machine.
234 5
Conclusions
An efficient algorithm to help solve specification problems in dynamic (and steady state) simulations has been presented. The algorithm is very efficient since it only deals with structural information. The algorithm is the basis of the automated specification analysis tool available in the commercial dynamic simulation package, Aspen Custom Modeler TM.
Acknowledgement: We would like to acknowledge the help received by our colleagues, Murray Laing in the ACM implementation and Steve Zitney in the preparation of the manuscript. References I. W. Morton, C. Collingwood, "An equation analyser for process models", Computers & Chemical Engineering, 22 (4-5) (1998) pp. 571-585. 2. I.S.Duff-"On algorithms for obtaining a maximum traversal", ACM TOMS, 1981 , _7(3), pp315-330 3. Harwell Library- Subroutine MC21A
Figure 1 : Example of use with ACM
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
235
Adaptive Neural Network Model Based Nonlinear Predictive Control of a Fluid Catalytic Cracking Unit
Z. Nagy, S. Agachi, L. Bodizs "Babes-Bolyai "University of Cluj-Napoca, Faculty of Chemistry and Chemical Engineering, 11 Arany Janos, 3400 Cluj-Napoca, Romania,Tel. 40 64 193833 / 23, Fax. 40 64 190818, E-mail: [email protected], [email protected], [email protected] Neural Networks are used for a wide variety of chemical applications because of their ability to learn system features. This paper presents the use of artificial neural networks (ANN) for dynamic modeling and nonlinear model predictive control of a fluid catalytic unit (FCCU). An off-]'ine trained ANN model based predictive control structure (NNMPC) and on adaptive neural network model based predictive control (ANNMPC) scheme were tested. Both control structures give a superior control performance compared to the classical proportional-integral (PI) controllers. To improve the convergence of the optimization process in both the off-line or on-line training of the ANN model and in the on-line control problem the use of genetic algorithm (GA) in combination with the classical optimization algorithms was proposed. 1. INTRODUCTION Model predictive control (MPC) is one of the most widespread advanced process control techniques in the chemical industry [1]. The main idea of MPC algorithm is to solve an optimization problem in order 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 the process model. Most of the predictive controllers use linear models for prediction because of the numerous techniques available for identification and control movement calculation. However, most of the chemical processes are highly nonlinear, with widely varied operating conditions, thus, LMPC usually fails. The drawbacks of the LMPC can be avoided using a nonlinear model of the process for prediction instead the linear one. In the nonlinear model predictive control (NMPC) techniques predictions are usually obtained by integrating the analytical model of the process, described by linear, nonlinear algebraic and differential equations. However, this approach has two main disadvantages compared to LMPC methods: a) it requires the elaboration of a complex, accurate analytical model of the process, which for most of the chemical processes can be very arduous; b) solving the optimization problem by integrating the analytical model for large scale, complicated processes might demand great computation effort and time. These shortcomings can be avoided using artificial neural networks (ANN) as the nonlinear model used in the control movement computation. Advantageous properties of neural networks, like parallel computation, nonlinear mapping and learning capabilities make them an alluring tool in many chemical engineering problems [2], [3], [4].
236 The Fluid Catalytic Cracking (FCC) is one of the main chemical processes in the petrochemical industry. The high nonlinearity of the process, as well as the strong interdependencies among the parameters have been demonstrated in the literature [5], [6], [7]. In this paper the application of a neural network based nonlinear model predictive control (NNMPC) scheme for the FCCU is presented. 2. DYNAMIC MODELING OF THE FCCU USING ANN The productivity of the unit depends on the reaction temperature in the riser. For high productivity one should operate close to the metallurgical limit of temperature of the riser. To increase the operation temperature one should minimize the variation of temperature around its nominal value. The objectives in the control of the unit was not only to increase the productivity by rising the reaction temperature but also to assure an environmentally safe operation of the unit by keeping the concentration of CO in the stack gas below a certain limit. The controlled and manipulated variables for the chosen control structure were: regenerator temperature (Treg), reactor riser temperature (Tr), concentration of carbon monoxide in stack gas (cCOsg), and the fresh feed flow to reactor riser (F3), flow of recycled slurry to reactor riser (F4) and lift air flow rate to the regenerator (F9), respectively. A Matlab version of Amoco Model IV FCCU was used to generate the training data necessary for obtaining the ANN model. To train the network we proposed a combined algorithm base on genetic algorithm (GA) and bayesian regularization (BR). The GAs function in a similar manner as optimization search routines; GAs search the solution space of the objective function by using simulated evolution, i.e., the survival of the fittest strategy or recipe. Generally, the fittest individuals of any population tend to reproduce and survive to the next generation, thus improving successive generations. However, inferior individuals can, by chance, survive and also reproduce. GAs have been shown to solve linear and nonlinear problems by exploring all regions of the state space and exponentially exploiting promising areas through mutation, crossover, and selection operations applied to individuals in the population. GA usually finds very fast the neighborhood of the optimal solution, but presents a slower convergence in finding the exact optimal value [8]. Using the advantages of both algorithms (the GA and BR) in the implemented learning algorithm the GA was used for "preoptimization", that is, to find to neighborhood of the global optima. Starting from the best solution found by the GA the BR algorithm has completed the optimization to find the final weights of the network. Table 1 shows the performance of the combined algorithm compared to the "pure" algorithms. Table 1. Performances of the learning algorithms Algorithm Learning Nr of generations in error GA/ Nr epochs in BR BR 945.7 0/57.5 GA 4.5 123.4/0 GA-BR 6.3 15/11.5
(average values for 200 different learning) % of training Best Average without learning learning success time [s] time [s] 17% 67.3 123.8 2% 320.8 512.2 4% 71.2 98.7
One can observe the faster convergence of the GA-BR algorithm and its ability to avoid to get caught in local minimums. Obviously, the performances of the algorithm depend on its
237 parameters. Increasing the number of population or generation gives a smaller percentage of training without success but on the detriment of learning time. To obtain the internal model of the MPC several ANN with different sampling time (ts=10, 20, 30, 40, 60, 120), different number of neurons in the hidden layer (nrhn=8-20), and different number of past inputs and outputs in the input layer (1-10) were trained and tested for 15 different simulation data. For each input sequence from the 15 simulation, the sum-squared-errors for each output parameters were computed with equation (1). ]
e
=
-
O
2
(y~o+, y ~ )
.
(1)
-
Q
i=I
These errors are represented in figure 1 for the four best ANN structures. The data from simulation 10 are used for training, this explains the minimum value of the errors for this data set. From figure 1 one can observe that the ANN no 4, gives generally the best prediction errors for each of the three outputs. This network has 10 neurons the hidden layer, uses a sampling time of 10 seconds and it has 18 neurons in the input layer. These neurons correspond to the current value (k) and to the past values (with two sampling period, k-1 and k-2) of the three process input (F3, F4, F9) and of the three process output (Tr, Treg, cCOsg). The output layer has 3 neurons, corresponding to the predicted values (k+l) of the three process output Tr, Treg and cCOsg. In the hidden layer as well as in the output layer the sigmoid transfer function was used. This ANN was used for prediction in the nonlinear MPC algorithm. In figure 2 and 3 the prediction performances of the selected ANN model are presented for the training data and one of the testing data, respectively.
140
Tr
,
105 9,,
,
]
cCO
,
,
\
120
\
/
25
t
+1
100
15
~
~
~
10t
+ ~
~
J
~
~
t t
2
~
i
4
' l
6
/I
/
~
~ )
8 10 Simulatiom
4ol
/
t
/
12
14
'/p-
'~t 2
4
6
~,
8 Simlati~s
'K
tO
12
\
..: 14
2
4
0
8 S~n~Uons
10
12
Fig. 1. The errors of the best 4 ANN model for 15 different simulations (1:ts=20 s, nrh,=20; 2:t~=20 s, nr~=15; 3:ts=20 s, nr~=10; 4:ts=10 s, nrhn=10)
14
238
3. ANN MODEL BASED PREDICTIVE CONTROL OF THE FCCU
In the MPC algorithm the future output deviation from the setpoints is minimized, whilst taking account of the control sequence necessary to achieve this objective. The predictive control algorithm used in this paper is based on the iterative solution of the following optimization problem:
rain {~v]' e [12j 9~j(k+llk)-r/(k+')ll 2+ ~" M [[),,-An, Au(k)...Au(k + M - 1) j i/=~ " i=~ /--~
(k + t-1)11 ~ l
(2)
where, y(k+llk) are the predicted outputs based on the information up to k, Au(k+/) future control moves, 2 and ), are the weight coefficients, P the prediction horizon, M control horizon, Ny number of process outputs, N, number of control variables. The predictions y(k+l[k), are obtained from the ANN model. For o prediction horizon P the same ANN model (with the structure of one step ahead prediction) is used recursively. Since,
239 control is based upon the prediction of future outputs obtained from the ANN model of the process, off-sets may occur due to disturbances, plant-model mismatch and noise on measurements. Using the MPC concept the discrepancy between model and process responses can be estimated at each sampling time as the difference between the measured and the predicted value, in the previous sampling time, of the outputs. In circumstances where the process noise is significant the estimate of the prediction offset can be filtered in order to reduce the effect of noise and enhance stability. In this case the prediction is obtained with the following equation:
y(k +zlk)= y (k + zl )+ gfilter
"(YANN(kIk--1)--Y .... "ured(k))
(3)
Using to the results obtained with the GA in solving the optimization problem in the training of the network, we implemented a similar combined optimization technique, GA and Sequential-Quadratic-Programming (SQP), to solve the on-line optimization in the control problem. To further improve the performance of the NNMPC, against unmeasured process disturbances, an adaptive neural network model based predictive control structure was proposed (ANNMPC). According to this approach the ANN is retrained on-line, during the control, using the past values of the controlled and manipulated variables. The simulation results with the off-line trained ANN based MPC (NNMPC), with the ANNMPC and with 3 independent proportional-integral (PI) controller are presented in figure 4. The disturbance simulated is a 5% step change in minute 5 of the coking factor in the feed flow rate.
Fig. 4. Simulation results of of the control for the simulated disturbance (a - uncontrolled process, b - PI, c - NNMPC, d - ANNMPC)
240 Both, ANN model based MPCs are superior to the PI control. The superiority of the ANNMPC compared to the NNMPC can also be observed. The parameters of the algorithms together with the integral of the square errors (ISE) are presented in table 2. Table 2. The parameters and performances of the simulated control algorithms Algorithm Parameters ISE PI TLI,Z3 = 1000, 2000, 3000; 87345.7 K1,2.3 "-0.05, 0.005, 0.005; T.~amp= 10 s NNMPC P = 20; M = I ; K~tter = [1 1 1]" 2 = [1 1 1]; 1348.2 7'= [0.001 0.001 0.001 ]; Tsamp = 10 s; topology described above; GA-SQP used for optimization, GA with 30 generation (20 population in each generation) ANNMPC Idem with NNMPC; Past 50 input and output parameters used to 234.1 retrain the network in each sampling period 4. CONCLUSIONS In this paper, we discussed the applicability of artificial neural networks in the control of a highly nonlinear chemical process. The ANN models can be identified without a priori knowledge from measured input/output data and used in a nonlinear model predictive control scheme. To improve the control performance, by minimizing the model/plant mismatch in the case of unmeasured disturbance an adaptive neural network based predictive control algorithm was proposed and tested. The best control performance was achieved with the ANNMPC, but both the ANNMPC and the NNMPC lead to a much better performance than the conventional PI control structure. Another main contribution of the research is the improvement of the optimization algorithms for both the learning process and the on-line control problem, by the combination of the classical optimization algorithms with genetic algorithm. REFERENCES
1. 2. 3. 4. 5. 6.
Garcia, C.E., Prett, D.M., and Morari, M., Automatica, 25, (1989), 335-348. Aldrich, C., van Deventer S.J., Ind. Eng. Chem. Res., 34 (1), (1995), 216-224. Baughman, D.R., Liu, Y.A., Ind. Eng. Chem. Res., 33 (11), (1994), 2668-2687. Bhat, N., McAvoy, T.J., Computers chem. Engng., 14 (4/5), (1990), 573-583. Cheng, Y., Karjala, T.W., and Himmelblau, D.M., Ind. Eng. Chem. Res., 34, (1995), 1735. Morari M., S. Agachi, I. Huq, J. Bomberger, B. Donno, A. Zheng, Joint research project between Chevron Research and Technology Company and Califomia Institute of Tehnology, Final Report, June (1993). 7. Morari M., S. Agachi, Modeling and Control of a Model IV FCCU, Joint research project Chevron Research and Technology Company and California Institute of Technology, Quarterly Report, March 1992. 8. Goldberg, D.E., Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wiley Publishing Company, Inc. (1989).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
COMPUTER DESIGN OF A SYSTEM OF PREDICTIVE CONTROL CONTINUOUS PROCESS PURIFICATION OF BIOPRODUCTS
241
FOR A
Mattedi, A.; Maciel Filho, R. Department of Chemical Processes - College of Chemical Engineering State University of Campinas - UNICAMP- PO BOX. 6066- ZIP CODE 13081970 Campinas-Brazil E-mail: [email protected] ABSTRACT In this work a strategy is proposed to the implementation of DMC predictive control in a biotechnological process of bioproducts purification. This process is composed by the adsorption, washing and desorption stages, for which a deterministic model was developed. Results of the model simulation in open loop showed the difficulties to control the system, such as: strong interaction among the input and output variables, non linearity and inverse response. The existence of the restriction of the fermented flowrate, in the adsorption stage, is solved through the strategy of switching of the fermented and recycle flowrate. The control actions are taken in values that guarantee the linearity of the system around the steady state operation point. One PID controller, acting in parallel to multivariable DMC, controls the purification factor through actions on the washing stage flowrate. The simulation results in closed loop of this predictive control system showed an excellent performance. Key words: predictive control, process control, purification process, bioproducts purification, adsorption 1.
INTRODUCTION
The studies and researches related to the development of processes of enzyme purification are due, mainly, to the industrial application and the aggregated value of such substances. The enzymes have the function of transforming some organic products in other more valuable of technological applications. Difficulties of continuous operation in large scale (Janson, 1984), took many researchers in the last decade to develop new technologies. One of those techniques with great perspectives of industrial application are the continuous extraction with recycle of the adsorvent, based on the principles of affinity adsorption, which is denominated process CARE ("Continuous Adsorption Recycle Extraction") originally proposed by Pungor et al. (1987). This process is constituted of two interlinked reactors, and in the first reactor the "reaction" of adsorption of the enzyme to be purified happens while in the second the desorption "reaction" occurs. The process operates continuously through the recycle of adsorvent. The kinetic parameters of adsorption and desorption, for the lisosyme enzyme, were obtained from the literature and they are well adapted to the present process (Chase et al. (1984) and Cowan et al. (1986)). The inclusion of a washing stage between the adsorption and desorption stage was proposed, seeking to increase the purification of the process, through the largest elimination of the contaminants. This CARE process of three stages is the case study of this work. The predictive control algorithm in which the proposed control strategy is implemented is based on the DMC approach (Dynamic Matrix Control) in multivariable structure, whose heuristics were developed by Cutler and Ramaker (1979) from the Shell Oil Company (USA) and Cutler (1983). The algorithm uses an internal model represented by response step and it is adapted to multivariable systems with restrictions in the control action.
242
@
THE CARE PROCESS
The CARE process of bioproduct purification, formed by the adsorption, washing and desorption stage, is schematized in the figure 1"
F~,~Co
F2
F3
F,,EI,Ct I
Adsorption Stage
..~i
"1
Washingst F2'age~ Ei__! C2
F,,E3,C3 Ft, El, Ct
w,-I DesorptiOnstage
'
F3,E3,C3
Figure 1. CARE process of three stages
The process operates in a continuous mode with the recycle of adsorvent. This is retained in the system due to the presence of a macroporous membrane contained in each one of the stages. The adsorption stage is based on the adsorption by affinity with high selective capacity and it is admitted as a reversible )wreaction )) of 2 nd order. The desorption stage is admitted as a irreversible "reaction" of 1st order. Each stage is considered as a perfectly mixed reactor and it is admitted the concentrations as aggregated properties (Rodrigues et al., 1992). The moving bed porosity is also considered constant (0.7). Through the mass balance for each component and from the laws of rate of the "reactions" for each stage, the equations of the model of the process can be developed as shown below (table 1). Table
dt
=
%,
1. CARE PROCESS MODEL OF THREE STAGES
+We'(E,-E,)+[k,q,-k,E,(q.-q, %,
dt
"[I
dt
xt
3.
"el
xl
dC,:~_~_(C,_C: ) C2 dt xI ~Ix~ Fp = (E3/Eo)/(C3/Co)
dC3._~JF,(C2_C3) C3 dt
Y~t
dq__A3= Ve (qz - q3) - k3q3
"q2 = ~_~__(q,_q2)+[k, E2(q" _q2)_ k2q2] dt %1 d C l ~- Co-Cl .[.. I~/E(C3 _ Cl) dt xI xt
The constants: "(=
~l
dqt= ~--~--(q, -ql)+[kiEl(qm -ql)-kzql]
dE, = Ve (E2_E3)_ E, 1-, dt ,, --~ + kaq3(---~)
5x I FI/F2
~/= FJFI
8=
FI/F3
%1= Vl/Fi
~2= Vl/F2
e = Vl/(Vl+Vs)
D E V E L O P M E N T OF THE CONTROL SYSTEM
For the design of the control system, it is interesting the study of the dynamic behavior, in open loop, due to the changes of the input variables. By means of that, it is possible to develop the control strategies for the system. In this work the following steps were considered:
243 3.1 The controlled variables and control action time The controlled variables are the enzyme concentrations in the liquid phase of the first (El) and third (E3) stages. Through the control of El, is possible to reduce the amount of eliminated enzyme of the adsorption stage, decreasing in this way, the system losses of the enzyme. The control of E3 is interesting because this variable is related to the final product of the process. Another controlled variable is the purification factor through the traditional controller PID (speed algorithm), that acts to DMC controller in parallel. The time of intervention is limited at the reading time of the chromatograph (0.4 hours) This time is reasonable when HPLC or other chromatography techniques are used. 3.2 Development of the control strategy The restrictions of some variable of the system were decisive in the development of the control strategy of this work. The fermented flowrate (F l) is limited to the production of the fermentation process. The maximum value of the fermented flowrate is 420 ml/h. Other limits are related to the actions of the manipulated variables (20, 2 and 5 ml/h for Fl, Fr and F3 respectively), due to two reasons: 1) to avoid abrupt actions on the manipulated variables, that are inappropriate in the practice; 2) to assure the linearity of the system, since the tests of linearity were accomplished for small variations. The strategy for the treatment of the restrictions is on-line tuned through changes in the suppression factors. The configurations used in the control system of this work are shown in table 2: Table 2. Denomination of the used control configurations
To deal with restriction in F l, the DMC multivariable controller implemented uses the configurations M13 e Mr3. The control system starts with the configuration M13, manipulating the flowrates Fl and F3. When FI reaches the maximum limit (420 ml/h), the system starts to operate in the configuration Mr3, manipulating Fr and F3 and fixing the flowrate Fl in the maximum limit. The manipulation of Fr is indispensable, because otherwise, there would be the problem of lack of freedom degrees for the control. The number of input variables needs to be the same to the output variables, or off-sets will appear in the output variables. In addition, the weak influence of the flowrate F3 on the exit El makes practically impossible the control of El using only the flowrate F3. As soon as the system in the configuration Mr3 is stabilized, the return to the configuration M13 takes place. This return condition can be equationed by means of the mass balance of the enzyme in the liquid phase for the process in steady state: the amount of enzyme in the input variables of the system should be approximately the same total amount of enzyme in the output variables: FIE o -_-FIE 1 + F2E 2 + F3E 3
in that way, as soon as: F, _--F,E,+FeE2+ F~E~ (it was considered a mistake equal 10"lml ) Eo
the system comes back to the configuration MI3 and the flowrate F~ stays near to the value that allowed the system to the new steady state operation condition. The control of the purification factor is accomplished by PID controller through the manipulation of the washing stage flowrate (F2). Through this flowrate is possible to eliminate most of the contaminants of the process. Simulations, in open loop, showed that a change in the flowrate F2 influences the purification factor significantly with a short response time; even so in a strongly no linear relationship. For this reason, it was decided to implement the PID controller for this loop, separately of the DMC controller. The actions of control of F2, produce not
244 significant effect in the output variables El and E3, so that they are admitted as small disturbances for the DMC multivariable controller 4. RESULTS OF SIMULATION In this section some results of evaluation of the performance of control system are presented, for changes of the set-points and disturbance in the enzyme concentration of fermented flowrate 4.1 Servo behavior Usually, changes of set-points are requested when there is a superior hierarchical level of optimization, that has the objective of driving the system to a condition high performance operation. The figures 2 and 3 show the dynamic behavior of the output variables when changes of the set-points are requested. It can be observed that the output variables El and E3 are satisfactorily controlled in order to follow the respective set-points. Due to the slow dynamics of the system, the alterations in response for the output are significant for each change, mainly with respect to the El. Other simulation results showed that the switching happens at around time equal 40 hours and it can be noticed that DMC controller, when operates in the configuration Mr3, presents lower performance for exit El and higher for exit E3. 4.2 Regulatory behavior The variable E0 is submitted to a periodic disturbance, according to the figure 4 to the period equal to 160 hours and variation of 20%. The time interval between the changes is equal to 40 hours, enough time for the system to come back to a new condition of steady state operation. In table 3 are presented part of the data of the simulation. In the time instant of 120.4 hours the change of the configuration M13 to Mr3 happens. The system starts then to operate the manipulated variables Fr and F3 until the instant 155.6 hours, when the system, through the return mechanism, operates in the configuration M13 again. Due to other disturbances, the system in the instant 292.8 hours, moves for the configuration Mr3. The figures 5 and 6 present the dynamic behavior of the output variables due to the periodic disturbance on enzyme concentration E0. The process is controlled satisfactorily, and the deviations of the controlled variables from the respective set-points are acceptable. The response time for the regulatory control is high due to the slow dynamics of the system. Table 3. Data of the simulation for the regulatorybehavior of control system
4.3 Results of the purification factor control In table 4, the values of the flowrate F2 and of the purification factor along the time are presented. A response time significantly short (approximately 5 hours) for the system to reach the new set-point (350) is noticed. For this, in the beginning, larger actions are applied on the flowrate F2 and later the system reaches the steady state. The actions of F2 for the control of the purification factor do not produce significant effects in DMC controller outputs variables. However, the DMC controller actions (except of the flowrate F3) cause a significant effect in the purification factor.
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246 Table 4. Simulation data of ourification factor control for the PID without action of DMC
5.
CONCLUSIONS
The developed control strategies made possible the implementation of the D M C control algorithm in a biotechnical process with non linearities and restrictions of variables. The presented results show the controller's excellent performance. The design methodology is interesting for another applications of industrial process. ACKNOWLEDGMENT Financial support of this research through the FAPESP is gratefully acknowledged. NOTATIONS Acronyms
CARE HPLC PID DMC SISO MIMO
Continuous Adsorption Recycle Extraction High Performance Liquid Chromatograph Proportional-Integrative-Derivative Dynamic Matrix Control Single-Input Single-Output Multiple-Input Multiple-Output
qm
Fv c ns nr nl SF
WE
Letters
K~
V1
Erm3,me concentration in the liquid phase Enzyme concentration in the solid phase Concentration of contaminants in the liquid phase Flowrate of the stages Liquid volume
Vs kl, k2,
Solid volume Constants of specific "reaction"
r
E q C F
1;i 1;D
Constant of maximum adsorption Purification factor Moving bed porosity Model horizon (DMC) Optimization horizon (DMC) Control horizon (DMC) Suppression factor on input F (DMC) Weighting factor on exit E (DMC) Proportional gain (PID) Integral time constant or reset time (PID) Derivative time constant (PID)
Subscripts
1, 2 e 3 0
Adsorption, washing respectively Recycle stage Alimentation flowrate
and
desorption
stages
k3
LITERATURE CITED Chase, H.A., "Prediction of the performance of preparative affinity chromatography", J. Chromat., 297:179-202, (1984a). Cowan, G.H.; Gosling, I.S.; Laws,J.F.; Sweethenham, W.P., "Physical and mathematical modeling to aid scale-up of liquid chromatography". J. Chromat., 363:37-56, (1986). Cutler, C.R. Ramaker, B.L., "DMC- A Computer Control algorithm". AIChE Annual Meeting, Houston, (1979). Cutler, C.R., Dynamic Matrix Control: An optimal multivariable control algorithm with constraints, Ph. D. Thesis, Univ. of Houston, (1983) Janson, J.C., "Large scale affinity purification-state of the art and future prospects". Trends Bio tech., 2:31-38, (1984) Pungor, E.; Afeyan, N.B.; Gordon, N.F., Cooney, C.L., "Continuous Affinity-Recycle Extraction- A novel protein Separation Technique". Bio-Technology, 5: (6), 604-, (1987) Rodrigues, M.I.; Zaror, C.A.; Maugeri, F.; Asenjo, J.A. "Dynamic modeling, simulation and control of continuous adsorption recycle extraction". Chemical Eng. Science, 47(1):263-269, (1992).
European Symposiumon ComputerAidedProcessEngineering- l0 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rights reserved.
247
Knowledge Based Modular Networks for Process Modelling and Control J. Peres, R. Oliveira and S. Feyo de Azevedo Departamento de Engenharia Qufmica, Faculdade de Engenharia da Universidade do Porto, Rua dos Bragas, 4050-123 Porto Codex, Portugal, E-mail: [email protected]* Abstract. In the present work we formulate a general structure for hybrid models. The structure is inspired in the Mixture of Experts (ME) Network developed by Jacobs and Jordan (1991). The expert modules are extended to formulate knowledge at different levels of sophistication. The resulting hybrid modular network can be given a probabilistic interpretation. In particular the learning algorithm is viewed as a maximum likelihood parameter estimation. The concepts are illustrated by the application to a baker's yeast production process.
1 INTRODUCTION Several authors have been proposing hybrid model structures based on the combination of Artificial Neural Networks (ANNs) with mathematical models and/or Fuzzy Systems (Schubert et al (1994)). The general idea is that the model should incorporate as much a priori theoretical knowledge as possible, leaving the ANN modules for mapping those parts of the system for which no other usable form of knowledge is available. The main advantage of such a strategy is that the model will incorporate all sources of information. The difficulty is that the hybrid model approach has not been, up to now, structurally defined, which means to say that a theoretical framework is still missing for such developments. Most of the work presented here is inspired on the papers of Jacobs et al. (1991) and Jordan and Jacobs (1994) who first developed the class of modular connectionist architectures termed as 'mixture of experts' and subsequently as 'hierarchical mixture of experts'. Basically they brought the concept of mixture of models into the field of connectionist systems. The main idea was to develop a connectionist architecture, which is able to learn how to partition a task into two or more functionally independent tasks and to allocate distinct networks to learn each task. The learning algorithm is viewed as a mixture estimation problem (E.g. Titterington et al., 1985). Other works extended the concept of 'mixture of experts' from which we distinguish the work of Weigend et. al (1995) where the expert networks used are non-linear (the architecture was termed as 'gated mixture of experts'), and the work of Ramamurti and Ghosh (1999) who suggested the use of localised gating networks based on radial basis functions. In the present work we want to explore the structure and properties of ME Networks for hybrid modelling of static/dynamical systems. This is discussed in detail in the following sections.
* This work was developed in the Research Unit 147/94 of the Portuguese Foundation for Science and Technology - Instituto de Sistemas e Rob6tica- Porto (URL: www.fe.up.pt/isrp)
248 2 KNOWLEDGE-BASED MODULAR NETWORKS 2.1 The architecture The training patterns {x,d}, denoted by the input vector x and desired response (target vector) vector d, are assumed to be generated by a set of k different regressive processes which are continuous non-linear and dynamical in nature. As such the expert modules are of the form dy! d t = fi (yi, x, wi)
(1)
being yi the output of expert module i, x the input vector, wi the parameters vector of expert i and fi a continuous non-linear function (notice that the expert modules suggested by Jacobs et al. (1991) were very simple linear relationships). The functional relationships fi are not the same for all the expert modules. We assume that the expert modules may include a priori knowledge and that this knowledge is available on different levels of sophistication. As such we have three main types of expert modules: i) white-box modules which express mechanistic knowledge by means of mathematical equations, ii) grey-box modules which express heuristic knowledge by means of Fuzzy Inferential Systems and iii) black-box modules which are able to capture information from process data. It is assumed that each type of knowledge represents better the process in some region of the input space. The job for the network is to partition the input space in three subregions, one region for each of the expert modules. With this configuration we are exploiting the best that each type of knowledge has to offer. The expert modules are mediated by a gating network as in the case of the ME network (Fig. (1)). There are two main types of gating networks mentioned in the literature (Haykin (1994)): i) based on 'softmax' functions, and ii) based on radial basis functions. It is important to mention here the importance of the relationship between the nature of the experts and that of the gating system. With very simple expert modules, say linear modules, there is the need of many experts for an acceptable mapping. With more sophisticated experts only a few experts are required. This does not mean that the gating system should divide the input space only in a few sub-regions. On the contrary, the gating system must be flexible enough to partition the input space in several sub-regions and to activate each expert in several different regions. Having these ideas in mind, we propose a gating system designed in 3 steps: x
Expert network 1
y ~
Expert network2
Y2
...
Expert network k
j / ~ gk i
Gating network
k Y= ~.]Yigi i=l Figure 1. Mixture of Experts network: the experts are linear and the gating network is based on a set of k 'softmax' processing units.
249
Step 1. Hyperspherical clustering of the input space with a set of C clusters distributed equidistantly through the input space. The clusters are based on radial basis functions ri ri (x, ci, ~.i) "- (2P) -n/2
I?Ei1-1/2exp {-1/2
(X-Ci) T ]~i -1 (X-Ci) }
(2)
being ei the clusters centres and Ei the covariance matrices. The covariance matrices are set diagonal ]~i : diag{•i2}. The standard deviations are computed by the nearest-neighbour algorithm. The set of clusters are divided into two subsets (Fig. (2)): The subset C of clusters within the training region and the subset B of clusters outside this region.
Step 2. Define randomly K subsets of C/K clusters {C}j within the training region. Each subset {C}j is associated with a specific expert module j. The number of clusters for each expert network is equal to C/K. The index vector I is used for assigning an expert j to each cluster.
Step 3. Given the input pattern x and the index vector I, peek for each expert j the nearest clusters q(x,I). The gating network outputs are then given by K
gj(x,I)= rj(x,I) ( ~ ri(x,I) )-1
(3)
i=1
2.2 Training algorithm for the Knowledge Based Modular Network The training algorithm should be viewed as a maximum likelihood parameter estimation problem. This is possible by giving a probabilistic interpretation to the architecture. Assuming gaussian distributions, the conditional probability of pattern y, given the input vector x and given the expert module i, is P(ylx,i) = (2p) -m/2 [Zi1-1/2 exp {-1/2 (y-yi) T Zi -1 (Y-Yi)}
(4)
being Zi the covariance matrix for expert i. The number of outputs is m = dim(y) = dim(yi). The output vector of the expert modules is interpreted as the mean of the gaussian distribution. The outputs of the gating network are interpreted as the conditional probability P(ilx) = gi(x,l) of peeking expert i given the input vector x for generating pattern y.
Figure 2: Schematic representation of hyperspherical clustering in a two dimensional input space. The region in grey defines a sub-space C of measured data
250 Jacobs et al. (1991a) applied a gradient ascendant weights updating algorithm where the weights of the experts and gating networks are updated simultaneously. Jordan and Jacobs (1994) applied the Expectation Maximisation (EM) algorithm for training the network, which proved to converge much faster then the gradient ascendant algorithm. Xu and Jordan (1996) studied the convergence properties of the EM algorithm concluding that the algorithm has a linear convergence rate. In the present work we adopt the EM algorithm. The algorithm is an iterative procedure consisting of 2 steps: the E-step and the M-Step. The E-Step computes the posterior probabilities hi (t) given by
gi(x(t),l)P(ylx(t), i)
hi(t)(p)_ -
k
(5)
gj(x(t),l)P(ylx(t)d) j=l The superscript index (t) represents the training pattern and (p) iteration. In the M-Step two separate maximisation problems are solved: w/~p)= argmax ( ~ hi (t) In P(ylx,i) ) t
(6)
I q') = argmax ( ~ ~ hj(t) In gj(0) t j
(7)
being w = [ w l , w 2 ..... Wk]T the parameters vector of the expert modules and I a set of integer
valued parameters of the gating system. 3 APPLICATION: PREDICTION OF BIOMASS CONCENTRATION IN A BAKER'S YEAST FERMENTATION PROCESS In the present application we develop a Knowledge Based Modular Network for predicting the evolution of biomass concentration in time as a function of glucose feeding rate profile F(t) which is the most important manipulated variable in a baker's yeast fermentation process. There are two main sources of knowledge for developing the prediction model: (i) Mechanistic knowledge - the most well-known mechanistic model for the Baker's yeast fermentation process is the bottleneck model of Sonnleitner and K~ippeli ( 1 9 8 6 ) - and (ii) Process data - for the present case, data of 5 fermentation runs are available which may contain additional important information not incorporated in the bottleneck model. The prediction Hybrid Modular Network will attempt to express in the most efficient way the two types of knowledge available. As such, the Knowledge Based Modular Network will be composed by two expert models mediated by a gating network. Expert 1 is a mechanistic (white-box) expert based on: i) mass balance equations on the relevant compounds ii) bottleneck kinetic model of Sonnleitner and K~ippeli (1986) and iii) general principles of macroscopic stoichiometry. The only unknown in the model is the input feed rate F profile. Expert 2 is a grey-box model based on the biomass (X) mass balance equation dX/dt = ( f l - F / V ) X
(8)
251
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F2 -'.
12
~ F1
1.0 /
/
,...
,,:
(a)
J
(b)
0.8
,'
/
;
0.6 84
/,
0.4 84 0.2
:i"
i
0.0
0
~0
20
30
4o
50
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6o
70
0
2 4 6 8 10 Glucose feed rate (g L -~ h ~)
Figure 3. (a) Biomass estimation as a function of sample number for fermentations F1 to F5: O, measured; ---, prediction results of Expert 1; ..., prediction results of Expert 2 ; - - , prediction results after training with the EM algorithm. (b) Gating network outputs as a function of glucose feed rate after training with the EM a l g o r i t h m : - - , gl; ---, g2. where the specific growth rate (/~) is modelled with a 3-layers Feedforward Neural Network with 1 input node, 5 hidden nodes, 1 output node and sigmoidal activation functions. The input to the network is glucose feeding rate per unit volume defined as Fs = F Si,/V being Si, the glucose concentration in the input feed F and V the broth volume. 4 RESULTS AND DISCUSSION
The final results of the EM training procedure are rather sensitive to the initial parameters' values. If one expert model is able to describe more accurately the dynamics of the process in the whole input space, the results after applying the EM algorithm are that only that expert model is used. As such, in a first step we adapt independently the two expert models to the available training data. The biomass prediction results are shown together with the measured values in Fig. (3a). The statistics used for comparing the models are the MeanSquare-Error (MSE). The plots in Fig. (3a) show that the mechanistic model (expert 1) predicts reasonably well the biomass measurements for fermentations F1 and F3, but badly for fermentations F2, F4 and F5. The initial MSE for the 5 runs is 15.00. The grey-box expert (expert 2) predicts reasonably well the biomass measured values for all five fermentation runs (MSE=l.28), however there are some measurements better predicted by the mechanistic expert. In a second step the Knowledge Based Modular Network is trained with the EM algorithm. The results obtained after 10 iterations are plotted in Fig. (3). The final MSE is 0.35 representing a big improvement compared to the initial MSE=l.28 of the grey-box expert. Only 3 out of the 33 clusters used to represent the measured input space are assigned to the mechanistic expert. This means that the grey-box expert is able to predict much more accurately the current set of biomass measurements than the mechanistic expert. In Fig. (3b) the gating network outputs are plotted as a function of the glucose feed rate. In the range F,=0-6 g L-~h-~ the gating output corresponding to the grey-box expert is almost always gl=l
252 while g2=O. However for values of Fs>6 g L-lh-I only the mechanistic expert is used (g2=l). This is in agreement with the initial design condition that outside the measured input space mechanistic models are the most reliable modelling techniques. 5 CONCLUSIONS Hybrid modelling is a very powerful tool for industrial process improvement. Its main advantage is providing a general framework for exploiting all available sources of knowledge such as mechanistic, heuristic and process data, for developing in a systematic way better process operating strategies. The central issue in hybrid modelling is efficient knowledge utilisation, i.e., how to combine different modelling methods while exploiting the best each type of knowledge has to offer. Hybrid Modular Networks may constitute a valuable method for hybrid process modelling. One of the most attractive points is that it can learn how to combine the various expert modules. The training is based on solid statistical concepts such as maximum likelihood parameter estimation, which is a very well studied subject. The biggest disadvantages lies on its complexity, which can be to some extent overcome with flexible and friendly software tools. REFERENCES
Haykin S., Neural Networks: A Comprehensive Foundation, Macmillan Publishing Company, Englewood Cliffs, NJ (1994). Jacobs R.A., M.I. Jordan, and A.G. Barto, Task decomposition through competition in a modular connectionist architecture: The what and where vision tasks. Cognitive Science lg, pp. 219-250 (1991). Jacobs R.A., M.I. Jordan, S.J. Nowlan and G.E. Hinton, Adaptive mixtures of local experts, Neural Computation, 3, pp. 79-87 (1991 a). Jordan M.I. and R.A. Jacobs, Hierarchical mixtures of experts and the EM algorithm. Neural computation, 6, pp. 181-214 (1994). Ramamurti V. and J. Ghosh, Structurally Adaptive Modular Networks for Nonstationary Environments. IEEE Trans. Neural Networks, 10(1), pp. 152-160 (1999). Sonnleitner B. and O. K~ippeli, Growth of Saccharomyces cerevisiae is controlled by its Limited Respiratory Capacity: Formulation and Verification of a Hypothesis," Biotech. Bioeng., 28, pp. 927-937 (1986). Schubert J., R. Simutis, M. Doors, I. Havlik and A. Ltibbert, Hybrid Modelling of Yeast Production Processes. Chem. Eng. Technol., 17, pp. 10-20 (1994). Titterington D.M, A.F.M. Smith and U.E. Markov, Statistical Analysis of Finite Mixture Distributions. New York: John Wiley (1985). Weigend A.S., M. Mangeas, A.N. Srivastava, Nonlinear gated experts for time series: discovering regimes and avoiding overfitting. International Journal of Neural Systems, 6, pp. 373-399 (1995). Xu L. and M.I. Jordan, On Convergence Properties of the EM Algorithm for Gaussian Mixtures. Neural Computation, 8, pp. 129-151 (1996).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
253
C o m p u t e r aided and control of a rotary kiln incinerator Souza, E.T. Inglez dea, Maciel Filho,
R. b ,
Tomas, E. c
a,b,c State University of Campinas- School of Chemical Engineering, Campinas, S P . Brazil, Postal Box 6066 - CEP: 13081-970 e-mail: etis@,lopca.feq.unicamp.br Abstract
The performance analysis of the incineration process, specially that carried in rotary kiln incinerator, is very important in social and environmental aspect because this technology treats the elimination of hazardous waste by thermal destruction, resulting in volume reduction up to 99,99%. However, for involving combustion and enormous operational expense of the equipment the incineration finds certain resistance to be more thoroughly accepted. Beating this in mind, it is important to have a reliable mathematical model as well as a solution procedure so that extensive simulation can be done. The objective is to find out the best operation strategy so that lower costs and safety can be reached. Taking this into consideration, this work seeks to contribute for the improvement, of the process representation, working with the simultaneous solution of the mathematical model that comes to predict satisfactorily the behavior of the main process variables in function of the operational parameters. Also the implementation of several control strategies, is carried out taking into account the industrial reality. Both studies have as objective the reduction of the operational costs always respecting the limits of pollution emission determined by the environmental protection agencies, including actions to avoid dioxins production. 1. Introduction
Due to the growing importance of the environmental aspect in the chemical processes, mainly in the subjects related to pollution emissions and the decrease of residues, the present work studies the reduction of solid and from hazardous waste with rotary kilns incinerator. As alternative to the landfill and other used in waste elimination, the incineration process has, capability to reduce the waste volume up to ninety percent due to the elimination of the water, combustion of the volatile matter and organic materials carried by thermal destruction. In fact, a process that involves combustion always present the disadvantage of the emission of pollution gases, such as the dioxins and the products of incomplete combustion. However with operational supervision of the main incinerator variables, this may be minimized and controlled to reach the environmental requirements. This work aims to contributes for the incineration be a safe and more economic process, which then can be widely used.
254
2. Process Description Model development consists of a series of equations, mass and energy balances for the main flows that occur in a combustion chamber: gases(g), jet(j) and solids(s). The chamber of the incinerator is divided in zones(i) to obtain the axial temperature profile. For each one of these zones, there are equations for the mass and energy balances for the three flows mentioned above plus the inner and outer wall temperature calculations. The dynamic modeling, used in control strategies, is represented by the same equations applied to steady state conditions, with time domain terms. Energy balance for the solid, gas and jet. Ts(i + 1) = Hs(i) + Ha(i) - ava(i) - nsv(i) + qs(i) + Tref Ms(i + 1). cps(i) + Ma(i + 1). cph2o(i)
(1)
Tg(i + 1) = Hg(i) + Hva(i) + Hsv(i) + D I ~ i ) - Hen(i) + qg(i) + Tref Mg(i + 1). cpg(i)
(2)
Tj(i + 1) = Hj(i)+ Hen(i) + Qgel(i)+ qj(i) + Tref Mj(i). cpj(i)
(3)
Mass balances the for solid, gas and jet. Ms(i) = Ms(i+1) +AMs(i) Mg(i+ 1)=Mg(i)+AMs(i)-AMa(i) Mj(i+ 1) = Mj(i) +AMen(i)
(4) (5) (6)
Inner and outer wall temperature
pw.Vw.cpw, dTw(i) _ qw(i) (in steady state dt Pew.Vow.cpew. dTinnerw(i) = qinner w(i) dt
dTw (i)
~=0) dt
(in steady state dTinner w(i) = 0) dt
(7) (8)
The solution of the model in steady state is not trivial, due to the non linearity of the model, and therefore it necessary to search which methodology is more suitable with a computational algorithm that is capable to solve the problem in a fast and robust way. The proposed solution procedure is an specific alternative to solve the equations showed above. The algorithm is relatively simple, it uses the equations (1) to (3), with initial guesses for all variables, and obtain estimates for the profile along the rotary kiln. The estimated values are the new initial guess and the calculation are repeated again, in the same way. The loop continue until the process reach the final convergence is reached. The new initial guesses calculation are as follow:
Te+l(guess) =
Tk (guess) + T(estimate)
(9)
255 The convergence criterion works in the sense to insure that the larger difference between a new initial guess and the estimated value is smaller or equal than error tolerance. All the mass balances do not need to be included in the loop, because the updated done in each iteration is enough to lead the convergence the mass variables. This is possible due to the weak influence of the temperatures in the mass balances. The dynamic model is build up considering the time variations in the energy balances only. The mass balance is considered as pseudo-steady state. Such hypothesis is quite reasonable, if it is considered that the delay time for mass flows dynamics is instantaneous or much faster than changes in temperature. The development of a fast dynamic model, in real time, need a efficient numerical method. In this case, the Gear method (LSOLDE subroutine) for a system of ordinary differential equation of first order is a good option and it was used in this work. Dynamic model Ps.Vs.Cps. dTs(i + 1) = -Hs(i + 1) + Hs(i) + Ha(i) - Ha(i + 1) - Hva(i) - Hsv(i) + qs(i) dt dTj(i + 1) Pi.Vj.cpj = -Hj(i + 1) + Hj(i) + Hen(i) + Qger(i) + qj(i) dt Ps .Vg.cpg dTg(i dt + 1) _- -Hg(i + 1) + Hg(i)- Hen(i) + AHr(i) + Hva(i) + Hsv(i) + qg(i)
(10) (11) (12)
2.1 Radiative, condutive and convective flows heat determination (qg, qj, qs, qw) The flow heat model is the most important part in an incinerator and it must be carefully analyzed. In this work, it was assumed that there are influences of radiative flows in adjacent zones. Some other hypotheses had also been used to simplify the flows of heat, to know, the gas is transparent to radiation and all the surfaces are black. The basic idea is represented by a balance of radiactive, convective and conductive flows. The total energy emitted by an element of the balance, wall, jet or solid is fractioned in amounts determined by the view factors and it reaches all the other elements. These balances are made for the wall, the external wall, gases, jet and solids The equations for the total heat exchanged by solids, gas, jet flow, intern and extern walls as follow: Qr~o. = ~ condutive(in / out) + ~ convectine(in/ ou) + ~2 radiative(in / out)
(13)
For example, in the jet: 0jet = Ajet.Fjs .cy.(Ts4 - Tj4) +Ajet.hjet.(Tg- Tj ) + Awan.Fwj .cr. (Tw 4 - T j 4 ) + Awau.Fjwp .Z. (Tj 4 - Twp4 )+ Aja.Fjwa .t~. (Tj 4 - Tw. 4 ) + Ajr .a. (Tj 4 - Tsv4 ) + A~olid.Fsjp .z. (Tj 4 - Ts 4 )
(14)
3. Control Strategies The control strategies make use of the feedback, feedforward and the combined feedback-feedforward concepts. The multivariable feedback-feedforward strategy shows to be very efficient and has a good potential to be applied successfully in real operation.
256 Controlled variable is the oxygen c o n t e n t in the exit of the kiln, but the gas exit temperature must be supervised, because it is also very important in the incineration process. The manipulate variable is the solid flow, that change your value to ensure the fixed conditions on the process. The disturbances are done in any inlet mass flow operational parameter, solid, air or fuel. 4. R e s u l t s
To be possible the evaluation of the prediction obtained by the developed mathematical model, the parameters used for the simulations were obtained from real conditions of the incineration process. Table 1 - Parameters used in the simulation Chamber parameters Zones number (i)
Intern and extern diameters
Chamber distance
1,40/1,25 m
12,0 m
10 Process parameters
Inlet temperatures(K)
Inlet flows rate (kg/s)
Solid (Ms): 0,20 primary air (Mj): 0,85 Solid (Ts): 303 primary air (Tj): 333 secondary air (Mg): 3,65 fuel" 0,10 secondary air (Tg): 333 fuel: 333 The results obtained are coherent with what is expected. It is observed that along the axial axis, the concentration of oxygen decreases. This is due to its consumption with solid waste burning. The concentrations of carbon dioxide, sulfur dioxide, increases along the axial axis. Its growing profiles depend on the generation of combustion products of the solid waste, that is being burned along the combustion chamber. - - . - - Wall ----.-- Solid ~ l i ~ Gas
1300 12oo 1100
[~ I
~---- Jel \
~
Extern Wall
ioo -'
0'14 t 0,12
.ot
7oo2
o,oe-]
8o'o,1 J
soo
F-
3O0 2OO
0,22 0,20 ~ , L 0,18
0,0
0,2
0,4 0,6 AXial distance
Figure 4 temperatures profile.
0,0
Rotary kiln
- - , - - CO 2 ---e w H 2 0
--~w 02
\ \
~ i
/
w~m~
,. - - ' - ' ' a ' ' ' ' ~
.~x:--
o=I. - - - -
1,0
0,0
0,2
0,4
0,6
0,8
1,0
Axial distance
axial
Figure 5 - Rotary kiln SO2,CO2 and 02 profile in the gas.
257
1
.:li l..1\ 992 ~
.~Ib-&-~k~b ~
991,0
-~ & & "-- -" ~-~ ~ ~ "-- ~ "--
980,5 980,0. seg,5 '
"1/
.,o,,
--.-+,o,I +
~.7-1 %,, .......................... 0
5
10
980,s~ 3. re, o;
20
-l-i
-I14 -i -l-i
4-1
--ii - i 4 - i - i
-i
/
;
i_._+,o~ I --,-.lo~ I
EQ ~-, 9e7,5 in m r 987,0
u~ m,s
986,0
15
i/ii ,_i -i -i-i
1"14--1444444--14--14--14444--14
25
Time - s
Time - s
Figure 6 - Exit gas temperature in open loop response for +10% step in the secondary air flow.
Figure 7 - Exit gas temperature in open loop response for + 10% step in the solid
flow
As observed, the dynamics of the process is fast. This characteristic is credited to the short residence time of the gases, showing the prediction capability of the mathematical model. The PID Controller is able to control the incineration process for load disturbances in the controlled variables as well as in the operational parameters. Its dynamics is also fast, facilitating that the incineration process operates in a range of operational parameters that are favorable to the environmental safety, with the best performance. The feedforward control, predicted with experimental planning, show to be efficient to adjust the process for larger load disturbances , since it appears an off-set due to the predictive statistical model error of the process. So, the best option is to use it only to reduce the initial disturbances and work together with a feedback controller, with PID, to refine the loop and to reach the set point, as it presents the figure 10. 0,02~ -
0,026 ]
]~ 0.0220,
0,024 t
+~
l V
to,o=tx / o,o,.t//
~o,0.
t
0,014 ~
+ 10%
.~
+,+,-o.+
o.
]IN
"
+oo.o
+ ..o,,.
8
~11 .
.
.
.
'
,
+
.....
0.0200
,
Time - s
Figure 8 - PID control response for figure 7 situation (step + 10%).
Tin'~ - s
Figure 9 - Feedforward control response for figure 7 situation(step +10%).
258 0.0230
.--.
9
0.0230.
/ \/-
0,0220
~ " ~
-
_
S~ p~-O.O~3
i~ ~176\ / o ...... t \ / !
,
,
,
0
, 10
9
,
.
,
20
30
Time
,,
,,
40
,,
! 60
- s
Figure 10 - Feedback-Feedforward control response for figure 7 situation(step +10%). 5. Conclusions
With a good and representative deterministic model describing the main characteristics of an incinerator in industrial scale, is possible to define the better operation conditions and control strategy. A very robust steady-state and dynamic model, able to represent the main phenomena taking place in rotary kiln incinerator, has been developed. It has some advantage when compared to the mathematical representation based on electrical circuits analogy, since it is possible to identify the terms which have more physical significance. With the model extensive simulation can be carried as well as the analysis of different control strategies, which appears to have large potential for industrial use, since it is relatively easy to implement. It was possible to explore the dynamic characteristics of the incinerator so that high operational performance with lower costs and safety have been obtained. References
1. Tomas E.; Mathematical modeling and simulation of an rotary kiln incinerator for hazardous wastes in steady and non steady state, PhD Theses, Chemical Engineering College ,Unicamp (1998) 2. Gorog J.P. et. al.; Radiative Heat Transfer in Rorary Kilns, American Society for Metals and The Metalurgical Society of Aime, Volume 12B (1981) 3. Dennis J.E. et. al.; Numerical Methods for Unconstrained Optimization and Nonlinear Equations, SIAM (1996) 4. Bazaraa M.S.; Sherali H.D.; Shetty C.M; Nonlinear Programming Theory and Algorithms, 2~ John Wiley & Sons Inc.(1979)
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
The use of controllers
259
process dynamic simulation for learning to design digital
Marta S. Basualdo a'b , Jos6 Salcedo Baand Diego Ruiz c a Depto. de Electr6nica. FCEIyA. UNR, Riobamba 326, 2000 Rosario (Santa Fe), Argentina b GIAIQ. UTN. FRR, Zeballos 1341, 2000 Rosario (Santa Fe), Argentina c Chemical Engineering Department, UPC, Av. Diagonal 647, E-08028 Barcelona, Spain The discussion presented in this paper draws on a comparative study based on several design techniques of digital feedback controllers implemented as SISO structure of a binary distillation column. By using rigorous process models, which serves as an authentic pilot plant, offers an attractive tool to learn more efficiently the "real" control problem. It must be noted that many textbooks have presented several design techniques about this subject but they did not compare over chemical processes. Generally the overall conclusions are based on linear transfer functions only. Chemical processes represent a real challenger for developing efficient digital control design techniques. The overall steps from the identification of the nonlinear, with dead time and inverse response system, including the design and tune of the controllers are presented. Hence, is easier to verify which is the "cost" of obtaining bad models which drive to bad controller designs. Several proofs, for load and reference changes are carried out by designing discrete controllers such as PID, Ragazzini and W transform methodologies by using MATLAB-SIMULINK software. The rigorous model of the distillation column is developed through an S-function of SIMULINK. 1. INTRODUCCION Simulation is the construction and use of a computer-based representation, or model, of some part of the real world as a substitute vehicle for experiment and behavior prediction. A pre-requisite, necessarily, is that the model be a good representation of reality and therefore an adequate predictor of what will happen. Several technical journals report simulation in a wide range of applications, based on that major decisions appear to be made using simulation and substantial benefits are claimed. This paper commences by examining the process simulation area of application in linear identification and digital controllers design. An analog publication was presented by Basualdo (1995) but using FORTRAN code for the simulator. The principal advantage of working in MATLAB platform is that the students can construct their own design and see clearer the differences. A feedback analog PID structure is replaced by digital one implemented by position algorithm. In addition Ragazzini and W transform philosophies are proposed as a practical work for controlling the top distillation column composition, in a single input-single output (SISO) configuration. Hence, it can be evaluated the impact of each control strategy, tuning parameters and the achievable performance with
* Author to whom all correspondence should be adressed, FAX: 54-341-4821772. E-mail [email protected]
260 each controller design. Therefore, better understanding about the trade off between servo and regulator problem and the incidence of the modeling errors for achieving a satisfactory performance can be obtained. In this work, the simulator mimics the nonlinear behavior of a binary distillation column. It can be used as an unknown plant, which need to be controlled. The mathematical model of the column consists in a set of differential equations, which represents the heat and mass transfer on each plate, condenser and reboiler. They are integrated by Runge-Kutta of 5 th order method. Since the program allows introducing step changes in the manipulated and disturbance variables, the problem begins by generating the transfer function (identification). Finally several tests with different digital controller designs, based on the identification done in the previous step, are analyzed. 2. S I M U L A T E D S Y S T E M In this section a brief description of the mathematical model implemented in MATLAB | 5.1 is presented. The system is shown in Figure 1 and consists in an adiabatic continuous column distillation used as a "real plant". It has 17 sieve plates and is fed in plate 7. The nonlinear dynamic model is developed under the following assumptions: in this example a binary mix of Benzene - Toluene is considered, vapor holdup is negligible compared with liquid holdup, each plate acts as an ideal stage; the system exhibits essentially ideal behavior; perfect mixing occurs in the reboiler, reflux accumulator and on each tray; negligible energy accumulation on all trays; perfect reboiler and accumulator level control; perfect pressure control. A total condenser is used and Francis weir Formula is employed. 3. P R O P O S E D APPLICATION WORKS The system shown in Figure 1 is represented by SIMULINK as it is shown in Figure 2. Making double click in the sub-diagram named "column" appears the second image shown in Figure 2. In this scheme the block called "colas2" is the s-function which contains the code for the rigorous simulation of the distillation column. The scheme shown in Figure 3 contains the block, which serves for simulating the controller. Here top Benzene composition is the controlled variable (CV1) and Reflux (R) is the manipulated variable (MV1). The perturbations considered here are the feed flow and feed compositions, F1 and XAF1 respectively. The system identification is done by doing step changes of + 1 % on variable R. Hence, by identifying both outputs plant, one can determine whether the system presents a non linear behavior. In Table 1 the identification parameters for both changes are presented, so the final transfer function is adopted as an average between the values corresponding to each step. For this example the output responses are over damped, so Cohen-Coon tuning method ()kstr6m and H~igglund, 1995) can be applied. The transfer function plant can be approximated as a first order plus time delay transfer function called "Gp".
Gp(s) =
1.192823 e
-2.5145795s
12.663015 s + 1
(1)
261
~
. 2 ........................ ~ : ~ ~"~':"
" xt~l
REFLUX ~st~
column
,.
I
~...~>1
]
,Y4kF I
boilup
Ii0
D~
IF
Oil
':s 9 ..4.~v ~
==i
,--,
:~=
.........
.-
F1
~
~em~
umn
Dem~
.-.g...
;0XlB
Rimlmr
Fig. 1" the binary distillation column (the "real system")
@
%
:EFLUX
Anal~
REFLUX
SPxab
~nbol
REFLUX
Fig. 2: SIMULINK diagrams corresponding to system shown in Fig. 1.
Table 1" parameters of the plant function Positive Negative R R step step Kp 1.108787 1.276859 r 2.563649 2.46551 rD 12.71601 12.61002
transfer Average 1.192823 2.514579 12.66301
Fig. 3: analog controller which is replaced for different digital designs 3.1. Digital PID design Firstly the system is controlled with an analog PID which must be replaced by a digital one in position algorithm given by (2). D(z) = Kc
[(1+--+ r, Y
z2
r, Y
/z+r,r
(2)
Z2 - Z
Therefore with the help of many scripts which can run in MATLAB the corresponding discrete transfer functions can be calculated for several time periods (T) and tuning parameters obtained for the rules of Cohen-Coon In this case T =lmin gave satisfactory
262 values. However, the final tune is done based on the minimization of the IAE (Integral Absolute Error) criteria. This is obtained by simulation of the rigorous model. The optimized parameters are given in Table 2, where can be seen the improvement through the IAE values. This exercise is useful for showing to the future engineers which is the "cost" of having a linear approximation of the plant for the control design problem and its influence over the selection of the best tuning parameters. In Figure 4 can be seen the dynamic behavior of the top composition for both sets of tuning parameters presented in Table 2.
0,~ . . . . . . . i ',(/C~- C(:x~ .
.
.
Table 2: discrete PID tuning. Method Parameters IAE% Kc = 4.905 Cohen TI = 6.761 9.82 Coon TD = 1.071 Minimum Kc = 3.971 TI = 15.236 6.34 IAE TD= 1.712
O,
ii
o,~
If'~.~l ~ ~.. y " ~ - ~ ; ......-~-"
\
.
r\ o,97~
nirirrt~ IAE
0,971 0,96 0
2O
4O
~rre(rrin)
Fig. 4: dynamic behavior of XAB for the tuning parameters given in Table 2. In addition, Ragazzini and W transform control structures are designed and evaluated for the same perturbation. The proposed design methods given in Franklin-Powell (1990) are used in this work. Again the IAE criteria is accounted for changing the initial conditions in order to obtain better dynamic behavior which option will be called "Dop". Therefore, the design specifications are adopted based on the dynamic characteristics given by the system controlled with the discrete PID. That is overshoot = 8.7% and settling time (5%) = 13 minutes. In the frequency domain those specifications mean' ~:= 0.614 MO ___-100 ~:= 61.4 ~ (3) 3 con- ~ trs~ = 0.375 rad / min 3.2. W transformed design Here is used the bilinear transformation between z and w given in eq. (4)
2 z- 1 W-Tz+l In this case Gp(z), obtained by the discrete equivalent of zero order hold for T= 1 is 4.486002E - 02(z + 1.019036) Gp(z) = 3 z (z - 0.9240675) Applying relation (4) Gp(w) is calculated as 4.438282E- 04(w- 212.1282)(w-2) 3 Gp(w) -
(w+ 2)3 (w+ 7.892914E-02)
(4) (5)
(6)
263 For the design conditions given in (3) the resulting controller is: 1 7 . 9 5 3 3 (w + 0.2)
D(w) =
2
(7),
w (w + 2 . 0 8 3 1 7 )
and transforming in Z domain 2 10.64052(z - 0.8182)
(8).
D(z) =
(z - 1) (z + 2.036898E - 02) An improved design is obtained by minimizing IAE criteria is, 22.8113(z - 0.718083)(z - 0.678083)
(9)
Dop(z) =
( z - 1)(z + 0553766) In Table 3 are presented the IAE values obtained with both controllers (8) and (9). Table 3" IAE values obtained with both controllers (8) and (9) Controller IAE% D(z) 15.95 ..................................p o p ( z )
.............................................7..2.3. .........
3.3. Ragazzini design methodology By applying Ragazzini methodology the following controller is designed: H(z) =
D(z).Gp(z) 1+ D(z).Gp(z)
(10)
then, D(z) results D(z) =
1
H(z)
Gp(z) 1 - H ( z )
(11).
Since Gp(z), given in (5) presents a delay it must be accounted by H(z). In addition there is a zero outside the unit circle of Z-plane, so it will provoke unstability when Gp(z) is inverted for the controller design. Another requirement is to have zero static error which means H ( 1 ) = I . The denominator of H(z) is determined in such a way that conditions (3) must be achieved. Therefore H(z) results: H ( z ) = 5 " 5 4 4 1 8 1 1 0 - 2 (z + 1 . 0 1 9 0 3 6 ) (12) 2
z (z
2
- 1.51837z+0.6303132)
Hence replacing (5) and (12) in (11) the transfer function for the controller is: (T=lmin) D(z) -
1.2359z 3 ( z - 0.9245)
(13)
(z- 1)(z + 0.2119)(z2 - 0.7302z+ O.2667) The final design is obtained, as in the previous cases over the rigorous model and searching for the minimum IAE. Hence the design is chosen as" Dop(z) = 4 . 6 3 5 9 ( z - O . 8 6 4 1 ) ( z - O . 4 ) ( z - O . 1 5 )
(14)
( z - 1)(z 2 - 0 . 7 6 0 2 5 z + 0 . 3 6 6 7 )
The achievable performance with both controllers in closed loop, for a set point change of 1% during 70 minutes can be evaluated from the IAE values presented in Table 4. Table 4: IAE values obtained with both controllers (13) and (14) Controllers IAE% D(z) 25.6 Dop(z) ........12.1
..........................................................................
.........................................................................................................
264
3.4. Comparisons among digital controller designs In this section discrete PID, TW and Ragazzini designs are compared for both servo and regulator behavior over the same process. In Table 5 are summarized the IAEs for each controller accounted load and reference changes. Table 5: Comparison amon~ different controller designs Controllers IAE% (Load) IAE% (Reference) PID (C-C) 0.3 28.1 PID (min IAE) 0.4 21.3 TW (rain IAE) 0.2 26.1 RAG(min IAE) 0.5 33.8 As can be seen PID design for minimum IAE presents the best performance for servo behavior and TW performance is superior for regulator behavior. Similar conclusions can be obtained from Figures 5 and 6.
05 o,
:#
0,966t ' i , / R a g azzini 0'966[~ t I 0,966
/PID
:'Y~ ,'|,
0,966
::,<
' :t'
:)l
o,~t
i
O,
..
0,966~
0,966!
Hagazztri
0,966,
50
100 150 time (rnin)
"
\/
-
0,966,
I
0
I:t / P'~
0,966
XD o'
0,966
200
250
Fig. 5" dynamic responses for reference changes
' ~o" go' ~o' ~o' ;oo','~o',,o Time (min)
Fig. 6: dynamic responses for load change
4. C O N C L U S I O N S Simulation has been recognized as a powerful tool to predict dynamic or static behavior. In this work it is used as a fast and effective way to teach the "real control problem" especially for digital controller designs applied over chemical process. Here the software serves as a pilot plant, which drives to learn how to handle the identification and many discrete control design techniques tasks over different control schemes. This technique have been applied satisfactorily to those engineering students who begin to learn control because it provides a good connection between theoretical and practical aspects of industrial control applications. REFERENCES Astr6m K. and T. H~igglund, (1995) "PID controllers". IS A. Basualdo M. S. "Dynamic Simulation of Chemical Process as a Tool to Teach The Real Problem of Identification and Control. (Engineering Education in the 21st. Century) FIE'95. IEEE.1-5/11/95. Atlanta GA. USA. Franklin G. F, J. D. Powell and M. L. Workman (1990) "Digital Control of Dynamic Systems". 2 nd ed., MA: Addison-Wesley
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
265
MODEL BASED CONTROL OF BATCH C H R O M A T O G R A P H Y Guido Dtinnebier* and Karsten-Ulrich Klatt a aProcess Control Laboratory, Department of Chemical Engineering, University of Dortmund, D-44221 Dortmund, Germany, e-mail: [email protected] This contribution proposes a new concept for the control of batch chromatography, consisting of an online model parameter estimation and an on-line optimisation of the operating parameters. The utilised process models are briefly introduced, followed by a description of the optimisation and parameter estimation routines. At the end of this paper, a simulation study of a process with a nonlinear adsorption equilibrium illustrates the capabilities of the proposed strategy. 1
INTRODUCTION
Life Science products are expected to have an growing importance in the chemical industry. Since pharmaceutical products, food and fine chemicals are subject to more and more complex standards and stricter legal restrictions, efficient methods for the separation of those sensitive products are needed. Chromatographic separation processes are an emerging technology for this type of task. The separation costs for chromatographic separations are very high and easily dominate the whole production costs. Since the economically optimal operation is close to an operation with impure products, large safety margins are usually included within the conventional operating strategy, leading to a suboptimal operation. A total automation of the process by using all available measurements exhibits a large economic potential, and should allow the operation of the process close to the cost optimal operating point while keeping the product specifications at any time. Chromatographic separations are conventionally operated in a Batch mode. In case of the elution mode considered in this work, one charge of the mixture to be separated is injected into the column together with a suitable solvent (Desorbent). This is usually realised with an injection valve, and the charge is carried through the column by continuously injecting additional desorbent. Due to the different adsorption affinity, the different components have different migration velocities and the mixture is gradually separated while moving through the column. The eluting solvent is analysed with a suitable detector at the outlet of the column, and a fractionating valve is controlled based on the measurement information to separate the mixture in its components (Guiochon et al. (1994)). The essential operating parameters of this process are the flowrate of the desorbent, the size of the feed charge to be injected and the cycle period until the next charge is injected. Conventionally, the process is operated with previously determined fixed values for these parameters, which are only modified manually in case of a non-satisfactory performance of the separation. This contribution describes a new control framework for batch chromatographic processes, based on parameter estimation and model based online optimisation, whose architecture is described in the next section. The control strategy is based on two main blocks, first the adaption of the process model to the measurements by a reduced online parameter estimation procedure, second, an online model based optimisation of the manipulated variables. A simulation study for an enantiomer separation concludes this contribution. *The financial support of the Bundesministerium ftir Bildung und Forschung under the grant number 03D0062B0 is very gratefully acknowledged.
266
Figure 1. Process Control Concept 2
MODEL BASED CONTROL OF BATCH CHROMATOGRAPHY
In the current industrial applications, chromatographic processes are not governed by an advanced feedback control. This contribution proposes a newly developed control structure for batch chromatographic processes, consisting of an online estimation of model parameters from the measurements at the column outlets and a model based online optimisation of the operating parameters (see Figure 1). The concentration of the single components is measured online at the column outlet (elution profile) and is used to control the fractionating valve and to adjust the model parameters. In the estimation procedure, the model parameters are adapted to gain an optimal fit of the model prediction to the measurements. The parameter estimation returns a current set of model parameters for the process model chosen, which is then used by the optimisation routine. The online optimisation routine calculates the optimal injection and cycle period and the optimal flowrate based on the adapted process model. Because the availability of a computationally efficient simulation model is an essential pre-requisite of this strategy, this issue is briefly sketched in the following section.
2.1
Process Modelling
The mathematical modelling of single chromatographic columns has been extensively described in the literature by several authors, and is in most cases based on a differential mass balance (see Guiochon et al. (1994) for a recent review). The modelling approaches can be classified by the physical phenomena they include and thus by their level of complexity. Many process models reported in the literature so far use an Equilibrium Transport Dispersive column model. It is based on the adsorption equilibrium isotherm and a linear driving force approach for the mass transfer from bulk to solid phase. Finite difference or collocation methods are used to solve the system of model equations. The computation times of these approaches are often within the range of the real process time. For the application within an online optimisation and control framework, computationally more efficient simulation models are required, which on the one hand still have a sufficiently good accuracy of prediction and on the other hand can be solved in magnitudes below process time. More details on the models and solution approaches developed in this framework, especially for the SMB process, can be found in Dtinnebier and Klatt (1999).
267 In case of a general nonlinear adsorption isotherm, a simplification of the model is not practicable without an unacceptable loss of accuracy. For the generation of an accurate and computationally efficient simulation model, only a suitable numerical solution strategy can be applied. Fortunately, there exists a very effective numerical solution for the complex General Rate Model. C3Ci 02Ci C3Ci 3(1-g.)kl,i (ci Cpi(rp)) O----t= Dax "-----TOx u Ox o~rp 9
OCpi
(1)
[ 1 63 (r2C3Cpill
incorporating arbitrary nonlinear isotherms proposed by (Gu, 1995). A finite element formulation is used to discretise the fluid phase and orthogonal collocation for the solid phase. A commonly utilised isotherm for enantiomer separations is the extended Langmuir isotherm: N2,i ci qi = Nl,i ci + ~ l + E k j cj J
2.2
(2)
On-line Parameter Estimation
Starting point for the online parameter estimation strategy is a set of model parameters previously determined in a number of independent experiments. A strategy to obtain those has been described in Altenh6ner et al. (1997). The complete set of model parameters should, especially for systems with nonlinear adsorption isotherms, be known with a reasonable initial accuracy in the range of 10- 15 %. The large number of parameters and their strong interactions does, especially in the case of nonlinear adsorption behaviour, not allow for the estimation of all parameters based on the measurement information available from the elution profile. Thus, the intention of our online parameter estimation strategy is not the determination of a set of consistent parameters to completely describe the system behaviour and to allow an extrapolation for a wide range of operating regions. In fact, for the application within the control framework proposed here, a set of model parameters which allows the extrapolation in a region close to the operating trajectory is sufficient. The model parameters can in principle be divided into two classes: I. Kinetic parameters: They describe the effects of mass transfer, diffusion and axial dispersion. II. Adsorption parameters: They describe the thermodynamic equilibrium of adsorption. For the reduced online parameter estimation procedure, one dominant parameter from each class is chosen for each of the nc characteristic components of the mixture. Though the classification of the parameters and the decoupling of their effects is only a rough approximation in case of nonlinear adsorption behaviour, this classification is a useful means for choosing the dominant parameters. The effects of the kinetic parameters are additive in a first approximation, therefore an experimentally determined elution profile can be approximated by only fitting one kinetic parameter and the adsorption parameters (Golshan-Shirazi and Guiochon, 1992). Simulation studies for some tested physical systems lead to the conclusion that for those systems the parameters K1,i and m2,iare the dominant parameters to be chosen for the parameter estimation (Dtinnebier et al., 1999) The reduced online parameter estimation problem therefore consists in both cases of 2nc parameters to be estimated. As soon as the set of peaks resulting from one injected charge is eluted, this data is used for a batchwise parameter adaption by adjusting the prediction of the model to the signals of the detector by means of a least squares type algorithm.
2.3
On-line Optimisation
Optimisation in the context of batch chromatography in the literature is normally not used in connection with model-based mathematical optimisation to determine the optimal design or the optimal operating conditions. Most of the work is concerned with the semi-empirical improvement of the system of desorbenVadsorbent chosen.. The design of a separation using rather simple mathematical models and a problem
268
formulation not suitable for an on-line optimisation framework can, e.g., be found in Felinger and Guiochon (1998). Considering a chromatographic column with given design parameters, the determination of the optimal operating point is conterminous to the solution of the following problem: a possibly large amount of raw material has to be separated into the desired components while strictly keeping the constraints on purity and recovery. The solution of the optimisation problem can therefore be determined by adjusting the following set of operating parameters: a) The throughput of solvent and educt, represented by the flowrate Q or the interstitial velocity u whilst respecting the maximum throughput allowed, limited by the efficiency of the adsorbent or the pressure drop. b) The injection period tinj , representing the duration of the feed injection as a measurement for the size of the feed charge. c) The cycle period tcyc, representing the duration from the beginning of one feed injection to the beginning of the next one. Furthermore, the switching points of the fractionating valve 72switch,i can be considered as Initialguessfor the 1 degrees of freedom for the optimisation problem. Flowrate Q In case of total separation, the valve is switched when pure desorbent is eluting between two product peaks. In order to avoid intermediate fractions, in case of a binary non-total separation, max (Pr) two switching times Z'switch,i have to be deterf mined per batch to maximise the product quality. Q The product requirements can usually be formulated in terms of minimum purities, minimum recoveries or maximum losses. In case of a binary separation, those constraints can be transInjectionperiodand formed into each other. In the sequel, we theredistancebetween fore use the product recovery Reci as a measure i twocharges for the product quality. The objective function for the optimisation is the productivity IJri = mproduct, i / mAdsorben t representing the amount of product produced. This formulation results in Simulation ofthe process ] the following nonlinear dynamic optimisation f Correction of] model and evaluation of ] injection perio~ problem: the elution profile
and dist.... J
max
Pr(u,tinj,tcyc,rswitch,i)
s.t.
Rec i > Reci,mi n,
(3)
O
This type of problem can in principle be solved with the optimisation software available today, but the calculation times to be expected are unsuitable for online applications. The complex optimisation problem should therefore be simplified and decomposed to allow a more efficient solution. We here exploit the fact that the recovery constraints can be treated as active at the optimal solution, since a solution can not be cost optimal if higher product qualities than required are obtained. We therefore consider those inequality constraints as equalities:
T
J
NO
I
NO
<
Calculatethe resultingproductivity
Maximumof ~ Y E S ""-...~oductivityf o u n d ~
Figure 2. Optimisation algorithm
END
269
Rec=~( u,tinj ,tcy c , rswitch.i),
(4)
where the operator 9 represents the respective system dynamics. If for a specified recovery the solution of (4) could be given in the following form:
Itinjl tcyc = f \ ~swtich,i)
( Recspec ,tt),
(5)
where f is a static mapping the range of which is restricted to the positive Euclidian space, then by inverting (5) into (3) one could formulate the equivalent optimisation problem with one degree of freedom max Pr( u ) s.t. 0~ U_
(6)
Because the objective function is unimodal, (6) can be treated as an unconstrained optimisation problem with one degree of freedom without considering the pressure drop constraint. If the solution found lies above the maximum throughput, the optimal solution is obtained at maximum throughput. Unfortunately, a closed form solution of the system dynamics as depicted in eq. (5) can not be given, but its structure allows a very efficient iterative solution within the optimisation problem. The resulting optimisation problem now consists of two stages, the iterative solution of the dynamic equality constraints (5) in an inner loop, and the solution of a now unconstrained static nonlinear optimisation problem (6) in an outer loop. By introducing the additional inner loop, the optimisation problem could therefore be substantially simplified. The resulting structure of the optimisation algorithm is given in Figure 2. To solve the equality constraints in the inner loop, the cycle time tcyc, the injection period tinj and the switching times rswitch,i for a given interstitial velocity u have to be determined such that the recovery constraints are exactly fulfilled. The iterative solution commences with a dynamic simulation of the process model for the respective u with initial guesses for tinj and tcyc. By integration of the resulting elution profiles, a set of switching times Zswitch,i is determined which does exactly give the desired product recovery. In general, these switching times are not feasible since the collecting intervals for the two fractions overlap, or not optimal, since a fraction of pure solvent lies inbetween the two collecting intervals. A simple gradient based search converges to the optimal and feasible solution in a few steps by adjusting the injection and cycle periods tinj and t cyc to force the collecting intervals to touch each other, and therefore gives the solution of the equality constraints for a given interstitial velocity u.
2.4
Simulation study
The experimental validation of the concept for a sugar separation with linear adsorption isotherms has been presented in Dtinnebier et aL (1999). In this contribution the possible extension to a more complex system with nonlinear isotherms is illustrated using a simulation study for an enantiomer separation. As an example, the pharmaceutical intermediate EMD53986 was chosen (Merck KGaA, Devant et al. (1997)). To illustrate the disturbance rejection capabilities, both the estimated and non-estimated parameters are assumed to be known with an error in the range of 2-30% only, while the product is required with 99.5% purity. Furthermore, white noise in the range of the expected measurement errors has been added to the simulated concentration values. After 35 injections, the real system parameters change as an effect of a new feed charge. In the upper part of Figure 3, the resulting product purifies for each injected charge are shown. The optimal interstitial velocity u and the optimal switching times for each injection can be seen in the middle and lower part of the figure, respectively. The slight offset between the desired and obtained purifies is a result of the different calculation procedures for the switching points in the optimisation routine and the plant and can not be avoided in case of noisy measurements. Both the initial and the intermediate disturbance could be rejected fast and reliably. The computation time for the simulation study was clearly below real time on a PC PII, and therefore the concept could be verified for
270
the example chosen. 3
system
CONCLUSIONS
~
99.5
99
Since the economic potential of chroma98.5 0 10 20 30 40 50 60 70 tographic separations can by far not be exploited by 0.06 ~- . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bpiimatintersti~al. Velocity ..... .! ........................... the conventional manual 0.055 operation, there is a grow.~ 0.05 ing need for a reliable and > stable process control and :~ 0.045 $ automation concept. This g 0.04 contribution presented a 0 5 10 15 20 25 new process control Switching times Time [h] [ I r r r T 1 [ : : ~ _-~:~ : / --~-Injection Period [s] I concept for Batch| ' ~ ~ : [ + Cycle Period [6 s] I Chromatography, con~ o 1 - ~ .... ~ ....... --~ I sisting of an online 200 .parameter estimation and a model based online optimisation routine, based 150 0 10 20 30 40 50 60 70 on an computationally Number of injection [-] efficient process model. A simulation study for an Figure 3. Simulation study for enantiomer separation enantiomer separation illustrates the applicability of this approach to a wide group of separations, and an implementation of this algorithm in an industrial standard control system is currently under way.
~
REFERENCES
Altenh6ner; U., Meurer, M., Strube, J. and Schmidt-Traub, H. (1997). Parameter estimation for the simulation of liquid chromatography. J. Chrom. A., 769, 59-69. Devant, R.M., Jonas, R., Schulte, M., Keil, A., and Charton, F. (1997). Enantiomer separation of a novel Ca-sensitizing drug by Simulated Moving Bed (SMB) chromatography. Journal ftir praktische Chemie - Chemiker-Zeitung. 339, 315-321. Dtinnebier, G. and Klatt, K.-U. (1999. Modelling and simulation of nonlinear chromatographic separation processes: A comparison of different modelling approaches. Chem. Eng. Sci., 55, 373-380. Diinnebier, G., Hanisch, F., Klatt, K.-U., and Engell, S. (1999). Model based control of batch chromatography, at-Automatisierungstechnik, 47, 466-476 [in German]. Felinger, A. and Guiochon, G. (1998). Comparing the optimum performance of the different modes of preparative liquid chromatography. J. Chrom. A., 796, 59-74. Golshan-Shirazi, S. and Guiochon, G. (1992). Comparison of the various kinetic models of non-linear chromatography J. Chrom. A, 603, 1-11. Gu, T. (1995). Mathematical modeling and scale up of liquid chromatography. Springer, New York. Guiochon, G., Golshan-Shirazi, S. and Katti, A. (1994). Fundamentals of Preparative and Nonlinear Chromatography, Academic Press, Boston.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
271
M o d e l Predictive Control of an Industrial Dryer V.M. Cristea, M. Baldea, ~.P. Agachi "Babe~-Bolyai"University of Cluj-Napoca, Faculty of Chemistry and Chemical Engineering, 11 Arany Janos, 3400 Cluj-Napoca, Romania, e-mail:[email protected] The paper presents the simulation results of an advanced control algorithm used for the control of the drying process of electric insulators. The industrial batch dryer is modeled and two different approaches are taken for its control. First, Model Predictive Control (MPC) is used for controlling the air temperature in the drying chamber. Results are compared with data obtained by using traditional PID control. In a second, more advantageous, approach a state observer is used for inferring the moisture content of the product, which is then controlled by means of the MPC controller. The linear model used by the MPC controller is periodically updated accounting for the non-linear behavior of the process. The requested drying program (both for temperature and moisture content control) consists of a rampconstant profile that is obtained by manipulating the air and natural gas flow rate. Simulation results reveal clear benefits of these MPC approaches over traditional control methods, and prove real incentives for industrial implementation. 1. INTRODUCTION The high-voltage electric insulator production implies a two-stage batch drying process. During the first step, the moisture content of the drying product is reduced from 18-20 % to 0.4% in special gas heated chambers. The second step is carried out in high temperature ovens, in order to achieve an even lower moisture content. The formed clay insulators are placed on special support and transport frames and then introduced in the drying chamber. An electric, motor driven, multiple fan provides the air flow through the chamber. The air inlet flow rate can be controlled by means of a butterfly valve. Gas and air flow rates are controlled according to a special program, during a period of about 100 hours, in order to obtain the desired moisture content and avoiding the risk of unsafe tensions in the drying products. An analytical dynamic model of the process is derived for model predictive control purposes.
2. MODEL DESCRIPTION Mass and energy balance equations are used to describe the dynamic behavior of the system. The main studied outputs of the model are: moisture content of the drying product X, outlet air temperature To and air humidity xo ; the input variables: natural gas flow rate V'F and mass flow rate of fresh air mai. The chamber is divided into three sections as shown in figure 1. Section 1 represents the air volume within the drying chamber, section 2 the direct surroundings of the drying product. Section 3 represents the drying product itself.
272 /
41
!
To
!
.?hai
X o
['~
7,~,
xf
Section 1
~,,,
~.-, t,,
at'_
..... x.o ......................................................
H
i ~ ~ ' ~ '[!~.......................... '~ :::::::::::::::::::::::::::::::::::::::::: ~ ~ ,, o :.!:"
ii"~~s~tion~3
~
L X ~
~ i ~
x ---Section 2
" .............
: ,, . . . . . . . . . . . . . . . . . .
",,', , ' , ' , ' " ' , ' , " , ' , ' , ' " , " , ' " ' ~ i ~ i i ~ i ~ i ~ i ~
-". . . . . . . . . . . . . . .
"
Va2
Fig 1. Description of the drying chamber. The mass balance of steam within section 1 and section 2 is described by the equations:
dxo
~no,"xs+mo.x-(mo +mo,)'Xo=Vo~"Pa"dt '
(1)
Fna .(Xo - X)-ms " dd,x : - ~d
(2)
(vo
. po . x) "
The last term of the equation can be neglected, which results in the differential equation:
dX _ (Xo _ x). tha dt C m "
(3)
As a result of differentiation of equation (3) and assuming that, d2X/dt2 ~ 0, equation (1) becomes:
dt
Xo).
V~c~ "p~
In section 3, behavior of the drying good is described with a normalized diagram by [ 1], [2]" dX rh~, ~=-~.A s. (5)
dt
ms
The drying velocity for experimental determined diagrams characterize the three periods of the drying process for the hygroscopic material, normalized according to equations [2]:
~'(~7) rhst = mL,.,i,
rl=
X -- Xequ " -Xeq u
(6)
Xc
It is assumed that X c is constant, not depending on the drying conditions, and that Xequ only depends on relative air humidity, but no other factors. It is also assumed that all diagrams of the drying velocity for different drying conditions are geometrically similar. The equilibrium humidity Xequ in dependence of the relative air humidity cp, for clay, was considered by a
273 correlation equation. The saturation humidity of the air, xsa, , is dependent of temperature To . For low partial pressures of steam, rhs,I was considered according to equation:
(7)
]~VlstI -- k'(Xsa , --X)
with the mass transfer coefficient k determined by experiment data. Two energy balance equations, for the chamber and for the bumer section, are used to describe the outlet temperature change:
mai.(Cpa.(Ti_To)+Xf.(hv+cpst.Til_xo.(hv+cpst.To))+ms = VaCh'iOa"
l dTo
dX v +Cpst'To)-CAMch(To -Te) 9---'~'(h
dXo )
a + Xo" pst]'---'~ +Cpst'--~ "To
(8)
(9)
A dynamic sensitivity analysis was carried out on this model, indicating the most important parameters and manipulated variables. According to this analysis, they are: mass transfer coefficient k, heat transfer coefficient of chamber walls CA, heating power of natural gas HF, mass of the drying product ms (clay without humidity), environment temperature of the inlet air Te, environment humidity of the inlet air Xe, volume of the drying chamber Vch, surface of the drying chamber Ach, surface of the drying product As, critical humidity of clay Xc and specific heat of natural gas CpF [3]. The scaled dynamic sensitivity analysis of the output variables with respect to the studied inputs pointed out the natural gas flow rate as the most important manipulated variable (about 10 times more important than the mass flow rate of fresh air). The control system was designed accordingly. 3. DYNAMIC SIMULATIONS OF MODEL PREDICTIVE CONTROL The first MPC approach obeys the current control practice, i.e. driving the evolution of the moisture content of the drying product in the desired way by means of controlling the air temperature inside the chamber. Usually, the desired decreasing profile of the drying product moisture content is obtained by imposing an increasing ramp-constant profile on the air temperature. A comparison was made between traditional PID control and MPC control of the air temperature. Both MPC [4] and PID (with anti-windup) control algorithms were implemented in a MISO structure with two manipulated variables: gas flowrate and air flowrate. For PID, a ratio air flow rate control was used. First, the setpoint following capacity was tested in the absence of disturbances. Afterwards, performance testing was carried out for three significant disturbances typically occurring in the industrial practice: a 10 ~ inlet air temperature Te drop (from 16 ~ to 6 ~ a 10 % heating power capacity HF drop of natural gas; a 20 % inlet air moisture content Xe rise. All of the three disturbances have been introduced as steps at time t=llO000 s. The simulation results (for case of the heating power disturbance) are presented in figure 2. The figure shows the response of the controlled variable over the entire time interval and a detailed representation of the period when the disturbance acts and is eliminated.
274
65
.
.
.
.
6O 55 so
45 40
35 C~ 30 25 2O 15
1
o
ols
~
~:s
;
T i m e [s]
1. 2.s
3
x lo ~
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fig. 2. PID ( '. ..... ') and MPC ( ' ') control of the outlet air temperature, for a given setpoint ( ' _ _ _ _'), when the heating power disturbance of the fuel, HF, occurs. The results revealed a very good behavior, particularly for MPC control. Although both methods exhibited good control ability, the setpoint tracking performance showed a zerooffset behavior for the MPC, whereas PID control proved to be less accurate (mainly for the ramp sections of the setpoint function). The control in the presence of the disturbance emphasizes the superior characteristics of the MPC, with shorter response time and smaller overshoot than the PID control. Taking into account that the target variable - the moisture content of the product - is not available for direct measurement, an inferential state observer is proposed for its estimation. The data provided by the state observer is used for direct model predictive control of the moisture content of the drying product, as shown in figure 3:
]
Setpoint
MODEL PREDICTIVE k CONTROLLER STATE OBSERVER
]
~.xX
I
.
/
m ai
Ceramic mass
/
]
Fig. 3. Structure of the control system for direct moisture content control. The selection of the setpoint for the moisture content is based on practical and theoretical considerations concerning the time evolution of the product drying rate. The conditions stated
275 by the above mentioned considerations are best fulfilled by a decreasing, seven segment ramp function, which is actually used as setpoint. Simulations were conducted using this control structure and the results (disturbances applied at t=20000 s) are presented in figures 4 and 5.
025
o2,
\
x
, o 0.15
" X
.~ o
E
/:::l " lJ {
0.15
~
o.1
0.2183
x
',,1-2181t . . . . 22
2.4
"t
2.6 xlO4
0.1
13_
/
/
~2'8'I ,2,~3[ '2'~2t ~.2181t
0.2
.
.
.
% \ ,
2.2
.
2.4
t I
\1 ,
]
2.6 xlO4
/ I / /
13_ 0.05
O0
0.05
1
2
Time [s]
3
4
5x 105
Fig. 4. MPC of the inferred moisture content of the drying product ( ' '), for a given setpoint ( ' _ _ _ _') when the heating power disturbance of the fuel, HF, occurs.
O0
1
2
Time [s]
3
4
5x 10~
Fig. 5. MPC of the inferred moisture content of the drying product ( ' '), for a given setpoint ('_ _ _ _') when the inlet air temperature disturbance, Te, occurs.
Again, setpoint tracking performance is very good. Moreover, the offsets introduced by the disturbances are eliminated by the MPC in a short time and with acceptable deviations from the desired trajectory. The MPC controller was tuned according to the dynamic sensitivity analysis and is based on the maximum allowed variation of both the controlled and manipulated variables. The applied model predictive algorithm has a few special features that make it more effective: it has an excess number of manipulated variables over controlled variables; in order to get the desired control performance, a constrained form of the MPC algorithm was used; finally, the linear model used by the MPC controller is periodically updated to account for the non-linear behavior of the process [5], [6]. 4. CONCLUSIONS The proposed Model Predictive Control of the batch drying process of electric insulators proves to be a good strategy for controlling the drying process. Its high performance is due to the use of the direct control of the product moisture content based on a state observer, to the updating of the model of the process on which MPC relies, and to the optimal manipulation of both the inlet air and gas flow rates. The results obtained simulating MPC control reveal a very good setpoint tracking performance as well as an effective disturbance rejection. In the industrial plant, this leads to increased energy efficiency, higher productivity and better product quality. The results entitle the application and the important incentives of this MPC control approach for industrial implementation.
276
NOMENCLATURE A
[m2] ~
CA Cp hv
[W/(m" ~ [J/(g ~ [J/g]
H
[Vd
k
[kg/(s. mZ)]
m
[kg]
,n
[kg/s]
thst
[kg/(s.
area of the surface heat transfer coefficient of chamber walls specific heat heat of vaporization (latent heat) - enthalpy flux -mass transfer coefficient -
-
-
-
-
-
m2)]
p R
[bar] [J/mol K]
r
loci
x
[kg/kg]
[]
-
mass (as far as masses of air are concerned, mass of dry air) mass flux drying rate
- pressure gas constant temperature - volume - humidity of drying good - mass of water per mass of dry substance humidity of air - mass of water per mass of dry air - normalized humidity of drying product -
-
-
r
[%]
p
[ g/m3]
- relative humidity of the air cp = X/Xs~ t - density
[]
- normalized drying rate (~ = rhst/rhsti
Meaning of the indices: a c
Ch e
equ I f
F
o S I, II, Ill
sat st
air critical - chamber - environment - equilibrium in - fresh - fuel - out - sample - number of drying period - saturation - steam -
-
-
REFERENCES 1. D.A. van Meel, Chem. Engng. Sci., 9, (1958), 36. 2. O. Krischer, W. Kast, Die wissenschaftlichen Grundlagen der Trocknungstechnik, Springer-Verlag, (1992). 3. R. Perry, C. Chilton, Chemical Engineers' Handbook, 5. Edition, McGrawHill,(1973). 4. C.E Garcia, M.P. Prett, M. Morari, Automatica, 25, no. 3, (1989), 335. 5. V.M. Cristea, $.P. Agachi., Revista Rom~.n~ de Informatic~ ~i AutomaticS, Vol. 7, no.4, (1997), 45. 6. M.V. Cristea, V. Marinoiu, $. P, Agachi, 2 na Conference on Process Integration, Modeling and Optimization for Energy Saving and Pollution Reduction, Budapest (1999), 223.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
277
A p p r o x i m a t e D y n a m i c M o d e l s Using Coarse Grid P a r a m e t e r Relaxation Victor J. Law University of Limerick, Limerick, Ireland Victor.Law @ul.ie
1. Abstract Previous work on methods for simultaneous parameter estimation and process simulation (Law, 1999) has led to a technique for generating approximate dynamic system models. The essence of the method involves the use of a coarse grid discrete representation of the model differential equations (in both time and space) and parameter estimation. The system parameters are allowed to deviate (relax) from their optimal values so that the coarse grid model response agrees with the fine grid results (using optimal parameter values) in a least squares or maximum likelihood sense. This method is called "Parameter Relaxation." Importantly, the physics of the model are retained. Several example systems are used to demonstrate the utility and fidelity of the approximate models. Systems studied range from simple lumped parameter ones to two-dimensional composite medium heat transfer systems.
2. Introduction Dynamic models in the form of differential equations are often solved using some form of discretisation (finite differences, finite elements). For complex systems such as oil and gas reservoirs, for example, the time required for solution can be extensive. In some applications, such as model reference adaptive control, the model solution must be at least as fast as real time. This paper presents a method for generating approximate models, which are inherently faster, based on a coarse grid (in both time and space) finite difference solution. The truncation errors introduced by the coarse grid are ameliorated by altering (relaxing) the system parameters so as to minimise an objective function that represents deviations from the fine grid model. The inherent physics of the model are retained. The parameter estimation problem can be stated succinctly as follows: minq = T ~,ri(u,v)2 = {u}
rTr
2i=1
subject to 3u
(1)
= f (u, v)
3t where
r V U
C w
t f nd
= = = = = = = =
nd-vector of residuals = C(u) - w p-vector of parameters to be estimated (control variables) m-vector of state variables observation operator; maps u into the space of v nd-vector of measurements. independent variable (typically time) m-vector of functions, which might contain spatial derivatives number of data points
(2)
278
The constraints are a finite difference or finite element form of the model differential equations and the "measurements" (w) are produced by solving the model with the proper parameters. The end result is a coarse grid model that mimics (at least in the vicinity of the "data") the precise one.
3. Solution Method The coarse grid parameter estimation problem can be solved by a number of methods (see, Gill, et al., 1981; Law & Fariss, 1972). Nested techniques minimise the objective function and solve the model each time the optimiser requires a function evaluation. Law (1999) presented a simultaneous approach that does not require repeated model solutions. In the simultaneous method, the parameters and discretised state variables are collectively modified iteratively to arrive at a solution. The results given here were produced using the simultaneous approach. 4. Examples 4.1: A First Order, Linear Differential Equation Model with Two Parameters Consider the very simplest of dynamic models du ~+vlu=0 dt
;u(0)=v 2
(3)
The control variables (parameters to be determined) are Vl, a coefficient in the model and v2, the initial condition of the state variable u. The first step of the SimReg algorithm is to discretise the differential equation by a suitable finite differencing scheme. For this simple example, consider a backward difference scheme: U k -- U k _ 1
At
(4)
+ VlU k = 0
Rearranging and writing explicitly the first few and the last of these equations yields C 1 = (1 + V l A t ) u 1 - v 2 = 0 C 2 = (1 + v 1 A t ) u 2 - U 1 = 0
Cm =
(1 +
VlAt)u m -Um_
(5)
1 = 0
For simplicity, assume that "perfect" data are available at every finite difference grid point. The solution to the model equation is simply u = v 2 exp(-vlt )
(6)
and this is used to provide the "exact experimental" data for the problem. Recall once again that this example is strictly to illustrate the algorithm; it is not meant to be highly realistic 9Data were generated using vl = v2 = 1. Using a very coarse differencing grid where At = 0.25, with data available at t = 0.25, 0.5, and 0.75, there are three state variables (m = 3) and two control variables (p = 2). When implemented in a computer program that solves the equality constrained sum of squares minimisation problem, the results summarised in Table 1 were obtained.
279 Table 1: Results for Example 1 Approximate Solution Exact Solution 0.7788 Ul 0.7788 0.6065 u2 0.6065 0.4724 u3 0.4724 1.0000 vl 1.1361 1.0000 V2 1.0000 The results of this very simple example are interesting from two points of view. First, the state variables agree exactly (to the precision shown) with the "perfect" data used. The optimal value of the initial condition (v2) is also exact. Only the model parameter represented by vl differs from its exact counterpart. The only source of error in this example is the truncation error introduced by the use of a very coarse finite difference grid. Therefore, the "relaxed" value for parameter Vl represents a compromise to accommodate the truncation error. Although the optimal value for Vl is somewhat different than its exact counterpart, the constraints are all satisfied and the sum of squares of residuals is small (essentially zero). The coarse finite difference model with the relaxed Vl can be used to simulate the system from any initial condition. For example, shown in Table 2 are the results of applying the approximate model from initial conditions of 0.5 and 1.5, respectively, with the "exact" counterparts also shown. Table 2: Results Using Approximate Model from Other Initial Conditions V2 = 0.5 V2 = 1.5 t Exact Model Exact Model 0.25 0.389400 0.389401 1.168201 1.168202 0.50 0.303265 0.303266 0.909796 0.909797 0.25 0.236183 0.236184 0.708550 0.708551 4.2: Application to One-Dimensional, Transient Heat Conduction Consider a one-dimensional composite solid consisting, for example, of two "zones" of equal thickness but with different thermal diffusivity (~i) in the z-direction as shown in Figure 1.
Step change from 0 to 1
(z2
Insulated
v
Z
Figure 1: Composite Solid with Two Zones. The heat conduction equation for this variable diffusivity problem can be written as follows: 0w b 0w 0t = 0---za(z) 0----z
(7)
where w(z,t) is the temperature in the solid at position z and time t. The initial and boundary conditions are specified as follows: w(z,O) = o
(8)
280 w(0,t) = 1 ~)w(1, t) 0z
(9)
= 0 (insulated)
(10)
Therefore, the solid dimension z ranges from 0 to 1; the solid is initially at zero temperature and at time 0+ the left hand boundary is changes abruptly to a temperature of 1 while the right boundary remains insulated. Temperature profiles computed using a fine grid (At = 0.001, Az = 0.01) backward difference formulation are shown in Figure 2, for diffusivities of 0.1 and 0.5, respectively. Using these fine grid results as a basis, errors attributable to grid coarseness are shown in Figure 3. That is, the diffusivities were maintained at of 0.1 and 0.5, respectively and results for different grid sizes were computed.
Figure 2: Fine grid profiles at selected spatial points with original diffusivities.
Figure 3" Errors using medium grid (At = 0.01, Az = 0.05) and coarse grid (At = 0.05, Az = 0.125) with original diffusivities.
281 The coarse grid parameter relaxation problem is to determine the thermal diffusivity in each zone with "exact" data supplied from a fine grid simulation of the system. In this case, the temperature "measurements" were made at the right hand boundary. Ten data points were generated using exact diffusivities of 0.1 and 0.5, respectively and finite difference parameters of At = 0.001, Az = 0.01. The optimal diffusivities for the approximate model with At = 0.01, Az = 0.05 were 0.09793 and 0.4178, respectively. For At = 0.05, Az = 0.125 the relaxed diffusivities were 0.09455 and 0.3269, respectively. Figure 4 shows the error profiles for z = 1 (the right hand boundary) in the solid as a function of time. The medium grid produced essentially error-free results while the coarse grid gave a maximum error of about 0.45%. These errors can be compared to those of Figure 3 to see how parameter relaxation improves the fidelity of the coarser grid models. It is important that the eventual steady state error is nearly zero.
Figure 4: Error profiles using the coarse grid model with relaxed parameters. Medium grid: At = 0.01, Az = 0.05; diffusivities = 0.09793 and 0.4178. Fine grid: At = 0.05, Az = 0.125; diffusivities = 0.09455 and 0.3269 4. 3: A Nonlinear Problem Consider a problem very similar to that of Example 1, but with a nonlinear term as follows: du ---+Vl u 2 = 0 dt
;u(0)=l
(11)
The control variable (parameter to be determined) is Vl, a coefficient in the model. The situation considered involves a "true" value of Vl = 5. This produces a problem with a high degree of nonlinearity. Shown in Figure 5 is a utilisation of the finite difference model (using the estimated value of Vl = 6.8106) from a different initial condition (5 versus 1). The time range in Figure 3 includes an extrapolation beyond that used in the parameter estimation phase (1.0 versus 0.5). Clearly, the finite difference model produces an improved approximate system response from a significantly different initial condition. In general, the coarse grid approximation should be made as close to the actual operating conditions as possible. This example merely illustrates that there are still potential advantages when the conditions are far removed from the ones used in the approximation.
282
Approx. Response from ~itial 5.00 4.00 :=
3.00 2.00 1.00
ndition f
- - e ~ - A p p r o x . v1 = 6.8106 - - l l - - Exact
k
--~k---Approx. v1 = 5
m
0.00
9 9
0.0
0.2
==
0.4
-=
0.6
==
1
0.8
7..[
1.0
Time
Figure 5. Exact and Approximate Model Results for Example 3 5. Conclusions Based on the examples presented here, the following conclusions can be drawn:
1. In all four of the examples presented, the coarse grid, relaxed parameter (CGRP) approximate models are of good fidelity. 2. Robustness is illustrated in Example 2, where data were available over the time interval 0 - 0.5 while excellent simulation results occurred in the extended time interval to 1.0. Furthermore, in this example, the results are shown from a significantly different initial condition than that from which data were generated. 3. A major attractiveness of the CGRP approximate models is that the "physics is not destroyed." That is, the underlying models, derived from basic principles, are still in effect. Only truncation error introduced by the coarse grid in time and space is present. The truncation error is ameliorated by relaxing the parameters to produce close agreement with data generated by a coarse grid solution of the model.
References Gill, P.E., W. Murray and M.H. Wright, Practical Optimization, Academic Press, New York, pp 1 3 3 - 139 (1981).
Law, V. J., Simultaneous Parameter Estimation and Process Simulation, Proceedings of the Fourth Italian Conference on Chemical and Process Engineering, pp. 123-126, Florence, Italy, May 2-5 (1999). V.J. Law and R.H. Fariss, Transformational Discrimination for Unconstrained Optimization, I.E.C. Fundamentals, 11 (1972), 154.
European Symposiumon ComputerAided ProcessEngineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
283
Analysis of different control possibilities for the Divided Wall Column: feedback diagonal and Dynamic Matrix Control M. Serra a, M. Perrier b, A. Espuna a and L. Puigjaner a aDepartment of Chemical Engineering, Universitat Polit6cnica de Catalunya, Diagonal 647, Barcelona 08028, Spain bDepartment of Chemical Engineering, t~cole Polytechnique de Montr6al, C.P.6079, succ. Centre-ville, Montr6al H3C3A7, Canada This work addresses the control of the Divided Wall Column (DWC). Different control structures of diagonal feedback control are compared using MIMO linear analysis tools in the frequency domain. A controllability analysis of the process is done for the separation of different mixtures and for different operating conditions, including optimal operation. Application of Dynamic Matrix Control (DMC) to the DWC is evaluated. Through simulation, the ability of DMC for disturbance rejection and setpoint tracking is studied and compared to that of the feedback diagonal control. 1. I N T R O D U C T I O N The DWC is a non-conventional distillation arrangement for the separation of ternary mixtures. Its interest is based on its potential to save energy and reduce investment costs. Its design was proposed almost 50 years ago [ 1]. Since then, many authors have addressed design aspects [2], but operation and control have received much less attention [3,4]. In this work, different control strategies are compared, and the influence of operating conditions on the controllability are analysed. 2. C O N T R O L L A B I L I T Y STUDY OF T H R E E D I F F E R E N T S E P A R A T I O N S In order to develop the controllability study, a specific DWC design has been selected. It has 13 trays in the prefractionator and 33 trays in the main column. Counting trays from the bottom, the feed tray is tray 7 of the prefractionator. In the main column, the side product tray is 17, the last common tray before the wall is 8 and the first common tray after the wall is 26. The separation of three ternary mixtures (components called A, B and C) into 0.99 pure products has been studied. Products are liquid saturated flows and feeds are liquid saturated equimolar flows. The relative volatility and Easy of Separation Indexes (ESI) of mixture 1 are ct=(1 : 1.85 : 4.65) and ESI=l.36. For mixture 2, or=(1 : 2.15 : 4.65) and ESI=I. For mixture 3, c~=(1 : 2.45 : 4.65) and ESI=0.77. In a DWC with the three product compositions controlled, two extra degrees of freedom remain for optimisation. For the three studied separations, the nominal steady state operations have been optimised to minimise the boilup rate. Linear analysis tools are used to compare different composition control structures. The considered manipulated variables are L, V, D, B, S, SPLITD and SPLITB, where L is the
284 reflux, V the boilup, D the distillate, B the bottom flowrate, S the side product flowrate, SPLITD the liquid split at the top of the wall and SPLITB the vapour split at the bottom of the wall. With closed inventory control loops, the system is linearised. Four inventory control structures are considered, which are called "DB", "LB", "DV" and "LV". The first letter of the name is the manipulated variable that controls the condenser level and the second letter is the manipulated variable that controls the reboiler level. In Tables 1, 2 and 3, the best composition control structures for the different stabilised columns and for the different mixtures are shown. Morari Resilency Index (MRI) and Condition Number (CN) values indicated correspond to a frequency of 0.04 rad/min. This frequency is the one corresponding to the main open-loop time constant divided by ten. Intersivity Index (II=MRI/CN) is used to classify the structures. The structure with largest II is the preferred one. Table 1. Preferred structures for mixture 1 (analysis at s=0.04 rad/min) "DB . . . . LB . . . . DV . . . . LV" DBS LBS VDS LVS MRI=0.12 MRI=0.14 MRI=0.14 MRI=0.16 CN=14 CN=35 CN=34 CN=264 Table 2. Preferred structures for mixture 2 (analysis at s=0.04 rad/min) "LV" "DB . . . . LB . . . . DV" DBS LBS VDS LVS MRI=0.34 MRI=0.28 MRI=0.29 MRI=0.25 CN=4.6 CN=I 1 CN=12 CN=91 Table 3. Preferred structures for mixture 3 (analysis at s=0.04 rad/min) "DB . . . . LB . . . . DV . . . . LV" DBS LBS VDS LVS MRI=0.17 MRI=0.19 MRI=0.18 MRI=0.20 CN=10 CN=18 CN=28 CN = 184 It can be observed from tables 1, 2 and 3 that for all inventory controls, the best set of manipulated variables does not depend on the mixture. For all mixtures, the preferred structure with "DB" is the worst of the four preferred structures, and the preferred structure with "LV" is the best of the four preferred structures. Specifically CN is very large for control structures with "DB" inventory controls. In none of the cases, the preferred control structures include SPLITD or SPLITB as manipulated variables. The analysis of controllability indexes has been done at frequency 0.04 rad/min but it should be done at the closed-loop bandwidth frequency. The preferred set of manipulated variables is found to have a small dependence on the analysis frequency but MRI and CN can vary considerably with the frequency.
3. C O N T R O L L A B I L I T Y AT DIFFERENT OPERATING CONDITIONS In this section, the controllability of the same distillation process at three operating conditions has been studied. The separation of mixture 2 described in the previous section has
285 been chosen. Optimal operation has been compared with two non-optimal operations, indicated as operation 1 and operation 2. Optimal operation has SPLITD=0.634 and SPLITB=0.500. Operation 1 was found fixing SPLITD at 0.614 and SPLITB at 0.500. Boilup increased by 3%. Operation 2 was found fixing SPLITD at 0.654 and SPLITB at 0.500. Boilup increased by 10%. In Tables 4 and 5, the preferred control structures for the different stabilised columns at the different operations are shown. Since SPLITD will be easier to manipulate in practice, when controllability with SPLID and SPLITB are similar, SPLITD is chosen. Comparing Tables 2, 4 and 5, it can be noticed that at optimal operation, the preferred sets of manipulated variables do not include the split variables. On the contrary, at the non-optimal operations, the preferred sets of manipulated variables include the split variables. For all inventory controls, nonoptimal operation preferred control structures have better controllability indexes. Table 4. Preferred structures at operation 1 (analysis at s=0.04 rad/min) "DB . . . . LB . . . . DV . . . . LV" B S SPLITD B S SPLITB D S SPLITD L S SPLITD MRI=0.67 MRI-0.61 MRI=0.69 MRI=0.70 CN=2.11 CN=3.86 CN=3.82 CN-23.18 Table 5. Preferred ;tructures at operation 2 (analysis at s=0.04 rad/min) "LV" "DV" "LB" "DB" B S SPLITD D S SPLITB D S SPLITD L S SPLITD MRI=I.03 MRI=0.71 MRI=0.91 MRI =1.07 CN=2.61 CN=2.04 CN=2.95 CN = 16.09 3.1. "DB" inventory control Relative Gain Array (RGA) of the preferred structures indicate L SPLITD S as the best pairing for the non-optimal operations and L S V for the optimal operation. According to RGA, L SPLITD S for operation 2 has better controllability than L S V for the optimal operation. PI controllers with P=I and I-0.0125 are implemented on each control loop and the bandwidths are determined. Indexes at the bandwidth indicate the same preferred structures indicated at frequency 0.04 rad/min. The CN of the optimal operation at the bandwidth frequency is very high. According to the results, controllability of the non-optimal operations is better. Simulations show good performance of the L SPLITD S control structure. Thus, it is possible to make a trade off between controllability and energy optimality. 3.2. "LV" inventory control RGA plots indicate that S SPLITD B, D S B and D SPLITD S are the best pairings of the preferred structures. With PI controllers of P-1 and I=0.0125 to all loops, the bandwidths are determined. At the bandwidth frequency, the preferred structure for operation 1 is D B S (MRI=2.7 and CN=2.4). For operation 2, it is D S SPLITD (MRI-1.3 and CN=2.7). At different frequencies, singular value decomposition indicates different preferred structures because of similar controllability indexes of the different control structures. For optimal operation, the best structure is D S B (MRI=0.7 and CN=I 8). Controllability indexes for nonoptimal operations are better. A stability analysis through w~*T~ maximum singular value indicate robust stability for the three operations with their preferred structures (w~ is the uncertainty in input channels and TI the closed-loop transfer function at the input) [5].
286 4. DYNAMIC MATRIX CONTROL Assuming that inventory control is solved at a lower control level, three manipulated variables are used for the control of the three product compositions. A 3x3 system has to be solved by the DMC. The tuning parameters are: At (sampling period), n (identification horizon), p (prediction horizon), m (control horizon) and k2 (move suppression factor). In some cases treated later, k2 is substituted by )~l, )~2 and )~3 (suppression factors for the individual inputs) [6]. System identification is performed applying step changes to the manipulated variables in open loop. The sign and the size of the steps have been found to have a very large influence over the identification of the DWC and over the DMC system derived from this identification. The main reason is the DWC non-linearity. Depending on the size and sign of the step change used for identification, the control converges or diverges. The observed inverse responses for several identification profiles are not responsible for control divergence, but changes in steady state gains. Small errors in the identification profiles can give the model the knowledge that when L increases the same amount than V, the purity of B decreases. But in the linear region, when L increases the same amount than V, B purity increases. This wrong sign makes the system diverge. Identification within the linear region is needed, which is not typically feasible in a real plant. In the previous sections, the "LV" D S B control structure was found to be advantageous. However, to apply DMC with this control structure has a major problem due to the open-loop instablility of the inventory control structure. The problem appears at the early stage of identification. Because of the instability, final responses do not exist. In this sense, PI control strategy is advantageous because it can be applied to an open-loop unstable system. To compare PI control and DMC strategies, discrete PI are considered and simulated. "DB" stabilised columns with L S V control structures are considered. At= 1 has been imposed to both control strategies. DMC could be in principle a better control approach because it takes into account interactions. However, the DMC depends highly on the identification of a non-linear system into a simple model. PI may have the advantage that interactions favour naturally the rejection of disturbances. Some distillation examples will help to compare these two control strategies.
4.1. Setpoint tracking The same separation described as mixture 2 is studied. With tuning parameters At=l, n=600, p=300, m=6, k2=100, a set point change of +0.001 in A purity is simulated. It is seen in Fig. 1 that convergence is extremely slow. Different move suppression factors in the different inputs are implemented. However, very small differences have been found. All the other parameters have been changed and a tuning achieving faster control and smaller overshoots has not been found. To compare DMC and PI control, the same setpoint change is simulated with PI control. Tuning of loops for composition of A, B and C is P=15, I=0.5; P=-2, I=-0.03; P=15, I=0.5, respectively. As seen in Fig. 2, at time=500 min, the set points of all three outputs are achieved (much faster than the DMC). On the other hand, overshoot of output 3 is larger than that of DMC. A tuning reducing this overshoot has not been found.
287 x 10 .4
X 10 .4
8
A
_o 4
B
-8 0
400
Time (min) 1400
2000
0
Fig. 1. Compositions with DMC
200
Time (min)
800
1000
Fig. 2. Compositions with PI
For set point changes in B and C purity, similar results are found: the PI can reach set points much faster but with larger deviation for the outputs that are not asked to be changed. Therefore, if the main objective is to achieve setpoint changes quickly, PI is better. If the main objective is to keep the other outputs constant when one output is changed, DMC is better. 4.2. D i s t u r b a n c e rejection
Rejection of a disturbance in A feed composition by the two control strategies is compared through simulation. The DMC tuning is At=l, n=600, p=300, m-6, )~1=14, L2=14, )~3=60. Reducing more the move suppression factors, the overshoots can still be reduced, but profiles begin to be very irregular. PI tuning for A, B and C purity loops is P=8, I=0.1; P=-8, I=-0.1; P=8, I=0.1, respectively. Greater overshoots are found with DMC and input variables vary more. As found for setpoint changes, the time response is shorter with PI. In Figures 3 and 4, the input profiles of the DMC and PI described simulations are shown. A similar behaviour is found for a disturbance in B feed composition: deviation is smaller and response time shorter with PI control. For the rejection of a feed flowrate disturbance, a greater overshoot is found with DMC, but it gives faster response. L and V increase with the same ratio. 0.3 0.02
0.2
0.01
0.1 -0.01
L
S 1000
Time (min)
4000
6000
Fig. 3. Input changes with DMC
-0.02
0
100 Time (min) 400 Fig. 4. Input changes with PI
500
288 Contrary to what happened for set point changes, for disturbance rejection, the overshoots of the variables are larger with DMC than with PI. Both strategies are affected by the directionality of the DWC system and have slow responses when L and V increase with the same ratio. However, PI deals better with the problem. DMC certainly uses the information of what is the influence of L, V and S over all the outputs when L, V and S play alone. When L is the only input variable that changes, B and D also change and the purity of the products is affected by these changes in D and B. However, when L and V both change at the same time, D and B can remain almost unchanged and the purity of the products is only affected by the change of the internal variables (L and V) and not by the external variables (D and B). In this case, much larger changes in L are needed to increase the purity of product A. The DWC non-linearity makes DMC limited. Non-linear models for Model Predictive Control should be compared to DMC in future work. 5. CONCLUSIONS Different composition control structures of diagonal feedback control are compared through MRI, CN and RGA. The separation of three different mixtures at optimal operation has been studied. The preferred control structures do not include the split variables and are the same for the three mixtures. DWC with "DB" stabilisation present very high CN. Simulations show that "LV" stabilisation and D S B composition control is a good control structure for the DWC. Three operating conditions including the optimal one have been compared. At optimal operation, split variables are not in the set of preferred manipulated variables but they are for non-optimal operations. Controllability of non-optimal operations is better, indicating a possible trade off between controllability and energy optimisation. With "DB', stabilisation, robust stability of non-optimal operations should be further studied. However, with "LV" stabilisation, all operation conditions present robust stability. A preliminary study is done applying DMC to the DWC. Some conclusions are obtained from the studied examples. "DB", L S V control structure is considered. For setpoint tracking, the DMC presents smaller deviations but longer response time. For disturbance rejection, PI presents smaller deviations and better response time. In general, DMC has been found to be quite limited for the control of the DWC. REFERENCES
1. Wright, R. O. U. S. Patent 2, 471,134, May 24, 1949. 2. C. Triantafyllow and R. Smith, The Design and Optimisation of Fully Thermally Coupled Distillation Columns, Trans. Inst. Chem. Eng. 70, 118-132, 1992. 3. E. A. Wolff and S. Skogestad, Operation of Integrated Three-Product (Petlyuk) Distillation Columns, Ind. Eng. Chem. Res. 34, 2094-2103, 1995. 4. M. Serra, A. Espuna and L. Puigjaner, Control and Optimisation of the Divided Wall Column, Chem. Eng. Process. 38, 549-562, 1999. 5. W.L. Luyben, Practical Distillation Control, Van Nostrand Reinhold, New York, 1992. 6. B. A. Ogunnaike and W. H. Ray, Process Dynamics, Modelling, and Control, Oxford University Press, New York, 1994.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
289
Control Strategies for Brine Electrolysis by Ion Exchange M e m b r a n e Cell Process ~. Agachi, A. Imre-Lucaci Department of Chemical Engineering, Faculty of Chemistry and Chemical Engineering, "Babe~-Bolyai" University of Cluj, Arany Jfinos 11, 3400 Cluj-Napoca, Romania
Ion Exchange Membrane process is the most effective in industrial chlorine production. This paper is a comparative study between traditional control (PID) and advanced control of the process (MPC). The presented results are based on mathematical modeling and dynamic simulation of the process.
1. INTRODUCTION
The importance of chlorine and caustic soda Chlorine is essential to world's chemical industry, with then 50% of all chemicals processing depending on this element. Chlorine was discovered in 1774 by the Swedish chemist Carl Wilhelm Scheele. During the last century, industrial users have found vast number of ways to take advantage of chlorine's useful properties in processes and products. Chlorine is a key building block of modern chemistry by being used in three principal ways: direct use (e.g. to disinfect water); as a raw material for chlorine-containing products (e.g. plastics, pharmaceuticals, pesticides) and as an intermediate to manufacture non-chlorinated products. Chlorine is produced by passing electric current through a brine solution (common salt dissolved in water). Essential co-products are caustic soda (sodium hydroxide) and hydrogen. Caustic soda (sodium hydroxide) is an alkali. It is widely used in many industries, including gold mining, food industry, textile production, soap and other cleaning agent production, water treatment and effluent control. Worldwide production of caustic soda is about 45 million tones [ 1]. It is used to produce a broad range of inorganic chemicals, but also for general manufacturing, mineral processing and water treatment.
2. INDUSTRIAL METHODS FOR BRINE ELECTROLYSIS Chlorine has been manufactured industrially for many years. During this time, the industry's firm commitment to the best safety, health and environmental practices has ensured continuous improvement. There are three main processes of manufacture: 9mercury cell; 9diaphragm cell; 9membrane cell; The mercury process. This produces extremely pure caustic soda; it accounts for almost two-thirds of European chlorine production. The process uses a mercury cathode and because
290 mercury is toxic, producers take measures to protect employees and maintain mercury losses to the environment at extremely low levels. During the last 20 years, emissions per tone of chlorine capacity have been reduced by more than 90 % following capital investment and continuing commitment to environmental improvement. The diaphragm process. This uses a porous mineral fiber as a separator, and produces less pure caustic soda, which is not suitable, applications. It accounts for almost one-quarter of European chlorine production. The membrane process. It is the result of recent advances in polymer chemistry. While the membrane itself is very expensive and requires very high purity brine, the process gives a high quality product with no environmental sensitivities during manufacturing. This method is used to produce about one-tenth of chlorine obtained worldwide. Two of those are used in our country: the amalgam cathode process and the ion exchange membrane process. Amalgam cathode and ion exchange membrane processes are used in plants from "Oltchim" Rfimnicu-Vfilcea and "Chemcomplex" Borze~ti [2]. In the membrane cell process, the anode and cathode are separated by a cation-exchange membrane. Only sodium ions and a little water migrate through the membrane. As in the mercury process, the brine is dechlorinated and recirculated for resaturation with solid salt. The life of the expensive membranes depends on the purity of the brine. Therefore, after initial purification by precipitation-filtration, the brine is additionally purified by ion exchange of higher valent cations (Ca 2+, Ba 2+ and Mg2+). The caustic solution leaves the cell with a concentration of 30 to 40 wt% and must be concentrated. The chlorine content of the sodium hydroxide solution is as low as that from the mercury process. The chlorine gas contains some oxygen and must be purified by liquefaction and evaporation. The consumption of electric energy with the membrane cell process is the lowest of the three processes, ca. 25% less than that for the mercury process. The amount of steam needed for concentration of caustic is relatively small. There are no special environmental problems. The cells are easy to operate and are relatively intensive to current density changes, allowing greater use of the cheaper off-peak-time electric power. Because of the multiple advantages of the membrane process, S.C. "Chimcomplex" S.A. Borze~ti (Romania) succeeded to acquire a license from Hoechst-Uhde (Germany) for a membrane cell plant. Having a capacity of 100,000 t/year, this plant is equipped with BM type bipolar cells. Main characteristics of the BM type bipolar cell (Hoechst-Uhde)[2] aScitedin literature 9 area of anode per element
2 m2
9 cell number in an electrolyzer
28-36 4 kA/m 2
9 nominal current density 9 inlet concentrations
brine: caustic:
9 cell temperature 9 outlet caustic concentration
90~ 40 wt%
9 chlorine current efficiency
~95 %
9 caustic current efficiency
~99 %
290 g/dm 3 12 wt%
291 3. MATHEMATICAL MODEL OF ION EXCHANGE MEMBRANE CELL The mathematical model of the membrane cell was presented in an earlier work [3] and it is an original model based on information obtained only from literature. This model has been developed for a Hoechst-Uhde BM cell, also used by the ion exchange membrane cell plant form "Chimcomplex" Borze~ti [2, 3, 4, 5]. This mathematical model is a dynamic model, developed for process control. Model structure
The model, which has been considered, is formed by a series of CSTRs having the following structure (figure 1). The basic element of the model is considered the pair of CSTRs situated on the same level in the anodic or respectively cathodic compartment of the cell. For each basic element of the model, mass, energy and voltage balance equations were considered as well as equations for physical properties. This structure of the mathematical model includes 70 differential equations and more than Fig. 1. Model structure for membrane cell. 160 nonlinear algebraic equations. 4. S I M U L A T I O N OF BRINE E L E C T R O L Y S I S
Solving the huge system of differential equation was possible only by numeric methods. MATLAB with SIMULINK software environment was used for this task (figure 2). It was possible to determine the inside profile of parameters like: concentration, temperature, flow, gas fraction, current (current distribution), voltage, pH, electrodic coverage, etc. by simulation [3, 4, 5].
Fig. 2. SIMULINK program for dynamic simulation of the membrane cell. A short selection of the simulation results is presented in the following figures (fig.3 a-e).
292 Brine concentration profile
300
Brine temperature profile
IO0
95
U'
ICn 280
P'
d t-
O U
._ L.
9O
/
.J
85
270
j
2400
\ 0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
_J
70
\
\
1.8
J
65 60
~,
0
cell height, [m]
/ d tO cj
25
J
20
J
J
J
/J
J
J
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.6
1.8
2
1.6
1.8
2
cell height, [m]
Caustic soda concentration profile
g
J
.J
=~ 72
260
f
J
J
Brine flow profile
162
/
160 158
J
\ X
\
154
J 152
X \
150 148
100
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
146
2
0.4
0
0.6
cell height, [m]
0.8
1
1.2
1.4
cell height, [m] Voltage profile
0
>
0
3.2
E~ 3.1 n3 0 >
3
2.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
cell height, [m] Figure 3. Parameter profiles based on simulations for IEM cell. a - brine concentration profile in anodic compartment; d - brine f l o w profile; b - brine temperature profile; e - current distribution; c - caustic soda concentration profile in cathodic comp.; f - voltage.
293 5. C O N T R O L OF M E M B R A N E C E L L Dynamic responses of membrane cell were used to test two different kind of control techniques: SISO (Single Input/Single Output) control structures using PID controllers, and MIMO (Multiple Inputs/Multiple Outputs) control structures based on Model Predictive Control[5]. For the SISO control structure in the case of a membrane cell, the following loops were selected: loop 1: controlled variable: brine concentration at cell outlet manipulated variable: brine inlet flow loop 2: controlled variable: caustic soda concentration at cell outlet manipulated variable: caustic soda inlet flow Two PI controllers were used for these two Table 1. Controller parameters loops. Controller tuning was made by simulation with the Ziegler-Nichols method. Parameters for Controller Type Kr T/[s] these controllers are presented in table 1. 1 PI 15 4500 In the case of MPC of the membrane cell, 2 PI 30 5200 when the same controlled variables and manipulated variables, the optimal values for the internal parameters of the controller were determined by simulation as follows: - model horizon T = 14400; - control horizon U = 2; prediction horizon V -- 10; weighting matrix for predicted errors V/~ = [0.05 0.05]; /47: = [1 1]; - weighting matrix for control moves sampling period At = ls. The controlled variables were subject to the following constrain: Ymin _
-
-
Inlet perturbation
Brine concentration control
468 417 ~
tnt---, 5 E 0 4.8 E
"
d 4.6 E O U 4.4
:....
/ / I ,LOpen
"O O t/3 4.2 52 .l..a u3 i~ 4 u 3.8
MPC
4
.
1
6
2
~
"~ 4 1 6 ~ ~ N ~ 7 , ~ 0
I 120
t i m e , [s]
IOO0
la
}
1500
4158 ~ 0
120
500
(
t i m e , [s]
1000
Figure 4. a - perturbation considered for testing the control structures SISO/MIMO); b - controlled variable" brine outlet concentration;
1500
294 Caustic soda concentration control
r~
E
[
0
SISO
_
./~-,,
.
3s .6
~c .4
A
.w_
/
~c.2
,-~-'"-'~-~
0 (,t')
Caustic soda flow
89
3C .8
/J
[
E ,~ 3.1 ~J (0 u
7]//
31
$0
u
g U
221
.8
, :/ra' -....
1
.6
L
2 c. .4
j"
A'
,'
2S .2
U
0
120
500
time, [s]
1000
1500
~.9 0
120
500
time, [s]
1000
1500
Figure 5. a - controlled variable: caustic soda outlet concentration; b - manipulated variable: caustic soda inlet flow. We could observe (figure 4.b and figure 5.a) that by an MPC Control structure a better process control can by achieved. The response of the controlled system it is almost instantaneous. By this fast and accurate response of the process, we could obtain an estimated 1-5 % of savings in energy and row material consumption. 6. CONCLUSIONS The importance of brine electrolysis consists in the importance of its products, chlorine, hydrogen, caustic soda, because these are row materials in almost any branches of chemical industry. These processes are also important, because of involved high-energy consumption. By use of simulation, it was proved the possibility of advanced control of the process. In order to achieve this purpose and taking into account the high (strong) interactivities between the parameters of the processes, a MIMO control structure called MPC was chosen. It was proven by simulation that the MPC is better then a PID controller structure because it allows a 1-5% cut down in specific energy consumption. We could estimate that by reducing with 1% the specific energy and material consumption by advanced process control, at the level of present chlorine production in Romania, the savings would be about 15-20 GWh/year. REFERENCES 1. C. Chao-Peng, M. Vreeke, Report of the Electrolytic Industries for the Year 1996, J. Electrochem. Soc., 144(1997), 10, 3674-3692 2. A.Szep, F.Bandrabur, I.M~nea, Brine electrolysis by ion exchange membrane process, Ed. CERMI, Ia~i, 1998, Romania 3. L. Oniciu, $.Agachi, J. Bugan, A.Imre, Process computer implementation at brine electrolysis, Research study, Cluj-Napoca, Romania, 1989 4. A.Imre, Dynamic simulation of Ion Exchange Membrane cell process, CHISA'98, Prague, aug. '98 5. A.Imre, Modeling and control of brine electrolysis by mercury cell process and ion exchange membrane cell process, PhD thesis, "Babe~-Bolyai" University Cluj, Romania, 1998
European Symposiumon ComputerAidedProcessEngineering- l0 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
295
A n e w m e t h o d o l o g y for the active control of the heat t r a n s f e r in Autoclave technology. V.Antonucci ~, M.Giordano b, S.Inserra ~ and L.Nicolais a
:~Department of Materials and Production Engineering, University of Naples "Federico II", P.ie Tecchio, 80- 80125 Naples, Italy ~'Institute of Composite Materials Technology, National Research Council, P.le Tecchio, 80 80125 Naples, Italy ~ALENIA S.P.A. Foggia
Abstract In this study a new methodology has been developed to analyse the heat transport and to control the processing parameters in the autoclave technology. The method, based on the use of the itself laminate-tool system as heat fluxmeter and the characterisation of the whole temperature field with the thermal profiles of selected points, allows to evaluate the convective heat exchange coefficient before the curing reaction starts and to predict the temperature evolution. The approach has been applied to patch of wing panels and validated by comparison with the experimental data. This work has been cofunded by European Comunity (BRITE Program: PERFECT )
Introduction The autoclave process is one of the most common technique for the manufacturing of high performance thermoset composite structures in the aeronautical and aerospace industry, both for military and commercial applications. In this process composite prepregs, woven fiber mats impregnated with resin, are cut and laid up in the shape of the part. Then the laminate is located on a mold, bagged and introduced into the autoclave, where a temperature and pressure cycle ix imposed. The process involves several phenomena: mass flow, heat transfer and exothermic non reversible chemical reactions associated to phase change. In fact, the exothermal curing reaction is activated by the heat transfer from the autoclave environment. A progressively denser polymeric network is developed until a critical degree of branching is reached and an infinite network is formed. The final properties and quality are settled by the accurate control of the temperature and degree of cure gradients within the parts. In the last few years, to design, control and optimize the manufacturing cycle of thermoset based composite materials, a new approach has been developed by integrating the use of process modeling, numerical simulation, advanced sensors and artificial intelligence. In fact, the virtual processing represents an useful tool to understand, control and manage the different parameters affecting the product quality and the manufacturing costs. Theoretically, in the autoclave process the knowledge of the mass rate of the autoclave fluid, the thermal mass of the tool, its geometry, the location inside the autoclave allows to calculate for each curing part the convective thermal exchange coefficient, the thermal and conversion maps. Commercial and industrial environments require continuos changes of the autoclave runs and so of the local fluodinamic, further the operator would manage many data. In this study, a new methodology has been developed to analyse the heat transport and to control lhc processing parameters, i.e. temperature fluid autoclave and pressure, based on: 9 The t~se o1 the itself laminate-mold system as heat fluxmeter;
296 9 The characterisation of the entire temperature map with the thermal profiles of selected points. Multiple or single loading are allowable in the autoclave technology. The manufacturing of many little parts is get putting them together in a single autoclave run, so the fluidodynamic is always ditferent inside the autoclave. In this case the need is to optimise on-line the heat transport belore the starting of the curing reaction in order to insure the respect of the specific requirements for each tool and modify the programmed autoclave fluid temperature. The manufacturing of big (very expensive) parts is obtained putting it in a single autoclave load. The tool location is always the same and in preliminary runs it is possible to define a fluidodynamic pattern, to select the representative points and their heat exchange coefficients and, finally, to optimise off line the time-temperature for the specific run. Starting from the most complex energy balance equation for a reactive mono dimensional rectangular system, simplified heat transfer models have been developed, applied and verified for single run of a complex tool with and without considering the heat generated by the curing reaction. The simplified equations have been used to evaluate the convective heat exchange coefficient before the curing reaction starts and to predict the thermal profiles for a settled programmed autoclave fluid temperature. The methodology has been confirmed by the comparison with the experilnental data.
Mathematical modeling The control and optimisation of a manufacturing process pass through the development of models ot: the physical phenomena involved in the production stage. In the case of autoclave technology the most relevant aspects are related to the evolution of temperature profiles along the parts during cure stage. The thermal problem has been analyzed with reference to a mono dimensional rectangular case, by considering the part thickness as characteristic length. To evaluate the relative importance of the heat transport terms, in the energy balance equation both the conduction flux and the heat generated by the exothermal reaction have been included: 2 cOT o3 T do~ pcp cO~ - K - - ~ + e p r H T~ (1) cOx dt , The curing system is made of e volumetric percentage of resin, p,Cp, K are the material properties of the curing system, p,. is the resin density, HT is the total amount of heat reaction, all lhese properties are temperature independent, o~ is the conversion degree. Initially, the whole system has been considered at the uniform temperature To. The external sections have been characterised by heat flux conditions, in particular at x=0 an adiabatic condition has been imposed and at x=L the conductive heat flux has been considered as: - K CO__TT-h[Ta (t)- T] /)x
(2)
T:,(t) is the autoclave fluid temperature, h is the heat exchange convective. The system has been analyzed in dimensionless form, choosing the following dimensionless variables: o: T-T 0 do(dt Length ~ - xT i ,9n e L t 3 - t - ~ Temperature "c---ATmax "Reaction rate ~ R - m a x ( d o ( d t )
297
Max(dcz/dt) is the maxil-num reaction rate and ATmax is the adiabatic temperature increase. In terms of these variables, the equation 1) and the boundary condition 2) become:
017
2 c-)z = - - + Da. ~R
~o a~ 2 O~ = a~
(3)
(o)]
(4)
Two dimensionless nurnbers; Damkohler and Biot numbers representing respectively the ratio between characteristic rates and temperature gradients, characterize the problem: The magnitude of the two dimensionless numbers determine two different limiting cases" a) Da<< I
b) Da>>l
C a s e a) The controlling stage is the conductive transport, the heat transport equation is simpler: 2
ao
(5)
a~ 2
The boundary conditions depend in turn on the Blot number, that determines other two limiting cases: Biot<>l. In the first case, the temperature gradient is in the convective layer and the part can be considered isothermal. These findings lead to the l:ollowin<,~ energy balance equation:
PCl" V 0q---i~ 3t - hS[T a ( t ) - T]
(6)
For Blot>> I, the part temperature gradient is relevant, but the boundary condition at ~:1 can be replaced with T=Tdt). Case b) The temperature eradients are neglectable. In this case a different dimensionless time has to pc V be introduced" O'= t p hS In the following the new balance equation is reported in term of the dimensionless variables
(9O
:
Da
Bi
(7)
The equation 7 shows another dimensionless group: the ratio between the Damkohler number and Biot number, i.e. ratio between the reaction and heating rates. D;i For - - >> 1, the system can be considered adiabatic once the curing reaction is activated. Bi
298 Da
For ~ < < 1
the limiting stage is the heat convective exchange, in this case the balance
Bi
energy equation reduces to equation 6 again.
The self-learning approach In the autoclave technology the heat transport between the curing parts and the autoclave environment depends on the local fluodinamic determined by the mass rate of the autoclave fluid, the thermal properties, the location and geometry of the parts. Since a theoretical approach would be complex to manage in an industrial field, in this study a semi-empirical method has been developed to control and optimise in real time the experimental thermal profiles. The new methodology is able to evaluate the effective convective heat transport coefficient during the first part of the autoclave run, when conversion is low. Then, in the last part of the cycle, when cross-linking reaction takes place, the method enables the real time prediction of the temperature profile. The self-learning approach is based on: 9 the use the tool itself as heat fluxmeter to evaluate the heat exchange coefficients and tool thermal inertia; 9 the characterization of the thermal field within the composite by the temperature profile of few points identified during the thermal tool qualification. The physical thermal model is based on the absence of thermal gradient through the composite thickness (Da>>l). With this assumption, the heat transport can be described by the equation 6, that together the kinetic model gives the following ordinary differential equations system:
dT dt
-
H(Ta(t ) - T ( t ) ) +
AT a& (8)-(9)
& = f(o~,
T, t)
The system has two parameters: H the global heat exchange coefficient and AT.~ the adiabatic temperature raise.
H=
h pccpcl c + ptcptlt
|zXT - ~ - - p r l - c E ~ -
(10) (11)
a - pccpcl c +PtCptl t The expressions of these parameters incorporate two unknown variables, the heat exchange coefficient h and the tool equivalent thickness lt, that can be evaluated separately. In fact, the autoclave cycle can be divided in two zones. The first part is a self-learning zone for the calculation of the global heat exchange coefficient H. Due to the absence of the curing reaction, the differential equations system reduces to the energy balance equation 6), that allows the calculation of the coefficient H by comparing the temperature raise of the part, T, and the autoclave fluid temperature, Ta.:
299
H -
1
dT
(12)
Tz~(t) - Y (t) dt
The heat exchanoe coefficient is assumed invariant with the temperature changes. In fact, the range of ll~e operating conditions of the autoclave doesn't induce relevant variations of the heat exchange coefficient h. The calculated value of H is, then, used to predict the temperature evolution of the curing part in the exothermal zone by integrating the differential systcm. By COml)aring the predicted and experimental profiles of selected points of the parts, the othcl parameter, the adiabatic temperature raise ATe, can be identified. Experimental validation The proposed methodology has been validated through experimental tests performed in autoclave with four patch of wing panels designed with an epoxy resin and different laminate thickness. For the all patch the global heat exchange coefficient has been evaluated before the curing rcaction starts with the equation 12). The table 1 shows the composite thickness and the calculated value of the coefficient H. Tablel Global heat exchange coefficients Test Composite thickness Ira] A B C D
H
[1/min] 0.034 0.056 0.031 0.048
0.0185 0.0075 0.0055 0
The calculated values of H have been used to predict the evolution of the temperature during the curiJ/g stage. Tile following figures report the comparison between the predicted and the experimental thermal profiles for the tests A and B. ok3 200 - 180 -~ 16o 140 120 t--
ooOo~176
100 8O 6O
./'
4O
~
q-emperature
..'_7
9 Autoclave
temperature temperature
0
50
100
150
200
250
300
350
Time, min
Figure 1" Prediction of the thermal profile for the test A.
300
? .
200
eeeee.ee~
180
;60 140
"9' S
'"~'~
120 100 9
8O 60
~
40
""7 .
20 0 0
50
100
'" 9
9 ~I
150
200
9 Experimental temperature 9 Autoclave Temperature Predicted temperature
250
~ i i i i
300
",
~
350
Time, min
Figure 2" Prediction of the thermal profile for the test B. The diagrams show a good agreement both in the heating and the curing stage, being the differences between the thermal profiles less than 6~ The discrepancy in the cooling stage is due 1o the pressure autoclave change, that induce variations of the global heat exchange coefficienl. The two dimensionless numbers, Damnkoler and Biot, have been evaluated too. The values, reported in the table 2, validate our assumption: Da>>l, i.e. the temperature gradients are neglectable. Table 2 Biot number evaluation
Test Da Bi ......A . ..................112105..............01428.... B 1.85104 0.213 C 1.05104 0.08
Conclusions To analyse the heat transfer between the curing parts and the autoclave environment, a new al-~proach has been proposed and verified. Starting from the most complex energy balance equation l:ora reactive mono dimensional rectangular system, simplified heat transfer models have been developed, applied and confirmed for patch of a wing panel with and without considering the heat generated by the curing reaction. The simplified equations have been used to evaluate the convective heat exchange coefficient before the curing reaction starts and to predict the thermal profiles for a settled programmed autoclave fluid temperature. The methodology has been established by the comparison with the experimental data. References I. Ciriscioli P.R., Q. Wang, G.S. Springer, "Autoclave curing-Comparisons of model and test results", Journal of composite materials, 26, 90-102 (1992). 2. S.l~lscrra, C.Voto, A.Trevisano, J.M.Kenny, L. Nicolais, "Control and optimization of autoclave processing of high performance composites" 37 th International SAMPE Symposium, Anaheimca (1992). 3. M. Pillai, A.N.Beris, P. Dhurjati, "Intelligent curing of thick composites using a knowledge-based system", Journal of composite materials, 31,22-51 (1997).
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
301
MODEL PREDICTIVE CONTROL: A MULTI-PARAMETRIC PROGRAMMING APPROACH Alberto Bemporad*, Nikolaos A. Bozinis t, Vivek Dua t, Manfred Morari*, Efstratios N. Pistikopoulos t *Automatic Control Laboratory ETH Zentrum, ETL I 29 CH-8092 Zurich Switzerland
tCentre for Process Systems Engineering Imperial College London SW? 2BY United Kingdom
b e m p o r a d , m o r a r i @ a u t , ee. ethz. ch
n. bozinis,v, dua, e. pistikopoulos@ic, ac. uk
In this paper, linear model predictive control problems are formulated as multi-parametric quadratic programs, where the control variables are treated as optimization variables and the state variables as parameters. It is shown that the control variables are affine functions of the state variables and each of these affine functions is valid in a certain polyhedral region in the space of state variables. An approach for deriving the explicit expressions of all the affine functions and their corresponding polyhedral regions is presented. The key advantage of this approach is that the control actions are computed off-line: the on-line computation simply reduces to a function evaluation problem. 1. I N T R O D U C T I O N On-line optimization is a commonly used tool in the chemical process industry for operating plants at their m a x i m u m performance. Typically, this issue is addressed via a Model Predictive Control (MPC) framework where at regular time intervals the measurements from the plant are obtained and an optimization problem is solved to predict the optimal control actions - for a recent survey on MPC, see [1]. In this work, we propose an alternative approach for the on-line calculation of control actions which requires a very small computational effort as an optimizer is never called on-line. This approach is based upon the fundamentals of parametric programming. In an optimization framework, where the objective is to minimize or maximize a performance criterion subject to a given set of constraints and where some of the parameters in the optimization problem are uncertain, parametric programming is a technique for obtaining the objective function and the optimization variables as a function of the uncertain parameters [2,3]. Here, we present a parametric quadratic programming approach to address linear MPC problems, where the state variables are treated as parameters and the control actions are computed as a function of the state variables. The rest of the paper is organized as follows. First a brief outline of MPC problems is presented and these problems are
302 formulated as multi-parametric quadratic programs (mp-QP). Next a solution approach for mp-QPs is presented, followed by an illustrative example. 2. M O D E L P R E D I C T I V E C O N T R O L Model Predictive Control (MPC) has been widely adopted by industry to solve control problems of systems subject to input and output constraints. MPC is based on the so called receding horizon philosophy: a sequence of future control actions is chosen according to a prediction of the future evolution of the system and applied to the plant until new measurements are available. Then, a new sequence is determined which replaces the previous one. Each sequence is evaluated by means of an optimization procedure which takes into account two objectives: optimize the tracking performance, and protect the system from possible constraint violations. In a mathematical framework, MPC problems can be formulated as follows. Consider the following state-space representation of a given process model:
x(t+l)
-
Ax(t)+Bu(t)
y(t)
-
Cx(t),
(1)
subject to the following constraints: Ymin <_ y(t) <_ Ymax, ttmin _< t t ( t ) _< ttmax, where x(t) E ~n, u(t) C ~m, and y(t) E ~P are the state, input, and output vectors respectively, subscripts rain and max denote lower and upper bounds respectively and (A, B) is stabilizable. Model Predictive Control (MPC) problems for regulating to the origin can then be posed as the following optimization problems: Ny -- 1
min U
s.t.
J(U,x(t)) = x't+N~ltPxt+N~,lt + Z
x't+kltQxt+klt + u't+kRut+k
k=O Ymin <__ Yt+klt <-- Ymax, k -- 1 , . . . , Nc Umi n < at+ k ~ Umax, k -- O, 1 , . . . , N c
(2)
x~l~ - x(t) Xt+k+llt -- Axt+klt + But+k, k > 0 at+ k -- lif xtTklt, N u ~ k ~ N y
where U A {ut,... , U t + N u - 1 } , Q -- Qt ~- O, [ ~ - ~ t }- O, P }'- O, (Q 89A) d e t e c t a b l e , Ny > Nu, and K is some feedback gain. The problem (2) is solved repetitively at each time t for the current measurement x(t) and a vector of predicted state variables, Xt+llt,... , xt+klt at time t + 1 , . . . , t + k respectively and corresponding control actions ut,... , ut+k-1 is obtained. In the next section, we present a parametric programming approach where the repetitive solution of (2) at each time interval is avoided and instead an optimization problem is solved only once. 3. M U L T I - P A R A M E T R I C Q U A D R A T I C P R O G R A M M I N G Parametric programming has largely been used for incorporating the uncertainties in the model, where (i) the objective function and the optimization variables are obtained
303 as a function of uncertain parameters and (ii) the regions in the space of the uncertain parameters where these functions are valid are also obtained [2-5]. The main advantage of using the parametric programming techniques to address the issue of uncertainty is that for problems pertaining to plant operations, such as for process planning [6] and scheduling, one obtains a complete map of all the optimal solutions and as the operating conditions fluctuate, one does not have to re-optimize for the new set of conditions since the optimal solution as a function of uncertain parameters (or the new set of conditions) is already available. In the following paragraphs, we present a parametric programming approach which avoids a repetitive solution of (2). First, we do some algebraic manipulations to recast (2) in a form suitable for using and developing some new parametric programming concepts. By making the following substitution in (2): k-1
xt+klt
-
A kx(t) + ~
(3)
A jBuk_i_j
j=O
the objective J(U,x(t)) can be written as the following Quadratic Programming (QP) problem:
min
1U'HU + xl(t)FU § x'(t)Yx(t)
s.t.
GU <_ W + I(x(t)
(4)
where U --A [ t t l t , . . 9 , ~tt+Nu_l] C ~s, S --A mNu, is the vector of optimization variables, H - H' >- 0, and H, F, Y, G, W, K are obtained from Q, R, and (2)-(3). With the A transformation, z -U + H - 1 F'x(t), where z E ~s, (4) can be written as the following Multi-parametric Quadratic Program (mp-QP)"
#(x)--
min
1 zIHz
s.t.
Gz <_ W + Sx(t),
(5)
where S A K + G H - 1 F ' , z represents the vector of optimization variables and x represents the vector of parameters. The main advantage of writing (2) in the form given in (5) is that z (and therefore U) can be obtained as an affine function of x for the complete feasible space of x. To derive these results, we first state the following theorem (see also
[7]).
T h e o r e m 1 For the problem in (5) let xo be a vector of parameter values and (zo,,~o) a K K T pair, where ~o = A(xo) is a vector of nonnegative Lagrange multipliers, A, and zo = z(xo) is feasible in (5). Also assume that (i) linear independence constraint qualification and (ii) strict complementary slackness conditions hold. Then,
z(/)] - -(M~176176 A(x)
z0]
(6)
304 wh e re~
H
aT
-A1G1
-V1
~Vr0--(Z,~lS1,... , ,,~pSp) T
m
-5~Gq
-Vq
where Gi denotes the i th row of G, Si denotes the i th row of S, Vi = G i z o - W i Wi denotes the i th row of W and Y is a null matrix of dimension (s x n).
SiZo,
The space of x where this solution, (6), remains optimal is defined as the Critical Region ( C R ~ and can be obtained as follows. Let C R R represent the set of inequalities obtained (i) by substituting z ( x ) i n t o the inequalities in (5) and (ii) from the positivity of the Lagrange multipliers, as follows:
cRR =
< w + Sx(t), a(x) > o},
(7)
then C R ~ is obtained by removing the redundant constraints from C R R as follows 9
cR~
(s)
where A is an operator which removes the redundant constraints. Since for a given space of state-variables, X, so far we have characterized only a sub-space of X i.e. C R ~ C_ X , in the next step the rest of the region C R ~e~t, is defined as follows [3]"
C R "~st - X - C R ~
(9)
The above steps, (6-9) are repeated and a set of z(x), A(x) and corresponding CR~ are obtained. The solution procedure terminates when no more regions can be obtained, i.e. when C R rr = ~). For the regions which have the same solution and can be unified to give a convex region, such a unification is performed and a compact representation is obtained. The continuity and convexity properties of the optimal solution are summarized in the next theorem. T h e o r e m 2 For the mp-QP problem, (5), the set of feasible parameters X / C_ X is convex, the optimal solution, z(x) " X / ~ ~s is continuous and piecewise affine, and the optimal objective function #(x) " X f ~-~ ~ is continuous, convex and piecewise quadratic. Based upon the above theoretical developments the solution of an mp-QP of the form given in (5), to calculate U as an affine function of x and characterize X by a set of polyhedral regions, CRs, can be obtained. This approach provides a significant advancement in the solution and on-line implementation of MPC problems 9 Since its application results in a complete set of control actions as a function of state-variables (from (6)) and the corresponding regions of validity (from (8)), which are computed off-line. Therefore
305 during on-line optimization, no optimizer needs to be called and instead for the current set of measurements the region, C R ~ where these measurements are valid, can be identified by substituting the value of these measurements into the inequalities which define the regions. Then, the corresponding control actions can be computed by using a function evaluation of the corresponding affine function. In the next section, we present an example to illustrate these concepts. 4. N U M E R I C A L
EXAMPLE
Consider the following state-space model representation:
{ x(t+l) y(t)
--
t0"7326 -0"0861 ] 0.1722 0.9909 x(t) + 0 1.4142 ]x(t)
0.0609 0.0064
]
u(t) (10)
together with the following constraints: - 2 <_ u(t) _< 2. The corresponding optimization problem of the form (2) for regulating to the origin is given as follows: 1
rain
x't+21tPxt+21t § ~
~,t,~,~+i
x't+kltxt+klt § .01ut2+k
k=o
- 2 ~ Ut_t_k ~ 2, k -
s.t.
(11)
O, 1
xtl~- x(t) where P solves the Lyapunov equation P - A ' P A + Q, Q - [1 o1 ], R - 0.01, Nu - Ny Arc - 2. The corresponding mp-QP problem of the form (5) has the following constant vectors and matrices.
H-
[
]
0.0063 0.0199 , F -
[ 014 0 0.11%] 0.1058 -0.0834
,G=
[1 01 [21 [001 10 0 0
1 -1
,W-
2 2
,K=
00
0 0 0 0
The solution of the mp-QP problem as computed by using the solution approach described in Section 3 is provided in Table 1. Note that the regions 3,4 and 7,8 in Table 1 are combined together and a compact convex representation is obtained. To illustrate how on-line optimization reduces to a function evaluation problem, consider the starting point x(0) = [1 1]'. This point is substituted into the constraints defining the regions in Table 1 and it satisfies only the constraints of the regions 7,8. The control action corresponding to the regions 7,8 from Table i is u = - 2 , which is obtained without any further optimization calculations. 5. S U M M A R Y
AND
CONCLUDING
REMARKS
In this work, linear MPC problems were formulated as mp-QPs. An approach for the solution of mp-Qps was proposed. It was shown that the solution (a set of control actions)
306 Table 1 Solution of the numerical example Region#
Region --6.3202 6.3202 --3.6447 3.6447
2
3,4
-7.5004 1 7.5004 6.5748 X -6.5748
[0.1123-0.0834 0.1470 0.1058] x ~ -
[
41968]
5.6485 -4.1968 0.1114 0.1322
[ 749003891] -0.0651 7.4906
6 '7,8
0.1174 5.3891
[ 2.0000 2.0000 2.0000 2.0000 [-0.0524] -0.0519
X < -X ~
[-o 4.-010 1 -0.1123 0.0834 J X
u
~ __
[2.6341] 1.3659 -0.0353
[,.3577] -0.0357 2.6423
[_oo 1 ] -0.0524
[_5.64854.1968] [1.3659] 5.6485 -4.1968 -0.1114 -0.1322
X < --
2.6341 -0.0353
7.4906 5.3891 ] [ 1.3577 ] 0.0651 --0.1174 X ~ -0.0357 --7.4906 -5.3891 -2.6423
-6.3202 -7.5004] X
2.0000
2.0000
[-7.4906 - 5 . 3 8 9 1 ] X Ar- 0.6423
--2.0000
-- 2.0000
--7.4906 --5.3891 ] X -- 0.6423
of mp-QPs is an anne function of parameters (state-variables) which is valid in certain regions of optimality which are described by linear inequalities. The main advantage of this approach is that control actions are computed off-line. The on-line computation thus simply reduces to a function evaluation problem. Current work focusses on the extension of the algorithms for multi-parametric mixed-integer programs [3] for hybrid control problems [8]. REFERENCES
1. 2. 3. 4. 5. 6. 7. 8.
M. Morari and J. H. Lee, Comput. Chem. Engng. 23 (1999) 667-682. J. Acevedo and E. N. Pistikopoulos, Ind. Eng. Chem. Res. 36 (1997) 717-728. V. Dua and E. N. Pistikopoulos, Ind. Eng. Chem. Res. 38 (1999) 3976-3987. A. Pertsinidis, I. E. Grossmann and G. J. McRae, Comput. Chem. Engng. 22 (1998) S 205. K.P. Papalexandri and T. I. Dimkou, Ind. Eng. Chem. Res. 37 (1998) 1866-1882. E.N. Pistikopoulos and V. Dua, Proc. 3rd Int. Conf. on FOCAPO, J. F. Pekny and G. E. Blau, Eds., pp. 164-169 (1998). E. Zafiriou, Comput. Chem. Engng. 14 (1990) 359-371. A. Bemporad and M. Morari, Automatica 35 (1999) 407-427.
European Symposium on Computer Aided Process Engineering - l0 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
307
Flowsheet Simulation for the Steel Industry- Using Experiences from Chemical Engineering and Modern Software Approaches H. M011er1, T. Peuker 2, G. Wozny 1 ~Technical University of Berlin, Institute for Process- and Plant-Design, Sekr. KWT-9, Stral~e des 17. Juni 135, 10623 Berlin, Germany 2Siemens AG, Industrial Projects and Technical Services, ATD MP TM ME, P. O. Box 3240, 91050 Erlangen, Germany The support of the process synthesis has been applied in the chemical industry for many years via the application from flowsheet simulators. With the assistance of these simulators the technological unit operations are usually graphically interconnected to a complete system and simulated. However, the flowsheet simulation in other industries did not become generally accepted yet. Now this proven idea can be put to use in steel industry, here in rolling mill technology. 1
Process Synthesis
The path from the decision-to-build to the I Project Planning Task commissioning of a new plant is an I extremely complex one. For this reason the procedure is divided into separate stages so Process synthesis that the problems to be dealt with are of I manageable proportions. The procedure is I 1I illustrated diagrammatically in Fig. 1. The Synthesis step I first step, synthesizing the process, is to I I ~r establish the layout and principal Evaluation step dimensions of the plant. This is followed by I ! the design of the machinery and automation J ~r systems. All three steps are very strongly interdependent so it would be really ~r Mechanical engineering design sensible to employ an iterative procedure. However, this is very difficult to do in actual ~r practice because the whole process is Automation design governed by a number of different decisionmakers and suppliers. ~r One important approach is the simulation of Plant whole plants in the way that has been commonplace in the chemical industry for many years making use of flowsheet Fig. 1. Synthesizing a process simulation [1, 2]. A comparable simulation tool for the iron and steel industry should be perfectly capable of providing support for technological decision-making. There is still a lot of innovative potential left in investigating a plant as a total entity. Whereas in the past individual technological functions have been optimized, often at considerable expense, and good success has been achieved, the study of
r
! ',
308
the plant as a whole - from raw material to finished product - would appear to promise even greater opportunities for improvement. In this way it is possible to deal with entirely new problems (e.g. optimizing the properties of materials taking into account the development of the grain structure through and beyond the rolling mill, optimizing rough rolled strip thicknesses .... ). Moreover, total decision-making also allows for the fact that the sum of individual optima is often not necessarily the total optimum. 2
Flowsheet Simulation
In a fiowsheet simulation, processes are represented as a network of material flows (Fig. 2.), with the nodes denoted as unit operations. This way of thinking corresponds to the phase model of production suggested by Polke [3]. According to this Unit3i model, process and state follow one another. A material flow in a particular state serves as the un,,, ~ 1 Unit1 input for a process (=element 1 of the system = unit operation2). L~ Unit4 I ~ I Unit5 The process influences the material flow in such a way that a process output generates a Fig. 2. A system as a network of nodes (units) and flows material flow with a particular state. This procedure is wellsuited to the engineer's way of thinking and can be applied to a number of different technical processes. This enables the modeling of complex plant structures. The complete plant model is made by the models of the process units, the material flow model, and the system structure model. 3
H Y B R E X - Modular System
In cooperation between the SIEMENS Company and the Technical University of Berlin a modern software tool (HYBREX) has been developed which supports the decision-making process in rolling mill technology. This tool enables the generation and evaluation of several promising plants within a reasonable time span. It enables the evaluation of a large number of aspects of plant design, leading to higher decision quality. The HYBREX tool for plant design and operation is developed with the form of modular design (Fig. 3.). It is based on a hybrid expert system. The core of the HYBREX tool is a flowsheet simulator, which provides a quasi-stationary simulation of the entire process. To keep abreast of the increasing importance of process optimization, a second component is the optimizer, which works closely together with the simulator. The optimization procedure can follow several objectives at once, such as maximizing throughput while minimizing energy consumption, so that more than just one optimization can be pursued (vector optimization). The module for compromise finding supports the user in the evaluation of individual alternatives. Beside the classic utility analysis, finding compromises involves strategic knowledge (operator knowledge and capability), which is available in the form of heuristic rules.
1 2 Concepts according to system theory
Concepts from a technological viewpoint
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Designed as a relational database, the central data administration serves as an interface among all modules and can even be accessed with standard software (Microsoft Access) from outside HYBREX. All modules communicate with the database via standard access methods (SQL, ODBC, DAO), read all necessary input data out, and write all data from the processing cycle back in. The flowsheet stimulator, for example, stores all r[ FlowsheetSimulator structure information, parameters of the technological elements, simulation results, and project information and calculation results in the database. In CentralData Optimizer ~ Compromise addition to linking the individual modules Administration -Finder together, the database also serves as a central pool of data for analysis and evaluation with standard software. The Materialflexible design of the database also DataBase permits further modules to be created and integrated into the overall system in the future work. Finally, the subordinate Fig. 3. Interaction of the different components material database contains the chemical composition of the steels; parameters describing the flow curve; and physical characteristics, such as density, thermal conductivity, tensile strength, and others. With the concept of the simulator proven mechanisms of commercial simulators for chemical process engineering are transferred to iron and steel industry. It concerns particularly the graphic creation of the system and its interconnection through technological unit operations and the application of a material data base.
i TM
t
4
General Software Approach
Integrated software development (IDE) systems have simplified the creation and implementation of complex programs. Advanced programming technologies, such as OOP, support convenient structuring of very complex technological information. Consistent use of the characteristics of object-oriented programming languages (capsuling, heredity, polymorphism) helps to provide a real-life view of processes and systems with their complex interaction. The HYBREX tool has been developed in the C++ programming language. Although the entire tool runs exclusively on the Windows 9x/NT operating system, selected components like the process models could be used on any other operating system that can compile ANSI C++ code. Fig. 4 shows the modular structure of the entire system with its open architecture. Each module is individually applicable. All interfaces correspond to international standards and guarantee the universal expandability of each section module. Beside the widespread CORBA (Common Object Request Broker Architecture) also the Windows specific COM (Component Object Model ) interface is used (e.g. for communication with Office applications). The open architecture permits the application of the simulator within other areas. Only the libraries and the data bases must be exchanged. The flowsheet simulator as the core of the HYBREX tool comprises the following components: 9 Model generator 9 Class library of the process models 9 Class library of the units
310
The Graphical User Interface (GUl) and the class library of the units are embedded in the MS-Windows operating system, which provides graphic functionality and the visual interface to the user. The class library of the process models, in contrast, is implemented in ANSI C++ Standard, independent of the operating platform, which enables universal application of the process models in other simulation environments. External Data
Material Database
IOSTREAM
DAO/ODBC
Process Models c
c
R B A
M
Flowsheet Simulator
Dynamic Link
External Lib.
(e.g. Math.-Library)
CORBA: CommonObject Request Broker Architecture COM: ComponentObject Model COM
Optimization Server
DAO: Data Access Objects ODBC: OpenDatabase Connectivty
Compromise Finder
DAO/ODBC
Central Data Administration
Optimization Client
Fig. 4. Modules and Interfaces of the HYBREX system The model generator is the direct user interface. With its help, a model of the plant can be created and visualized, comprising individual units, such as rolling with the fiat stand, temperature changes in the roller table and others. A distinction is made between technological and non-technological units. Technological units correspond to the real world, e.g. the flat stand unit is modeled directly on the real fiat stand in the rolling mill. An example of a non-technological unit is the iteration unit which monitors the output values of a particular technological unit. If the monitored value deviates from a pre-set target value, then this value is iterated. The entry point for this iteration loop can be any previous unit. The following units are currently available: flat stand, edging stand, miler table, interstand cooling, caster, furnace, coilbox, coiler, shear, cooling section (technological units), strip editor, loop editor and graphical monitor (non-technological units). Individual units are combined into a plant (rolling mill) by selecting the units from a menu and placing them as needed onto the worksheet. Using the mouse, the units then are linked to the final layout (Fig. 5). A double-click on a particular unit takes the user to an entry template, which displays all parameters and settings for the simulation. While the mill structure is being created on the display, working memory is allocated to the process model of the mill. During the simulation procedure, the simulator accesses this internal plant model and calculates the entire plant, block-by-block. In another mode, the simulator provides automatic calculation of e.g. an entire annual production. Parameterization of the units and the strips to be rolled takes place through external files. The templates for the parameter files are generated by the simulator according to the current mill structure and exported in a standard format, such as ASCII. External program tools (e.g. Microsoft Excel) are used to adapt variable parameters, which include not
311
only the strip parameters, but the unit parameters such as the roll diameter or the pass schedule (distribution of thickness reduction over all stands). In this way, any number of variants can be drawn-up without having to manually parameterize individual units.
Fig. 5 Rolling Mill layout from within the flowsheet simulator
4.1
Class Libraries
The basic technological operations, such as rolling in the horizontal stand or cooling in the roller table, include core models that range from simple equations to complex rules for calculation. The same core models are used for individual operations. One example is the operation rolling in the horizontal stand, which comprises the core models for calculating the rolling force, the change in temperature, the material microstructure, and others. The core model for calculation of temperature change is also found in the rolling in the vertical edger operation. To avoid multiple implementation, a class library has been developed for the core models. Individual core models are encapsulated in classes and arranged hierarchically. All classes are implemented in ANSI C++ Standard, making them independent of operating platform. In effect, the class library is a collection of offline process models that can not only be used in HYBREX, but in other simulation tools as well. Online use of this library in process automation is planned. Similar to the class library of the core models, the units are also collected in a class library. These classes are not independent of the operating platform since graphics functionality needed to generate the flowsheet- and the mill structure - had to be implemented in them. The individual classes represent technological (e.g. stand, roller table) and non-technological (e.g. iteration unit) elements. All elements are derived from a common class, for which basic functionality has been defined including drawing on the display, the acceptance/handover of current values for the strip to be rolled and methods for loading and storing of visual information (position, size, arrangement, symbols). Added to these are the handling routines for particular user reactions (mouse click, shift, zoom, etc.). From this basic class elementspecific basic classes are derived for each unit type (e.g. stand, roller table), for which element-specific characteristics and methods have been defined, e.g. input templates for
312
parameterizing the units, functions for loading and storage of unit parameters, and database interfaces. The basis classes are abstract, meaning that no concrete object can be generated from them. Classes derived from the basic classes include the characteristics and methods of the special technological and non-technological units. Within them, the member variables are defined, ensuring the functionality of the core models. 5
Results
Today, the HYBREX simulator is being used for several tasks. Equipped with convenient options the tool can evaluate a large number of plant design aspects. In combination with the optimizer different parameters can be optimized to achieve a given set of production goals. By modeling and simulation HYBREX generates a set of solutions dependent of the boundary conditions. As an example of a practical problem to be solved we ask for the optimal pass schedule. That means how to distribute thickness reduction over the individual stands. Consider a five stands finishing mill with an overall thickness reduction from 30.0 mm to 2.0 mm, an entry strip temperature of 965 ~ and an entry speed of 0.95 m/sec. We start with a 36.32 percent relative thickness reduction in each single stand and obtain 73,504 kW as the sum of motor power. After running optimization with minimizing motor power as the objective we obtain a pass schedule that leads to a motor power of 70,677 kW. It's an improvement of exactly 4.0 percent. 6
Conclusion
With the described tool flowsheet simulation is being introduced into the steel industry. In cooperation between university and industry an efficient tool for decision making support of construction, modernization and operation of plants in the metallurgical industry has been developed. Substantial principles of the flowsheet simulation in the chemical industry have been used. It concerns particularly block-by-block linkage of individual components to a system and their sequential calculation as well as the application of a material data base. Thereby the system layout develops graphically. In order to use the new possibilities of modern software creation, a strictly modular concept is pursued with the implementation of the simulation system. The individual modules are separately applicable. They can be used in heterogeneous Client/Server systems. The tool is still in the development phase. Simulator and data bases are already implemented. At present the optimization component is developed further into a general applicable component. First experiences in interaction between simulator and optimizer are to be presented in the lecture. References:
[1] [2]
[3] [4] [5]
Blal3, E.: Entwicklung verfahrenstechnischer Prozesse. Methode - Zielsuche -
L0sungssuche- L0sungsauswahl, Otto Salle Verlag, Verlag Sauerl~nder, Frankfurt, 1989 Lohe, B.; Futterer, E.: Station~ire Fliel~schema-Simulation. In: Schuler, H. [Hrsg.]: Prozel~simulation. VCH, Weinheim, 1995 Polke, M.: Proze~leittechnik, Oldenbourg, 1994, 2. erw. Auflage MOiler, H.; Peuker, Th.; Wozny, G: Advanced Software Improves Decision Quality in Rolling Mill Technology. In: Scholz-Reiter, B.; Stahlmann, H.-D.; Nethe, A. (Publ.): Process Modelling, Springer Verlag Berlin, Heidelberg, New Yorck, 1999, pp.370-382 Peuker, Th.; S5rgel, G.; MOiler, H.: Flowsheet simulation - a new tool for rolling mill technology. In: MPT International 3/1999, pp. 155-161
European Symposiumon ComputerAidedProcess Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
313
Some aspects of rate-based modelling and simulation of three-phase distillation columns* E. Eckert and T. Van6k Department of Chemical Engineering, Institute of Chemical Technology, Prague, Technick~i 5, 166 28 Prague 6, Czech Republic For a long period the description of phenomena observed in separation equipments, which with respect to various phases taking part in the separation process could be symbolised as one of the V(G)-L, L1-L2 or V-L1-L2 types, is based on the concept of an equilibrium stage using various types of corrections to fit the real performance of the equipment. At the same time, it is well known that this concept produces more or less unreliable results and, moreover, it is not fully suited for the investigation of the dynamic behaviour of separation columns. In the area of mathematical modelling of these equipments more and more the non-equilibrium concept prevails even if the dimension of problems solved would be considerably increased [1, 2]. The non-equilibrium concept has been gradually incorporated into most commercial simulation systems in the form of standard modules, e.g. RATEFRAC module in ASPEN PLUS system. 1. M O D E L L I N G
CONCEPTS
The problem with the design of a mathematical model of a distillation stage with potential occurrence of two liquid phases (V-L1-L2) using the non-equilibrium concept (i.e. rate based approach) is that its practical formulation and usage is disabled according to missing values of a number of model parameters. The potential existence of three co-existing phases implies the potential existence of three phase interfaces: V-L1, V-L2, L1-L2. In order to describe the processes on each interface it is necessary to have methods for the estimation of mass and heat transfer coefficients on both sides of the interface and methods for the estimation of the interfacial area. Alternatively, a method calculating directly the product of the interfacial area and the heat or mass transfer coefficient could be used as usual for these types of processes. Up to now some methods have been published only for counter-current two-phase V(G)-L (distillation, absorption) and L1-L2 (liquid extraction) processes. In our case the flow of liquid phases across and between column plates is co-current and the estimation methods employed in case of liquid extraction are useless for our purpose. As it could be expected that the second liquid phase is present in the form of a small drops dispersed in the prevailing continuous liquid phase, eventually splitting or coalescing, it is necessary to know the appropriate rate constants if we would wanted to develop a more reliable rigorous model of the process [3]. Unfortunately, no methods are known for this
Authors appreciate the support of the Volkswagen-Stiftung (Germany) (Proj. No. 1/71413Ak) and of the Grant Agency of Czech Republic (Proj. No. 104/97/0916).
314
Fig. 1: Simplified scheme of a three-phase (VLL) stage.
Fig. 2: Scheme of the model of a three-phase batch distillation column operating at the total reflux conditions.
purpose and we could only choose some compromise until the knowledge of such phenomena increases. Obviously, certain possibilities would appear if we take into account some model simplifying assumptions, i.e. a) In the case when both liquid phases are present their composition is in equilibrium and their temperatures are identical. b) The volume occupied by any phase is always ideally mixed, i.e. the phase temperature and composition are uniform in the entire volume and at the same time identical to outlet values. c) The heat exchange with the surrounding is provided only from the continuous liquid and vapour phases. d) No heat nor mass transfer between dispersed liquid and vapour phases takes place as their contact time is likely to be very short. These assumptions served for the development of a combined non-equilibrium and equilibrium model of a separation stage depicted in Fig. 1 and in detail described e.g. in [4, 5]. According to the shortage of space we only present the overview of model equations in Tab.1.
315 Table 1 Overview of model equations (I = number of components)
Subsystem
Equation type
Dim.
Comments
Vapour phase
Component balances Energy balance Internal energy holdup Pressure drop Component balances
I 1 1 1 I
Differential form - changes in holdups Differential f o r m - change in holdup Algebraic equation Algebraic equation Differential f o r m - changes in holdups
Energy balance Internal energy holdup Tray hydraulics Flash calculations
1 1 1 2/+2
Stage volume balance Mass transfer rates
1 2/-2
Differential f o r m - change in holdup Algebraic equation Algebraic equation Algebraic equations- Heinichen's method [8] Algebraic equation Algebraic equations - Maxwell-Stefan equations Algebraic equation Algebraic equations
Cont. and disp. liquid phases
Stage volume V-L1 interface
Energy balance 1 Thermodynamic I equilibrium Interface mole fraction 2 Algebraic equations summations Dispersed liquid Flow rate 1 Algebraic equation Total eqs per stage 71+11 Differential and algebraic equations 2. S O L U T I O N M E T H O D S
When developing more complex models it is necessary to have powerful and reliable numerical tools for their solution. Chemical engineers can profit from the existence of a number of simulation programs as SPEEDUP, gPROMS, etc., which usually contain the following important subsystems: a) A thermodynamic database and a library of thermodynamic functions for the calculation of properties of pure components and their mixtures. b) A library of robust and stable numerical methods capable to solve large systems of algebraic or differential and algebraic equations (DAE) even in case when the model has discontinuities or some other numerically difficult features. Nevertheless, our experience with the SPEEDUP system has shown some numerical and technical problems. For example, SPEEDUP usually requires an extremely good starting point which is in most cases unavailable. In case of complex processes it is necessary to build the model gradually and to utilise previous solutions as starting points. If we use the homotopy method to move from one solution to another it is impossible to continue the solution across the binodal curve. On the other hand, this problem is possible to overcome if we formulate the dynamic model and use integration methods for obtaining the new stationary solution.
316
3. EXAMPLES The suite of problems we have used for testing of the proposed approach concerns the dehydration of ethanol using cyclohexane as the entrainer in various continuous or batch columns. This system exhibits two- as well as three-phase regions when changing the composition and thermodynamic conditions. There has been an extensive experimental work done using a real laboratory column [6, 7]. The examples used here are based on experiments No. 281195 and 201295B documented in [7]. The scheme of the batch laboratory column with 8 real stages equally used as the structure of the column model is depicted in Fig. 2. In case of rate based modelling the number of stages in the model and the column are of course the same. Above stage 1 is a total condenser from which both liquid phased are entirely returned back to the column, i.e. during the experiment the column was performing at conditions of a total reflux. For modelling purposes the condenser is treated as a stage at equilibrium conditions without any holdup at which the incoming vapour condenses and instantly forms one or two liquid phases at the bubble point. For the decision about the number of liquid phases and their outlet flow rates from the condenser or any other regular stage a modification of the Heinichen's algorithm is used [8]. The vapour flow rate entering the column was estimated from the measured heat duty of the total condenser. Examples 1 and 2 differ in the feed composition and therefore in the presence of second liquid phase on stages. While in Example 1 there the results prove two liquid phases on all stages in Example 2 only some of stages appear to be three phase. The input data are summarised in Table 2. Table 2 Specifications for Example 1 and 2 Value or source Example 1 Example 2
Parameter type
Components Vapour feed
Tray
Flowrate Temperature Pressure Composition- mole fractions Type Construction and performance parameters Activity coefficients
Thermod. model Mass Binary mass-transfer transfer coefficients Heat transfer Heat-transfer coefficients
ethanol (1), water (2), cyclohexane (3) 8.09 10.3 mol/s 10.98 10-3 mol/s dew point (calculated) 1.013 105 Pa <0.007,0.239,0.754> <0.926,0.049,0.025> cap tray See [4, 5] for details Modified NRTL method, data obtained from
[91 Estimated by AIChE method for cap trays [10] Estimated by Chilton-Colburn analogy
317 3.1. Discussion of simulation results. The simulation results using the classical equilibrium and the combined models are compared with experimental data in Fig. 3. In Example 1 it can be observed that the equilibrium model predicts the existence of the heterogeneous ternary azeotropic point already on first five trays of the column while the non-equilibrium model approaches this point gradually. The experimental data are situated approximately in the middle between these two Example 1" Temperatures (deg C)
72,0
r Interface ---o-- Liquid = Vapour [] Equilibrium 9 Experirnenta! y
70,0 68,0 66,0 64,0
.~
~ f f
/~j ,,~"/
62,0 60,0 58,0
1
2
3
4 5 Stages
6
7
8
Example 1" Mole fractions of ethanol
-'43-- Equilibrium .... ~ ~Perirnental 2
0,8
3
4
"~
5 6 Stages
~
~ ; ~
1
7
8
9
Example 1" Mole fractions of cyclohexane
78,0 76,0 74,0 72,0 70,0 68,0 66,0 64,0 62,0 60,0 58,0
0,7 0,6
2
3
_~~ .,~ /'/; ~//
S
4 5 Stages
6
7
]
8
1
{
k 9
Example 2: Mole fractions of cyclohexane
0,5
0,5
0,4
0,4 0,3
0,3 02
02 0,1 0,0
r Interface +Liquid r Vapour [] Equilibrium
Example 2: Mole fractions of ethanol 1,0 0,9 [ r Nonequilibrium . ~ ~ 0,8 j-'E1-- Equilibrium ~ ~ 0,7 0,6 0,5 0,4 0,3 02 0,1 , , . . . . , 0,0 l 1 2 3 4 5 6 7 8 Stages
0,6
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example 2: Temperatures (deg C)
I
2
3
4
5 6 Stages
7
8
9
.
2
.
.
3
.
.
.
4
5 6 Stages
,
7
8
9
Fig. 3: Comparison of temperature and vapour phase concentration profiles for Examples 1 and 2 - non-equilibrium model, equilibrium model and measured data.
318 profiles. The difference between the models is in values of heat and mass transfer coefficients indefinite in case of the equilibrium model and calculated in case of the combined model. Example 2 is closer to conditions used for industrial dehydration of ethanol. The heterogeneous azeotropic point is situated again to the top of the column. The equilibrium model shows four three-phase stages at azeotropic conditions while the combined model finds this only on the first stage. The match between the experimental data and results of nonequilibrium modelling in the bottom of the column where only one liquid phase exists is surprisingly good. The reason might be that in the lower part of the column the dispersed liquid phase disappears and the model reduces to an entirely rate-based model, i.e. the equilibrium L1-L2 part of the model might be inadequate. Also the AIChE method for estimation of mass transfer coefficients could be unfeasible for two liquid phases. These reasons could also account for the differences between experimental data and simulation results in Example 1. The discrepancy in temperature profiles is more obvious in Example 1 and could be explained by many factors - subcooling of the reflux, inaccurate measurement of the pressure in the laboratory column, etc. -
4. C O N C L U S I O N S The detailed modelling of separation processes have proved to be useful for the modelling and design of real complex industrial separation columns. The resulting solution of the model, if reached, can influence the column design and performance with direct economical effects. Moreover, for this type of column model the use of stage or column efficiencies is not needed, thus overcoming any ambiguous determination of global column efficiency, usually taken as 70 % [6]. There are certain difficulties resulting from our insufficient knowledge how to calculate certain physical and transport properties of complex mixtures and how to estimate the performance of column trays with various constructions. This should be the theme for further investigations. REFERENCES 1. Wesselingh A., Trans IChemE, 75 (1997) 529. 2. Lao M. and Taylor R., IEC Res. 33 (1994) 2637. 3. Eckert, E., Collect. Czech. Chem. Commun. 60 (1995) 2085. 4. Eckert E., Van6k T.: Simplified rate-based modelling and simulation of three-phase distillation columns. Proc. of 18th lASTED Int. Conference MIC '99, p. 223, Innsbruck, February 15-18, 1999, Austria. 5. Eckert E., Van6k T.: Rate-based modelling and simulation of three-phase distillation. Proc. of 45th Conference CHISA'99 (on CD ROM), Srnf, October 18-21, 1999, Czech Republic. 6. MUller D. et al., ICE Symp. Ser. 142 (1997)149. 7. Klein W., Diplomarbeit, LFPT, RWTH Aachen, 1996. 8. Heinichen H., Diploma project, ENSIGC Toulouse/RWTH Aachen, 1994. 9. Connemann M. et al., Fluid Phase Equilib. 60, (1990) 99. 10. AIChe Bubble Tray Design Manual, Am. Institute of Chemical Eng., New York, 1958.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScience B.V. All rights reserved.
319
Modeling and Simulation Tools for Supercritical Fluid Processes S. Diaz, S. Espinosa and E. A. Brignole Planta Piloto de Ingenieria Quimica- PLAPIQUI (UNS-CONICET) Camino La Carrindanga Km 7 - 8000 Bahia Blanca - Argentina e-mail: [email protected], [email protected], [email protected]
ABSTRACT The application of an upgraded group contribution equation of state combined with rigorous simulation and optimization to supercritical processes is reviewed and new applications are presented for vegetable oil purification and separation of fatty acid alkyl esters. 1. INTRODUCTION Chemical processes with supercritical fluids have received increasing interest during the past decade. Experimental data are scarce and difficult to obtain. Several laboratory experiments with supercritical fluids have been reported, but there is still much research to be done on near critical fluid property predictions, rigorous unit simulations and synthesis and optimum design of these processes. Commercial process simulators provide a wide range of thermodynamic models for nonideal mixtures at high pressures. However, in many cases these models fail to give a realistic and quantitative description of the phase behavior, for example in mixtures of gases with associating components (Gros et al., 1997) or with a wide range of component molecular weights (gases with tryglicerides) (Bottini et al., 1999). Gros et al. (1998) demonstrated the potential of rigorous process optimization and simulation to the synthesis and optimization of process schemes and conditions for oxychemicals recovery and dehydration from aqueous solutions, using the dual solvent effect (extraction and dehydration) of near critical light hydrocarbon gases. The synthesis problem of the extraction-dehydration has been solved as a mixed integer nonlinear programming problem, where the objective function represents the process energy consumption. Optimum process schemes depend on the oxychemical to be recovered, supercritical fluid used as solvent and feed composition (Diaz et al., 1999). Amaro et al. (1998) have applied a similar approach for the synthesis of operating schemes and selection of process conditions for the Gas Antisolvent (GAS) Crystallization Process of organic salts from aqueous solutions. In GAS a near critical gas is dissolved in the solution and the solute precipitates due to the gas strong antisolvent effect. However, organic salts are soluble in aqueous solutions, but nonpolar gases are not. Therefore a cosolvent is required to increase the gas solubility in the mixture. An operating scheme is found to achieve a high loading of salt in the crystallizer and complete miscibility of water in the gas antisolvent - cosolvent mixtures. Several cosolvents were identified and process conditions that maximize the water solubility in the crystallizer were proposed. The results were confirmed by crystallization experimental studies. In this work the purification of vegetable oils from liposoluble contaminants is discussed; near critical propane and CO2 are analyzed as potential solvents. Furthermore, the separation of fatty acid esters is studied with the use of supercritical CO2 as a high-pressure entrainer. The process is based on the high solubility of long chain fatty acid alkyl esters in supercritical CO2. The process conditions depend on the main column pressure, temperature and CO2/feed ratio. The column reflux is obtained by heating and pressure reduction of the extract. The selection of optimum process conditions for this
320
separation process is illustrated for the separation of methyl myristate (C14:0) and ethyl stearate (C18:0) and for the recovery of EPA (C20:5) from a fish oil fatty acids mixture. The observed behavior may be applicable to other problems of purification or refining of fatty oils like removal of cholesterol from butter and milk fat. 2. THERMODYNAMIC MODELING OF SUPERCRITICAL PROCESSES The GC-EOS model was proposed by Skjold-Jorgensen (1988) to study gas solubilities in nonideal mixtures at high pressures. This model was applied for the prediction and correlation of solubilities of solvents in supercritical fluids by Brignole et al. (1987). The original model takes into account only repulsive and dispersive interactions. Gros et al. (1997) have extended the capability of this model to treat associating mixtures (GCA-EOS) in mixtures of water and alcohols with non polar gases, like propane or CO2. Natural oils and derivatives are complex mixtures of glycerides with fatty acids of different chain length and degree of saturation. However, their molecular structure can be characterized with a few functional groups. The development of supercritical processes for the purification of triglycerides or derivatives requires the prediction of phase equilibrium compositions and conditions for mixtures of high molecular weight components with gases at high pressure. The use of the SRK equation or similar equations of the van der Waals family has proven to be unsuccessful to predict and correlate the complex phase equilibrium behavior observed in those systems. An upgraded version of the GCA-EOS model has been developed for the correlation and simulation of vegetable oils and derivatives processing with near critical fluids (Bottini et al., 1999). In the application to high molecular weight compounds, the component critical diameter is adjusted from activity coefficient data at infinite dilution of alkanes in the heavy compounds. In the oil molecules, the number of methyl plus methylene groups are in the range of 40 to 60. For molecules of this size, even with moderate energetic interactions, it is very important to revise the binary energy parameters of CO2 with (CH2) and (CH3) in order to avoid degeneration of the predictions with a large increase in molecular size (number of CH2 groups). The same observation is valid for other interaction parameters like aromatic groups with (CH2) and (CH3). Bottini et al. (1999) and Espinosa et al. (1999) give revised sets of parameters for the application of the GCA-EOS for natural oils and derivatives. The model is able to describe multicomponent vapor-liquid, liquid-liquid and liquid-supercritical fluid equilibria. 3. MATHEMATICAL MODELING OF SUPERCRITICAL PROCESSES The determination of operating conditions for different supercritical processes has been solved as Nonlinear Programming (NLP) problems. Design variables have been selected to represent main continuous decisions associated to each particular process. Equality constraints represent the process mathematical model and they are solved within a sequential process simulator. This program includes rigorous models for a high-pressure multistage extractor (Kehat and Ghitis, 1981), low and high pressure distillation columns (Naphtali and Sandholm, 1971), and a multiphase flash (Michelsen, 1982). The GCA-EOS has been integrated as thermodynamic support for these model unit simulation routines. Inequality constraints include process specifications, operating bounds and bounds on potential units. Depending on the type of process, different optimization goals have been analyzed. The optimization program interacts with a rigorous sequential modular process simulator in a black box way and non-linear programming problems have been solved with OPT (Biegler and Cuthrell, 1985). The use of a black box simulator for function evaluation does not guarantee problem convexity and may converge to locally optimal solutions. Reported numerical results have been obtained running the program with different initial points that converged to the same optimum; this fact does not guarantee that the solution is the global optimum, but it enforces the fact that it is a strong local minimum.
321 4. ANALYSIS OF SUPERCRITICAL EXTRACTION PROCESSES 4.1. Removal of contaminants or valuable products from natural oils
Natural oils are prone to contamination with liposoluble pollutants. The use of CO2 for the removal of pollutants or valuable substances from natural oils is a promising approach. CO: is an environmentally benign solvent and fatty oils have very low solubility in liquid or near critical CO:. Propane has greater solvent power than CO2 for organic solutes, but has the disadvantage of being flammable. Besides, the liquid-liquid immiscibility between vegetable oils and propane is observed under a limited range of pressures and temperatures (Bottini et al., 1999). Under these conditions the solubility of the oils in the propane phase is also very low. The removal of pollutants with near critical solvents from vegetable or animal oils or oils containing natural matrices, also requires the determination of optimal operating conditions. In this case, the solubility of the pollutants in the near critical phase should be increased keeping the solubilities of the TAGs in the extract phase at very low values. In addition, the density difference between the oil and the extract phase should be adequate when countercurrent operation is F i g u r e 1. NCF extraction process flow diagram. required. The extraction cycle and the solvent recovery system are shown in Fig. 1. An isothermal extractor of 15 stages is used. The concentration of solutes in the oil is studied in the diluted range: 1% for hexane and 0.01% for other pollutants. The objective is to minimize solvent-to-feed ratio (S/F) subject to an oil weight concentration in the extract lower than 0.05% and purity specifications (less than 5 ppm for hexane, less than 0.01 ppm for aromatics and chloroaromatics). Extraction with C02: The system vegetable oil - CO2, shows phase immiscibility under near critical and supercritical conditions. This immiscibility region extends to very high pressures. Therefore, the two-phase behavior, required for countercurrent removal of pollutants with CO2 is observed over all foreseeable operating conditions. The optimization of operating conditions is performed for temperatures in the range 303 <_T (K) _<340 and for pressures 7 _
322
isotherm there is a maximum pressure at which complete miscibility is obtained. Therefore, to keep the solubility of the oil in the propane phase at very low values, operating pressures should be below the maximum pressure for each isotherm. The process behavior is similar to the one with CO2. Optimal conditions for the removal of different pollutants with propane are also given in Table 1. The optimal pressure and temperature combination is the same for each solvent and it is independent of the extracted pollutant due to the very low pollutant concentrations and the tight constraint imposed on oil loss in the extract. Table 1. Optimal operating conditions for the removal of pollutants with propane as solvent. Pollutant Hexane Benzene Toluene Chlorobenzene Naphtalene Dichlorobenzene Biphenyl
Textr (K) CO2 C3 333.7 399.4 333. 400 333. 400 333. 400 332.8 400 n.a. 400 n.a. 400
Pextr-(MPa) COz - C3 16. 7.0 16. 7.15 16. 7.15 16. 7.15 15.9 7.15 n.a. 7.15 n.a. 7.15
S (Kmol) CO2 C3 376. 18.50 70. 55.83 88. 67.88 200.5 107.15 206.3 200.23 n.a. 215.90 n.a. 322.17
S/F CO2 2.6 4.58 5.75 13.08 13.44 n.a. n.a.
C3 1.28 3.65 4.43 6.99 13.05 14.06 20.98
Pollutant ppm 5. 0.01 0.01 0.01 0.01 0.01 0.01 .
.
4.2. Supercritical Fluid Fractionation of Fish Oils Nilsson (1996) has recently reviewed the supercritical fluid extraction and fractionation of fish oils. Fish oils typically contain straight-chain fatty acids (C14-C22), often unsaturated having from one to six double bonds. The high degree of unsaturation of these oils, of great pharmaceutical interest, precludes the application of vacuum distillation because these components are highly thermally labile. Among the components of interest derived from fish oils are concentrates of EPA (eicosapentanoic acid) and DHA (docosahexaenoic acid) in the form of ethyl esters. Eisenbach (1984) proposed the fractionation of fatty acid ethyl esters using CO2 as a high-pressure entrainer in a semi batch fractionation process. In this scheme, a hot finger in the top of the column decreases the esters solubility in the gas phases and acts as a partial condenser. A continuous high pressure (extraction and fractionation column) of the process is also proposed by Eisenbach (1984). Brunner (1994) discusses this separation problem in detail, however he points out that the selection of process conditions by rigorous computational solution is not available. The upgraded version of the GCA-EOS is able to predict with good accuracy the high pressure vapor - liquid equilibria of fatty acid alkyl esters with CO2 in the entire range of operation for fractionation, solute recovery and CO2 separation process. In this work, we have determined design and operating conditions for different optimization goals such power consumption, product specification or a combination of them. The approach is illustrated for the separation of methyl myristate (C14:0) and ethyl stearate (C18:0), straight chain fatty acids ethyl esters and for EPA recovery from a C 14, C 18, C20 mixture. 4.2.1. Separation of a methyl myristate- ethyl stearate mixture An extractor-separator system is studied for the separation of a methyl myristatc - ethyl stearate mixture (54 and 46% weight, respectively) with carbon dioxide. The objective function is the maximization of C14 recovery. Main optimization variables are: CO2 recirculation rate, column operating pressure and temperature, the solvent recovery unit temperature and reflux ratio, subject to given constraints on components recovery and purity. Optimization results are given in Table 2 for an extraction temperature of 340.15 K.
323 Table 2. Optimal operating conditions for the fractionation of methyl myristate (C14:0) and ethyl stearate (C18:0) Variables Pextractor (Mpa) Trash (K) Reflux to Column/Liq. Top Product Solvent/Feed (wt) Solvent Purity (%) C14 Purity (%) C18 Purity (%) C 14 Recovery (%)
Lower bound 12.00 378.00 0.40 17.00 99.98 80.00 99.00
Upper bound 17.00 390.00 0.80 22.00 100.00 100.00 100.00
Optimum 16.80 378.00 0.57 21.85 99.98 84.30 99.87 98.80
4.2.2 EPA recovery from fatty acid alkyl esters EPA (eicosapentanoic acid or 20:50)3) is a fish oil derivative that has pharmaceutical value. In this work, a urea-adducted ethyl ester derived from fish oil (C14, 14.1%; C18, 9.7%; C20, 74.9%, wt.) is fractionated in a supercritical fluid extraction train, as shown in Fig. 2, to obtain an EPA concentrate. Optimization variables are: extractors pressure, solvent flowrate in each column, separators temperature and refluxed fractions. Additionally, an infeasible path strategy has been applied for the convergence of recycles of separators liquid streams to the extractors. Results are shown in Table 3.
Figure 2. Fatty acids alkyl esters fractionation process.
Table 3. EPA recovery with supercritical carbon dioxide Variables Pextr (El) (Mpa) Trash (S1) I(K) Solvent/Feed E1 (wt) Solvent Purity (S 1 & $2) (%) C 14 Recovery (%) EPA in Raffinate (%) EPA Recovery (%)
Lower bound 12.00 358.00 17.00 99.98 90.00 92.00
Upper bound 17.00 390.00 60.00 100.00 100.00 100.00
Optimum 13.83 377.3 45.05 99.99 100.00 92.01 90.00
324 A 92% EPA stream (solvent free basis) is obtained as the raffinate in the first extractor with 90% EPA recovery. These results are comparable to experimental values reported by Nilsson (1996) with an increase in recovery and purity. 5. CONCLUSIONS Optimum operating conditions for the purification of vegetable oils with near critical CO2 and propane have been identified with the help of reliable predictions of phase equilibria using the GCAEOS model. The problem of fractionation of fish oil fatty acids ethyl esters has also analyzed. Rigorous thermodynamic modeling and process optimization have been applied to a great variety of problems for processing, refining and purification of natural oils with high-pressure gases. In this way, a better understanding has been gained on the selection of process conditions for these nonconventional separation processes. Acknowledgement. The authors gratefully acknowledge financial support from CONICET and Universidad Nacional del Sur, Argentina. REFERENCES
Amaro Gonz~ilez, D., M. Zabaloy, G.D. Mabe, E.A. Brignole, "Mixture properties of solvents for Gas Antisolvent crystallization", AIChE 1998 Annual Meeting, Miami, USA, 15-20 November 1998. Biegler, L., J. Cuthrell. "Improved infeasible path optimization for sequential modular simulators.II: The optimization algorithm", Comp. Chem. Eng., 9, 257-265, 1985. Bottini, S., T. Fornari, E. A. Brignole, "Phase equilibrium modeling of triglycerides with near critical solvents", Fluid Phase Equilibria, vol. 158-160, 211-218, 1999. Brignole E.A., P.M.Andersen, Aa.Fredenslund, "Supercritical fluid extraction of alcohols from water", Ind. Eng. Chem. Res. 26, 254, 1987. Brunner, G., "Gas extraction" in "Topics in Physical Chemistry," edited by H. Baumgartel, E. Franck, W. Grunbein, 1994. Diaz, S., H. Gros, E. A. Brignole, "Thermodynamic modeling, synthesis and optimization of extraction- dehydration processes", submitted for publication to Comp. Chem. Eng., 1999. Eisenbach, W., "Supercritical fluid extraction: A film demonstration", Ber. Bunsenges. Phys. Chem., 88, 882-887, 1984. Espinosa, S., S. Diaz, T.Fornari, "Group Contribution Equation of State GC-EOS: Extension and revision", Proc. EQUIFASE 99, V Iberoam. Conf. on Phase Equilibria for Process Design, 274-280; Spain, June 1999. Gros, H.P.; Bottini, S.B.; Brignole, E.A. "High pressure phase equilibrium modeling of mixtures containing associating compounds and gases", Fluid Phase Equilibria, 139, 75-87, 1997. Gros, H., S.Diaz, E.A.Brignole, "Process synthesis and optimization of near critical separations of aqueous azeotropic mixtures", J. Supercritical Fluids. 12, 69-84, 1998 Kehat, E., Ghitis, B., "Simulation of an extraction column", Comp. Chem. Eng. 5, 171, 1981. Michelsen, M."The isothermal flash problem. Part II: phase split calcul.", Fluid Phase Equilibria. 9, 21, 1982. Naphtali, L., Sandholm, "Multicomponent separation calculations by linearization", AIChE J. 17, 148, 1971. Nilsson, W. B., in "Supercritical fluid technology in oils and liquids chemistry", Cap.8, edited by J. King and G. List, A.O.C.S. Press, 1996. Skjold-Jorgensen, S., "Group Contribution Equation of State (GC-EOS): a predictive method for phase equilibrium computations over wide ranges of temperatures and pressures up to 30 MPa". lnd.Eng. Chem.Res. 27, 110, 1988.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
A computer
325
aided tool for heavy oil thermal cracking process simulation
R. Maciel Filho a and M. F. Sugaya b
aFaculty of Chemical Engineering, Unicamp, CP 6066, Campinas 13081-970, Brazil. Email: maciel @feq.unicarnp.br bpetrobras R&D Center, Ilha do FundS.o QD7, Rio de Janeiro 221949-900, Brazil
This study proposes a dual plug flow reactor representation for the light pyrolysis of petroleum distillation residues in coil-type reactors. It consists of two parallel plug flows: one vapor and the other liquid, traveling at different speeds in a coil. Reaction is assumed to be the rate controlling process, with equilibrium between the phases. A heuristic lumping approach supported by pilot plant data is used and the pseudokinetic scheme derived presents a certain feed independence within the range of stocks available for the study. An industrial case study (delayed coking) is explored to provide insight into the problem of reconciling the kinetics of pyrolysis and carbonization for the upgrade of distillation residues. 1. I N T R O D U C T I O N The presence of metals and asphaltenic molecules in distillation residues has been a serious obstacle to the use of catalytic processes. Progress in this area has been significant but most processes are still limited to atmospheric residues of light petroleums or mixtures of vacuum gasoils and distillation residues (Le Page, 1990). Hence, thermal cracking remains the major option. Unfortunately, a severe carbonization accompanies all thermal upgrade processes, specially when heavy oil fractions and large conversions are involved. The usual practice in such situations is to split the conversion between a coil reactor and a soaking vessel downstream where coke accumulates, as in the delayed coking process (Fig. 1). In this process, distillation residues are typically mixed with a recycle stream in the bottoms of a fractionator and the combined feed is sent to a fired heater where the pyrolysis reactions start. By carefully designing the tubes for high velocities and by using large surface to volume ratios, the reactions can be delayed until the reactants reach the coking drums downstream, thus restraining the coke deposition in the furnace as much as possible. The effluent from the coking drums is then fractionated and sent to other parts in the refinery for further treatment. The process is semi-continuous because condensation reactions cause coke to accumulate in the operating drum. When it becomes full, the output from the furnace is diverted to a parallel drum which has already been cleaned, tested and pre-heated. The full drum is steam
326
Fractionator /._~ A,
naphtha
Cokedrums
:_'.]
51.......
\
gasoils
heavy gasoil
I--i
/
F...... ~[[
Feed ___.___~
....
a
.....
Fig. 1. The delayed coking process.
stripped, quenched with water, drained and cleaned. 2. PROCESS MODEL
One of the major problems in modeling the pyrolysis and carbonization of petroleum distillation residues is the low compositional knowledge of the feed. Sophisticated analytical techniques contribute marginally in describing a few structural properties of the charge and they provide only average parameters. The feed characterization adopted here uses the methodology developed by Bozzano et al. (1998) which fairly predicts distributions of fundamental properties such as molecular weight and boiling point, as well as some structural characteristics of the feed from simple properties (namely, initial boiling point, sulfur content, average molecular weight and density). For the pyrolysis products, characterization is based on chromatographic analyses and common ASTM distillations (D-86 and D-1160). Because the thermal cracking of distillation residues at low temperatures is essentially a liquid phase phenomenon it is important to estimate residence times for this phase properly. The reaction is assumed to follow the first order, 16-lump scheme in Fig. 2 and Equation 1:
dPi - h L A T P L
dx
kiCR, L
(1)
WL
where Pi is a mass fraction or a fundamental property (SPGR or MW) of species i, h is the hold-up and W is the mass flow rate. Subscripts L and T refer to the liquid and total streams, respectively. Kinetic parameters are determined from pilot plant data obtained by Bria and Filgueiras (1982). The conversion process is assumed to be controlled by the reaction in the liquid, with a thermodynamic equilibrium between the phases. However, the vapor phase moves through the
327
R
G
SPGR
350-400
N
SPGR
400-450
SPGR
450-475
L 350-400
R
SPGR
475-500
400-450
SPGR
475-500 500-550
MW 550+
450-475
SPGR
500-550 550+
Fig. 2. Kinetic scheme. R = residue (550~ G = gases, N = naphtha (C5-204~ L = light gasoil (204-350~ 350-400 = yield of gasoil boiling between 350 and 400 C, SPGR 350-400 = specific gravity of gasoil boiling between 350 and 400 C, MW = molecular weight.
coil at higher speeds than the viscous, heavy liquid so the hold-up of the phases is different from the fraction calculated based on flow rates and densities of the two phases (Hughmark, 1962). The flow regime naturally changes as the reaction proceeds, with the increasing vaporization in the coil (e.g. Griffith and Wallis, 1961). Pressure drop is computed following Dukler et al. (1964). Phase equilibrium predictions are based on the Redlich-Kwong equation of state, as modified by Soave (1972). Physical properties are estimated with the methods listed in Table 1. The effect of the temperature over the liquid viscosities is computed with the ASTM method modified by Wright (1969). Table 1 Methods for physical properties used in the model. ................................................................ Density Thermal conductivity V!sc_os:!tY........
Vaporphase Liquid phase . . . . . . . . . . . Equation of state (SRK) Gunn-Yamada(1971) Stiel-Thodos(1964) Mallan et al. (1972) Dean-Stiel(!:965 )....................T~____(. . :1985__ ) ..................................................................................
While in the pilot plant reactor the processing conditions are essentially isothermic, tube wall temperatures for industrial furnaces can be estimated from the API Recommended Practice 530 document (1988), given the heat flux. Enthalpies are calculated from a modified BWR equation (Lee and Kesler, 1975) using a reference state of saturated liquid at -129 C. A heat of reaction of 800 J per g of products boiling below 204 C (ASTM D-86) is assumed. 3. R E S U L T S AND DISCUSSION
3.1 Pilot plant results The composition of the light gases (H2-C4) and naphthas (C5-204) produced by the pyrolysis reactions are nearly independent of the feed and extent of conversion, for 5 different
328
0.100
3O
25 20
0.010
--.I ..... VR-2 I ---~-- VR-3 [
5
I
0
0
I
I,
I
15
20
25
I
5
10
0.OOl
o.ooo 0.00126
(wt.%)
Conversion
l
Gases N a p h t h
I
0.00128
I
0.0013
I
0.00132
I
a
Yield
GasoU 350-400
Yield
400-450
Yield
450-475
Light
.......~i~,....... ~ VR-4 I ~VR-5 I
10
*i
0.00134
1/T [K-l]
Fig. 3. Ethane in the light gases.
Fig. 4. Kinetic constants for feed VR-4.
vacuum residues (Fig. 3 is typical). Because of this, average values can be used to represent the internal composition of these lumps. For the light gas oils, composition and quality are also independent of conversion. As the temperature is increased, the pyrolysis reactions increase in rate, but the residence time of the liquid phase decreases because of vaporization. The first effect is superior and the calculated kinetic constants display a monotonic tendency to increase with the operating temperature. However, as the temperature is raised, the conversion gradually increases to a smaller extent and high coil conversions are ultimately limited by the available reactor length at constant pressure. As a consequence of this behavior, reaction constants (kc, kN, kL, etc.) can be obtained by curve fitting the data using a multivariable Newton-Raphson procedure. Robust methods for calculating equilibrium are essential. Convergence problems often occur near the inlet, where the reactions produce small quantities of supercritical gases. Small changes in the amount of gases can cause large variations in the bubble pressure. To ensure the absence of grid resolution problems thousands of flash calculations are necessary. Typical calculated kinetic constants are presented in Fig. 4 while equivalent sets have been obtained for other feedstocks, revealing a certain feed independence. As can be seen, the activation energies are similar. Moreover, linear relationships are obtained in spite of the large differences in pressure between some runs. For VR-4, for example, run number 4 has been performed at a much lower temperature. Because the pressure employed during this run is much higher, the residence time increases and compensates for the lower temperature. The overall result is an intermediate conversion (Table 2). Table 2 Conditions employed during processing of VR-4. .............................. : ............. ::::::::::::::::::::::::::::::::::::::........ :::-:-: . . . . . . .---:........ : ............. : ..................... : ................... -::. . . . . . . . . . . . . :........................................... :: ................ : = = : : ......... :-::=:--:~=: :: . . . . . . . . . . . . . .
run
1 2 3 4
number
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
....Temperature (C) 490 495 500 475 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Pressure ( M P a ) 0.2 0.2 0.2 2.0
.
.
.
.
.
.
.
.
:...
~
_
.,
............
Conversion(wt.%) 13.2 15.0 18.7 16.6 , ..............
_. . . . . .
,,,
329
3.2 Case Study: Delayed coking The model is evaluated by using it to examine the performance of and industrial delayed coking unit operated by Petrobras at Betim in Brazil. The heat duty and fuel gas consumption match the industrial data very well, as can be seen in Table 3. Changes in temperature in the tubes predicted by the model also compare with plant data. The reaction is significant only in the last third of reactor, where the temperature is above 400 C. No significant reaction occurs in the convection zone. Table 3 Results used to compare model predictions with industrial data. Pressures in bar, temperatures in C, duties in GW and fuel gas consumption in Nm3/d. . ............................. 52-F-1 A Model 52-F-1 B Model Ti . . . . . . . tion 238 238 238 238 Tout . . . . . . . tion 396 396 372 372 Dutyconvection n.a. 7.2 n.a. 6.0 Tout, radiation 502 502 502 503 Dutyradiation n.a. 6.6 n.a. 7.6 Dutyfu. . . . . 12.9 13.8 13.4 13.6 Fuel gas 1380 1470 1420 1450
4. CONCLUSIONS The linear relation observed between the kinetic constants derived from the proposed heuristic lumping methodology and DPFR representation indicates a pressure, temperature and even feedstock independence of the corresponding Arrhenius parameters. Therefore, the results indicate that coil reactor modeling at low conversions can be performed with a single set of Arrhenius parameters within a limited feedstock range. Some pyrolysis processes for the upgrade of distillation residues are performed in two sequential units in order to optimize the delicate balance between selectivity and campaign time caused by the parallelism between radical decomposition and condensation reactions (e.g. delayed coking). Since differences in product quality at high conversions are known to be feed and conversion dependent, the results obtained suggest that the changes responsible for the differences take place mainly in the downstream unit, in spite of the fact that substantial conversion can occur in the upstream unit. While coke formation can be successfully delayed to the soaking reactors downstream, the pyrolysis reactions cannot. This is because the energy required for the endothermic reactions and vaporization of products in the soaking vessel is supplied by the furnace. If conversion is delayed too much the vaporization is low and this energy will have to be supplied as sensible heat, to keep the distillate yields in the process at the same level. This implies higher furnace process temperatures, higher tube wall temperatures and lower velocities, which will effectively increase the plugging tendency in the furnace. Moreover, this may lead to the formation of shot coke in the soaking stage (Ellis and Bacha, 1996). Reducing the total pressure, the pressure drop in the system or the hydrocarbons partial
330 pressure (increasing steam) will improve the vaporization of products, thereby increasing the transfer of latent heat to the soaking reactors at the expense of sensible heat. Unfortunately, any of these options represents a substantial increase in costs which severely limits choices. Nevertheless, the case study results indicate that most of the conversion and pressure loss in the system takes place at the few last tubes in the furnace and at the transfer line to the soaking reactors. Clearly, design efforts to reduce the pressure loss should be directed to these sections, in particular by minimizing the length of the transfer lines and the number of pipe fittings, which are determined by plant layout. Therefore, furnace tube plugging should be minimized by using high velocities while not reducing residence times, e.g., by using small diameters, long lengths and low heat fluxes. Unfortunately, these give rise to an undesirable pressure increase which has to be handled by using multiple passes and by reducing pressure losses in critical areas. The choice of appropriate operating conditions is crucial in seeking to increase the production of liquids and needs to be anticipated at the design stage because more options are then available. The model derived here can be a useful tool to study the effect of some design parameters. REFERENCES M. Bria and R. Filgueiras, Relat6rio Diter 18, Petrobras R&D Center (1982). G. Bozzano, M. Dente, M. Sugaya and C. McGreavy, 216th ACS National Meeting, Division of Fuel Chemistry Preprints, 43(3), 653 (1998). Dean D. E., and Stiel L. I., AIChE J., 11, 526 (1965). Dente M., Bozzano G., and Bussani G., A, Comput. Chem. Eng., 21, 1125 (1997). Dukler A. E., Wicks M. III, and Cleveland R. G., AIChE J., 10, 44 (1964). Ellis P. J., and Bacha J. D., Light Metals, 477 (1996). Griffith P., and Wallis G. B., Journal of Heat Transfer, 83, 307 (1961). Gunn R. D., and Yamada T.,AIChE J., 17, 1341 (1971). Hughmark G. A., Chem. Eng. Prog., 58(4), 62 (1962). Lee B. I., and Kesler M. G.,AIChE J., 21, 510 (1975). Le Page J. F., Chatila S. G., and Davidson M., Raffinage et Conversion des Produits Lourds du Pdtrole, Editions Technip, Paris (1990). Mallan G. M., Michaelian M. S., and Lockhart F. J., J. Chem. Eng. Data, 17, 412 (1972). McGreavy C., Sugaya M., 216th A CS National Meeting, Division of Fuel Chemistry Preprints, 43(3), 684 (1998). Soave G., Chem. Eng. Sci., 27, 1197 (1972). Stiel L. I., and Thodos G., AIChE J., 10, 26 (1964). Twu C. H., Ind. Eng. Chem. Proc. Des. Dev., 24, 1287 (1985). Wright W. A., J. of Materials, 4(1), 19 (1969).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
331
NATURAL GAS FIRED POWER PLANTS WITH CO2-CAPTUREPROCESS INTEGRATION FOR HIGH FUEL-TO-ELECTRICITY CONVERSION
EFFICIENCY
Hanne M. Kvamsdal a, Thormod Andersen b and Olav Bolland b aSINTEF Energy Research, N-7465 Trondheim, N o r w a y bNorwegian University of Science and Technology. N-7491 Trondheim, N o r w a y A concept for capturing and sequestering CO2 from a natural gas fired combined cycle power plant is presented. The present approach is to decarbonise the fuel prior to combustion by reforming natural gas, producing a hydrogen-rich fuel. The reforming process consists of an air-blown pressurised auto-thermal reformer that produces a gas containing H:, CO and a small fraction of CI-14 as combustible components. The gas is then led through a water gas shift reactor, where the equilibrium of CO and 1-t20 is shifted towards CO2 and H2. The CO2 is then captured from the resulting gas by chemical absorption. The gas turbine of this system is then fed with a fuel gas containing approximately 50% ~. In order to achieve acceptable level of fuel-to-electricity conversion efficiency, this kind of process is attractive because of the possibility of process integration between the combined cycle and the reforming process. A comparison is made between a "standard" combined cycle and the current process with COz-removal. This study also comprise an investigation of using a lower pressure level in the reforming section than in the gas turbine combustor and the impact of reduced steam/carbon ratio in the main reformer. KEYWORDS: Gas fired power plant, CO2 capture, Process integration, Process simulation 1. I N T R O D U C T I O N The emission of CO2 from combustion of fossil fuels may contribute to undesired climate changes through an increase of the atmospheric greenhouse effect. In order to reduce the CO2 emission from natural gas based power generation plants, three different main types of concepts have emerged as the most promising. A) Separation of CO2 from exhaust gas coming from a standard gas turbine combined cycle (CC), using chemical absorption by amine solutions [ 1,2]. B) Gas turbine CC with a close to stoichiometric combustion with oxygen (97%+ purity) from an air separation unit as oxidising agent, producing CO2 and water vapour as the combustion products. [3,4]. C) "Decarbonisation", in which the carbon of the fuel is removed prior to combustion, and the fuel heating value is transferred to hydrogen [5,6]. These concepts have been compared in several papers [e.g. 7,8]. Concept A is regarded as the most mature of the three, though there seems to be some remaining development work with respect to the chemical absorption process. Concept B has been described through numerous studies in the last decade and opposed to concept A and C about all the CO2 gas produced by the combustion can be captured. However, it has been regarded as the least attractive of the three concepts from a commercial point of view [9]. The concept C has been known for years mainly related to studies for CO2 removal in conjunction with coal gasification integrated with CC. However, the production of electricity from decarbonised hydrogen is unlikely to be competitive with concept A unless some synergy effect can be achieved by integration between the different process steps [ 10]. In the present work, focus is put on concept C; decarbonisation prior to combustion.
332
2. R E F O R M I N G O F N A T U R A L GAS The technologies that are most relevant for the production of synthesis gas are: 1) Conventional steam reforming (tubular fired reforming), 2) Partial oxidation of natural gas (NG) with oxygen or air, 3) Autothermal reforming (ATR) and 4)Variations of gas heated reforming. In combination with a NG fired CC plant the autothermal reforming (ATR) method might be even more attractive for three main reasons [10]. First as this method actually is a hybrid combination of methods 1 and 2, the heat generated from the exothermic oxidation reaction is directly exploited in the endothermic steam reforming reactions (the reactions are given by [11 ]). Secondly, preheated air is supplied from the air compressor of the gas turbine and steam is supplied correspondingly from the steam turbine or boiler in the CC plant. Another important aspect here is the fact that an air-blown ATR, together with water gas shift reactors and CO2 removal process, produce a fuel with no more than about 50% hydrogen. Modem gas turbines with low-NOx combustors are restricted regarding the hydrogen concentration of the fuel. Traditional steam reforming processes would, in this application, produce a fuel to the gas turbine with significantly higher hydrogen content. 3.
PROCESS
DESCRIPTION
Figure 1 shows the process configuration of case 2. The hydrogen-rich reformed gas is combusted in a gas turbine (GT), which is integrated with the decarbonisation process [5]. A model of the gas turbine type GE9351FA from General Electric was used in the simulations. This gas turbine represents modem technology of today, and it is used in a large number of plants built in the last few years. The considered steam cycle, Heat Recovery Steam Generator (HRSG), steam turbine (ST) and seawater-cooled condenser (COND) is an advanced process with three pressure levels and steam reheat. The reforming process is supplied with high-pressure air (3) and medium pressure steam (31) from the gas turbine compressor and the HRSG unit, respectively. There is also integration between the power plant and the reforming process with respect to preheating of feed streams in the reformer. This requires supplementary firing of the gas turbine exhaust (a small fraction is removed from the decarbonised fuel) approximately from 600 ~ to 750 ~ The steam production based on exhaust gas heat, below 600 ~ is very much the same as without any supplementary firing of the exhaust gas. This means that the supplementary firing does not increase steam production, as all the heat from the supplementary firing is used for preheating of the reformer feed streams. ,
,
lil J "
p12
~-~ 4 ~
I~r,--" I IA r~ ~it ~ '
I
~il
"
I]]
t
I
Figure 1: Process flow diagram, case 2
,,
NPD
i
333
The high-pressure air extracted from the gas turbine compressor exit is typically 25% lower than the required gas turbine fuel nozzle pressure. Thus an extra pressurisation either of the air to the ATR or the fuel back to the gas turbine is needed. NG (1) mixed with the medium pressure steam (31) is preheated to 500 ~ in the HRSG unit prior to the prereformer (PRE). The steam to carbon ratio is approximately 2. The air extracted from the gas turbine compressor (3) and the prereformer products (7) are preheated to 600 ~ upstream the ATR unit. In the prereformer (PRE) most of the heavier hydrocarbon components are converted to H2 and CO while methane is converted in the ATR unit. The ATR outlet temperature was set to 900 ~ It is assumed that the PRE and the ATR reactors are operated at equilibrium conditions. The steam cycle takes advantage of the reforming process by utilising the cooling process of the reformer products downstream the ATR to generate additional saturated high-pressure steam (25, 26). This saturated steam (27) is further superheated in the HRSG unit, and fed into the High Pressure (HP) steam turbine (28). The produced CO is converted to CO2 in the high and low temperature shift reactors (HTS, LTS), according to Eq. 3. Most of the water is removed in the water removal (WR) unit by condensation at 25 ~ It is assumed that 90% of the CO2 content is removed (38) in the absorber unit (ABS). The removed CO2 is assumed compressed to 100 bar for storage (not shown in Figure 1). A fraction (7.6-10.3%, see Table 2) of the resulting fuel is used for supplementary firing (20) of the gas turbine exhaust at the hot end of the HRSG. The rest is compressed (FC) to about 20 bar (18), heated by the feed stream (12) to the lowtemperature shift converter, and then fed to the gas turbine combustor (19). It is assumed a pressure drop of 3% in the prereformer, heat exchangers and shift-reactors, while 6% is assumed for the ATR. 4. C O N C E P T U A L V A R I A T I O N S In addition to a natural gas fired combined cycle used as a base case, three cases including fuel decarbonisation are investigated in the present work. The main difference between the case 1 and cases 2&3, is the location of the required pressurisation in the loop going from the gas turbine compressor extraction (3) to the fuel feed of the gas turbine (19). In case 1, the air stream (3) is cooled and compressed before the preheating in the HRSG. Around this compressor the streams are heat integrated, though the temperature upstream the HRSG preheating section is reduced compared to the temperature of the gas turbine compressor exit. The compression of the air is increasing the heat demand in the HRSG preheater. In cases 2&3 the fuel gas (17) is compressed (FC) downstream the absorber unit (ABS), giving a lower pressure through the reforming process. Steam is continuously supplied to the reforming process and used both in the reformers and the shiftreactors. In the first cases l&2 all the steam flows through the entire reforming process. Since a large portion of this steam is used in the shift-reactors (LTS and HTS), a large flow of steam is flowing through the reformers as "inert" material, in excess of what is required to avoid coking of the ATR catalyst. The steam that does not take part in the reforming reactions is heated up to the ATR operating temperature and then cooled before the shift-reactors. This heating and cooling of the "inert" steam represents a thermodynamical loss. The influence of splitting the steam supplied to the process, so that 50% of the steam is fed upstream the pre-reformer and the rest is fed upstream the shift-reactors, is investigated, as case 3. The pressure level in the reforming process of case 3 is the same as in case 2. 5. RESULTS Simulations are made with a combination of the computational tools GTPRO (Thermoflow, Inc.) and PRO/II (SimSci, Ltd.). Table 2 presents a comparison of the three different concepts for the integration of the natural gas reforming and the power cycle. In addition, data from a conventional modem combined cycle power plant is presented (Base). A comparison of the different cases shows that all the three cases with fuel decarbonisation, as expected, results in efficiencies well below that of a conventional combined cycle (Base case). The cases with removal of COz give efficiencies in the range 46-47%, compared to about 57% for the natural gas fired base case combined cycle. This is a reduction of 10-11%-points (compression of CO2 for deposition is included).
334
Table 2 Computational results for the different cases. Stream numbers Base 1 Natural gas LHV Input [MW] (1) [AI 684 874 Total air flow rate GT [k~/s] (2) 630 630 Air extraction to ATR [kg/s] (3) 83.1 ATR inlet pressure [bar] (4) 23.8 ATR outlet temperature [~ (9) 900 Flow rate inlet Pre-ref [kg/s] (6) 55.0 Flow rate inlet ATR [kg/s] (4)+(8) 137 Fuel composition % HE 0 55.6 % N2 1.6 40.9 % CO 0 0.3 % CO2 0.6 2.0 % CH4 93.2 0.5 % CEH6 3.7 0 % H20 0 0.2 % Ar 0 0.5 % other 0.9 0 Fuel flow GT [kg/s] (17) 14.3 67.8 Fuel flow supplementary firing [kg/s] 0 7.6 (20) Power output GT [MW] 257 255 Power output ST [MW] 148 179 Gross power output [MW] [BI 391 421 Air & fuel compression [MW] [C] 5.5 fOE-compress!on [Mw] [D] 14.2 I Net power output [MWl [B-C-D1 391 401 Net efficiency 1%1 I(B-C-D)/AI 57.2 45.9 I CO2 emissions [g/kWh el.] 355 64.0 COs reduction [%! 0 82.0
(#) refer to Figure 1 2 3[ Explanation 864 8431Base case: Natural gas 630 630l fired CC, ISO conditions, 15 82.5 78.11 ~ cooling water. 13.8 13.SlCase 1: Compressor on the 900 900lair (3) supplied to the ATR. 57.4 36.91Case 2: No extra 140 115|pressurisation of the air to 56.0 56.31ATR, but a compressor on 40.8 40.2|the fuel stream (17) to the gas 0.41 turbine. 0.35 2.0]Case 3: As case 2, but the 2.0 0.15 0.4|MP-steam (31) is split, and 0 0l partly supplied both upstream 0.2 0.21 the prereformer (31) and 0.5 0.51downstream the ATR 0 01(mixed with 10). 68.2 66.31 7.0 5"5I 256 257 190 180] 432 4241 10.9 10.7I 14.2 13.7! 407 4oo1 47.1 47.41 57.0 60.4I 83.9 83101
In Figure 2 a graphical presentation of computational results are shown and as can be seen seen, the "corrected efficiencies" for cases 2 and 3 are higher than for case 1. This indicates that it is more favourable with respect to efficiency to maintain the lowest pressure (approx. 14 bar) through the reforming process, and instead pressurise the reformed fuel before it enters the gas turbine. When the reforming process is operated at 14 bar less supplementary firing of the HRSG is required than in the case with 24 bar. The reason for this is explained in the following. In order to minimise the compressor work, the air stream (3) in case 1 is cooled before it is compressed to 24 bar. The consequence of this is a lower HRSG preheating inlet temperature and thus more supplementary firing is required. This leads to a lower ratio between the flow of fuel supplied to the gas turbine and the flow of natural gas required as input to the process, resulting in a lower corrected efficiency. Another reason for the better net efficiency of the cases 2 and 3 compared to case 1 is related to the pressure of the MP steam extracted from the steam turbine (lefimost column group in Figure 2). In cases 2 and 3, the pressure of the MP steam is 15 bar compared to 25 bar for case 1, resulting in better steam turbine performance and a lower efficiency penalty for this extraction. An apparent benefit of' case 1 is that the compression of the air upstream the reforming process leads to a lower efficiency penalty than the fuel compression of the other two cases does, due to a lower mass flow. The difference is, however, not large enough to compensate for the drawbacks of case I, mentioned above. As seen from Figure 2, the "corrected efficiencies" for cases 2 and 3 are higher than for case 1. This indicates that it is more favourable with respect to efficiency to maintain the lowest pressure (approx. 14 bar) through the reforming process, and instead pressurise the reformed fuel before it enters the gas turbine. When the reforming process is operated at 14 bar less supplementary firing of the HRSG is required than in the case with 24 bar. The reason for this is explained in the following. In order to minimise the compressor work, the air stream (3) in case 1 is cooled before it is compressed to 24 bar. The consequence of this is a lower HRSG preheating inlet temperature and thus more supplementary firing is required. This leads to a lower ratio between the flow of fuel supplied to the gas turbine and the flow of natural gas required as input to the process, resulting in a lower corrected efficiency. Another reason for the better net efficiency of the cases 2 and 3 compared to case 1 is related to the pressure of the MP steam extracted from
335
the steam turbine (leRrnost column group in Figure 2). In cases 2 and 3, the pressure of the MP steam is 15 bar compared to 25 bar for case 1, resulting in better steam turbine performance and a lower efficiency penalty for this extraction. An apparent benefit of case 1 is that the compression o f the air upstream the reforming process leads to a lower efficiency penalty than the fuel compression of the other two cases does, due to a lower mass flow. The difference is, however, not large enough to compensate for the drawbacks of case 1, mentioned above.
Figure 2 Graphical presentation of computational results. The left figure shows the most important factors affecting the plant efficiency. The right figure shows a corrected efficiency (see text) and the resulting net system efficiency for the different cases. E...xplanati•n of the different columns in Figure 2: 1 = Corrected efficiency (%). This is the efficiency of a standard CC fired with decarbonised fuel supplied at 250~ and including the air extraction necessary for the reforming process (12-13% of the compressor inlet flow). A standard CC operated this way, gives a net efficiency very close to that of a natural gas fired CC (57.2% in this work). The corrected efficiency is, however, related to the flow of natural gas necessary for producing this hydrogen-rich fuel (1). Thus the gap between 57.2% and the corrected efficiency represents mainly the loss of heating value in the reforming process. This loss includes use of fuel for additional firing of the HRSG unit for the purpose of preheating the reforming feed streams. 2 = Efficiency change (%-points) due to extraction of MP-steam from the steam turbine. 3 = Efficiency change (%-points) due to HP-steam generation within the fuel reforming process. 4 = Efficiency change (%-points) due to the work required to compress either air or fuel in the reforming process. 5 -- Efficiency change (%-points) due to the work required to compress CO2 from atmospheric pressure (it is assumed that the pressure out of the CO2-absorber is atmospheric) to the pressure required for storage (100 bar). 6 = Net efficiency (%). This is the total system efficiency, calculated as: 6 = 1+2+3+4+5.
The results for the cases 2 and 3 are not very different, though a slightly better efficiency for case 3 (0.3%points difference). This indicates that the splitting of the steam supplied to the reforming process more or less is insignificant by means of energy efficiency. There are, however, other beneficial aspects as the volumetric flow through the prereformer and the ATR is considerably lower than in case 2. This implies both lower investment and operating costs for this section. The reduced flow o f inert steam through these reactors also reduces the duty required to heat inert components, and the effect of this can be seen from the slightly higher efficiency number of case 3. The steam to carbon ratio in the ATR reformer feed is well
336
below the recommended value (1.5-2) which means that coking might be a consequence. However, this conceptual change was examined in order to check the potential of energy efficiency improvement. The degree of CO2-reduction for the three cases are very much dependent on the assumption of a 90% removal of CO2 in the absorber unit, and thus not much focused here. However, the numbers calculated in Table 2 gives an indication on what degree of total CO2 emissions that can be expected from such a plant compared to a conventional combined cycle plant, when the difference in net efficiency is compensated for. It is known, however, that it might be possible to remove up to 99% of the CO2 fed to an absorber unit, which for the present cases would mean approximately 90% reduction of CO2 emissions compared to a standard combined cycle power plant.
6. C O N C L U S I O N S A concept for removing CO2 from a natural gas fired combined cycle power plant is presented. This concept implies removal of the carbon in a process combining auto-thermal reforming and combined cycle power production. It was found that these two main parts of the total plant should be tightly integrated (air, steam) in order to achieve an acceptable fuel-to-electricity conversion efficiency. However, compared to a natural gas fired combined cycle power plant, the removal of CO2 implies a reduction of efficiency of about 10-11%-points, or increased natural gas consumption of about 21-25%. It was found advantageous, from a thermodynamic point of view, to keep the pressure low in the reforming process (below that of the gas turbine), and to use a compressor to increase the fuel pressure prior to the gas turbine. This conclusion may not be the same with respect to the economy as lower pressure indicates larger equipment costs. Compared to a standard combined cycle, this technology of removing C02 from a power plant implies a rather complex plant. It would be possible to build such a plant with a dual fuel firing, which means that the gas turbine can be operated using either natural gas or reformed fuel.
REFERENCES [1] Erga O., Juliussen O., and Lidal H., 1995, "Carbon dioxide recovery by means of aqueous amines", Energy Convers. Mgmt., Vol. 36, No. 6-9, pp. 387-392 [2] Feron P.H.M. and Jansen A.E., 1997, "The production of carbon dioxide from flue gas by membrane gas absorption", Energy Convers. Mgmt., 38, Suppl., pp. $93-$98 [3] Hendriks C.A. and Blok K., 1992, "Carbon Dioxide Recovery using a Dual Gas Turbine IGCC Plant", Energy Convers. Mgmt Vol. 33, No. 5-8, pp. 387-396 [4] Mathieu Ph., 1999, "Presentation of an Innovative Zero-Emission Cycle for Mitigating the Global Climate Change", Proceedings of the 4th International Conference on Greenhouse Gas Control Technologies, Interlaken, Switzerland, pp. 615-620 [5] Steinberg M., 1995, "The Hy-C Process (Thermal decomposition of Natural Gas). Potentially the Lowest Cost Source of Hydrogen with the Least CCh Emission", Energy Convers. Mgmt., Vol. 36, No. 6-9, pp. 791-796 [6] Audus, A., Kaarstad, O. and Skinner, G., 1999, "CO2 Capture by Pre-Combustion Decarbonisation of Natural Gas", Proceedings of the 4~h International Conference on Greenhouse Gas Control Technologies, Interlaken, Switzerland, pp. 557-562 [7] Bolland O. and Mathieu P., 1998, Comparison of Two CO2 Removal Options in Combined Cycle Power Plants, Energy Conversion and Management, 39 (16-18), pp.1653-1663 [8] Akai M., Kagajo T. and lnoue M., 1995, "Performance evaluation of fossil power plant with CO2 recovery and sequestering system", Energy Convers. Mgmt., Vol. 36, No. 6-9, pp. 801-804 [9] Bolland O., Undrum H., 1999, "Removal of CO2 From Natural Gas Fired Combined Cycle Plants", proceedings of the Power-Gen '99 Europe, Frankfurt, Germany, 1-3 June [10] IEA Report PH2/19, 1998, 'Precombustion Decarbonisation", Study by Foster-Wheeler for lEA GHG and Statoil [11] De Groote, A. and Froment, G.F. 1995, "Reactor Modeling and Simulations in Synthesis Gas Production", Reviews in Chemical Engineering, 11 (2), pp. 145-183
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
337
Simulation of convective drying of multicomponent moisture in a computer code M u l t i d r y P A K Z. Pakowski Faculty of Process and Environmental Engineering, Lodz Technical University, 90-924 Lodz, ul. Wolczanska 213, Poland. Drying of solids containing multicomponent moisture is common and important in hightech industries like production of pharmaceuticals, magnetic storage media etc., yet it is neglected in present day process simulators. The reason is that the process is poorly understood due to it's complexity and the fact that it requires specialised tools for solving problems encountered in this area. On the other hand drying in neglected in specialised texts on multicomponent mass transfer (e.g. Taylor, Krishna, 1994). The diffusional resistance characteristic to evaporation to inert gas influences selectivity of the process otherwise governed by relative volatility alone. This makes the predicting of the process more difficult as a new type of azeotropes i.e. dynamic azeotropes emerge, defined not only by the relative volatilities but also by the multicomponent diffusional mass transfer. Main goal of calculations is either to simulate the process of drying for specified initial conditions or to predict stationary points and their zones of influence (selectivity maps). These goals can be reached with computer code MultidryPAK (MdP) that was written for this purpose. Construction parts of the program are described below. 1. DATABASES
1.1 Gas and liquid libraries The structure of databases used for single components is identical to that of a computer code dryPAK (dP) for single component drying simulation presented earlier (Pakowski, 1999). This block contains separate files with properties of gases and liquids. Liquid components additionally contain decomposed chemical structure data for UNIFAC application. Separate files with binary data for calculation of activity coefficients of binary pairs by Wilson and NRTL methods are also enclosed.
1.2 Solid library Solid data are limited to thermal data. At present no multicomponent sorption isotherm equations are built in and the area of sorption limited drying is excluded from calculations.
1.3 Property functions This block includes procedures to calculate physicochemical properties of single components and their mixtures, including VLE and gas and liquid phase diffusivities. At present only systems with no immiscibility gap can be considered. The diffusivities in infinite dilution are computed by the Fuller's method for gases and several methods for liquids (Siddiqi and Lucas, Nakanishi, Vignes etc).
338 2. EVAPORATION (DRYING) RATE CALCULATION Calculation of the evaporation rate of each evaporating component is based on the solution of the rigorous Generalized Maxwell-Stefan (GMS) equation in the following matrix form: N = I~ECk(y* - y )
(1)
where [3 is the bootstrap matrix containing the determinancy condition, E is the correction matrix for finite molar fluxes, k is the matrix of mass transfer coefficients at zero molar fluxes and y* and y are vectors of equilibrium and bulk molar fractions of each diffusing component. For the evaporation of a n-1 component liquid to inert gas (nth component) eq. (1) was converted to the following working equation:
(2)
y * . y : [exp((i))_ i]-I (y * _tp) where (9 is defined as Ni(k-1i
(I)ik = C--
n
1 1 for i=l,2..n-1
(3a)
kik
q) ii = Ni + s .:NJ for i~j=l,2..n-1
Ck~n
(3b)
j-1 Ckij j~i
Matrix I is the unity matrix and matrix T is defined by eq. (4). The exponent function of matrix d) is calculated by series expansion.
V= Ni
(4)
N 3. FINDING STATIONARY POINTS Stationary points in evaporation are also termed dynamic azeotropes. These are points in a ndimensional phase diagram where evaporation takes place without changing neither composition nor temperature of the evaporating liquid. Their composition and temperature is greatly affected by concentration of the evaporating components in the gas phase. When a dynamic azeotrope exists the volatile component may not evaporate preferentially in a certain initial liquid composition range, which may mean that the process gets out of control. For stationary point analysis the following system of equations provides a model of the process: a I q - ~n-1 NiM i (hgi - h,i )1 0x i _
1 NA(x i _ Ni/N)
-----o~ vIC 1
0T___s=
t~
i=l n-1 ZmliCl i i=l
(5,6)
339 3.1. Binary l i q u i d s The phase diagram for evaporation of a binary mixture is illustrated in Fig. 1 for the case of isopropanol-water system. I
l
t /
t
::: / / o., ,/ /
;I ;~
o.e
I
',,
::: fi
I I
JI
15.0 I
,/
I 10.0
i
5.0
0.0
/
/ 0.0 0.1
0.2 0.3
0.4
I
/
/
, \
)
O.a
p.-
o.1
J
/
0.5 0.6 0.7
0.8
0.9
1.0
0.1
I i I
/'
\
! ! ! |
,. ! |
o.o
0.0
,,///
\\
'/
/
~
O3
InN
//I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x I
X1
Fig. 2 Selectivity curves for 2-component liquid
Fig. 1 Phase diagram for 2-component liquid
The lines marked TN are the lines at which the RHS of eq. 5 is zero and the line marked TW is where the RHS of eq. 6 is zero. The values of Ni in the above equations must be found by solving eq. 2. The dynamics of heating that can be obtained from eq. (6) may be influenced by the solid heat capacity that can be added in the denominator of eq. (6). It is, however, normally excluded from calculations. Fig. 1 shows such phase diagram generated by MdP. Please note that not all points where two conditions are fulfilled are stationary points as proved in Fig. 2. 1.(Po 1
0
x1=0.99
.
9
~
~
o o ~ ~ ~ . ,"~o o.~o 9 II~
/
o7-H - - ~
/~/
..~o.~o
IJ ~ / "--L o.o~ - . ~ / - . . < / ~ ~ ~/ o\ . .0.50 o o ~11~ / ~ ~ ~ ~~176
oi~ o:
~
~
~/~.~~
~
o
:
1
0.10 0
o o0.0
O.
14.
~0.2
18"0
.
.
.
o
__ _ "~'0
2"0
.
A o.~o
.
Fig. 3 Stationary point map for ethanolwater system in air at 30~
0.0
/ 0.01
~.i d.8'd.9 ~'om2 . . .
Fig. 2 is the selectivity diagram corresponding to Fig. 1. Two stationary points are visible at Xl-0.03 and x1=0.79. Dotted line corresponds to the third intersection point of TN and TW lines and is a separatrix of"zones of influence" of the two azeotropes. The third type of graph produced for binary liquids is the map of stationary points shown in Fig. 3 for ethanol-water system. The method of mapping is described in Pakowski, 1994.The region in which isosters do not intersect is free from dynamic azeotropes.
340
3.2. Ternary liquids Phase diagram for a ternary liquid is 3D and not easy to visualise, however MdP still offers selectivity curves as shown in Fig. 4. The case shown in Fig 4 indicates one dynamic azeotrope. nonane
methanol
/ /j
..... acetic acetone
chloroform pyridine
Fig. 5 Selectivity curves for 4-component liquid
Fig. 4 Selectivity curves for 3-component liquid
3.3. Quaternary liquids For quaternary liquids only selectivity curves can be easily visualised as shown in Fig. 5. The case shown indicates presence of three stationary points. methanol c~,1.00
q~l= 0,100 q~2=0,100 = ,
/
0.80
0.60
\\
0.40
Methanol
\
0.20 Acetone
chloroform
Fig. 6. Entropy production 3D graph for 3-component liquid evaporating to air.
acetone
chloroform
Fig. 7.2D map of entropy production for 3-component liquid evaporating to air with superimposed selectivity lines.
3.4. Rational method of mapping stationary points Pakowski, 1994 proved that stationary points in multicomponent systems evaporating to inert gas are identical with local minima of entropy production generated in gas boundary layer. This entropy production can be calculated form the following equation
341
q gradT-R'~lNi gradxi (3 = -- ~
i=l
(7)
Xi
It is then necessary to map the entropy production at given external conditions for the whole range of concentrations in the liquid phase. This can be easily visualised for binary liquids. For ternary liquids either 3D wire cage maps are available (Fig. 6) or 2D maps showing isoentrop contures (Fig. 7). For more than 3-component systems a general optimisation algorithm is available which seeks for a minimum of the cy function (eq. 7) in the vicinity of a guess value of the liquid composition vector. 4. DRYING PROCESS SIMULATION
4.1. Continuous processes with resistance in gas phase Liquid phase here is assumed to be perfectly mixed. The role of solid can only be accounted for by declaring the initial moisture content. The solid will then be considered in heat balances of the liquid phase.Simulation of gas-liquid (solid) contact can be performed in co-current, counter-current and cross-flow. In cross-flow the solid phase is assumed to be perfectly mixed in the direction of gas flow. This assumption can be used in fluid-bed dryer design. Evaporation history including compositions of the gas phase and liquid phase and their temperatures are computed.
4.2. Continuous processes with resistance in liquid phase This case does not exist in the single component drying. Here liquid which may form the second phase alone (evaporation of droplets) or in mixture with solid (suspension) can provide an additional resistance to diffusion. The process can be described for three one-dimensional geometries (plate, cylinder and sphere) with the following model
r n Or
rnDiCl
~-
-N l
~-
=C 1
&
(8)
where n--0 for plate, n=l for cylinder and n--2 for sphere. For evaporating liquids additional equations for velocity of the moving interface are necessary. The matrix Dl of multicomponent Fickian diffusivities is related to the GSM binary diffusivities by D:B-1F where gik = x
(9)
(11 / Xn
1
Xk
s
B ii = D m + j=l "a~lj jr
for i= 1,2..n- 1
(10a)
for iCj=l,2..n-1
(10b)
342 and F is the thermodynamic factor matrix defined as
OlnYi ]
Fik-Sik +Xi
~:k
for i,k = 1,2..n- 1
(11 a)
P,T,xj~=k,n
(c31ny n ] r'nk ----1 -/- X n
0Xk
P'T'Xj~k, n for k=l,2..n-1
(1 lb)
where 8 is the Kronecker symbol. This procedure requires further work. At the moment it can calculate the process of evaporation of pure liquids and suspensions for three 1D geometries and for up to 3component systems. 5. CLOSURE The resulting software MultidryPAK, which is still in development, can be used for designing of drying and evaporation encountered in many processes e.g. in spray drying of pharmaceuticals. In future it is planned to be enlarged with a solid sorption part. REFERENCES 1. Taylor R., Krishna R., Multicomponent Mass Transfer, J.Wiley, NY, 1994 2. Pakowski Z., Stationary States in Evaporation of Multicomponent Liquid Droplets to Inert Gas Streams (in Polish), Z.N.PL, 716 (1994) 3. Pakowski Z., Comp. and Chem. Eng., 23 (1) 1999, $719-$722 NOMENCLATURE A - interfacial area, m 2 A - in graphs denotes dynamic azeotrope C - total concentration, kmol/m 3 5~j - GMS binary diffusivity, m2/s c - heat capacity, kJ/kgK h - enthalpy per component, kJ/kgK k - mass transfer coefficient, kg/m2s M - molar mass, kg/kmol m - mass, kg n - total number of components in gas, N~ - molar flux of i-th component, kmol/mZs N - total molar flux, kmol/m2s q - convective heat flux, W/m 2 p - total pressure, Pa R - universal gas constant, kJ/kmolK
T- temperature, C x - molar fraction in liquid phase, y - molar fraction in gas phase, t - time, s 7 - activity coefficient, -
Subscripts and superscripts
*-
at equilibrium i,j,k,n, 1,2 - component number g - gas 1 - liquid phase v - vapor
Matrices and vectors 13, E,k,~, y, x,W,D, B, F - explained in the text
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
AN ALGORITHM FOR LIQUID EXTRACTION ORGANIC ACIDS
ANALYSIS PROCESS
343
OF ELETROLYTIC LIQUIDFOR CONCENTRATION OF
Pinto,R.T.P.a. Lintomen, L. a ; Meirelles,A.J.A. b e Wolf-Maciel,M.R. a a Chemical Engineering Faculty / Separation Process Development Laboratory b Food Engineering Faculty / Physical Separation Laboratory State University of Campinas - CP 6066 - CEP 13081-970 - Campinas - SP - Brazil Telephone: +55-021-19-7883900 / +55-021-19-7883971 - Fax: +55-021-19-7883965 e-mail - [email protected] [email protected] 9
An algorithm for analysis, simulation and optimisation of liquid-liquid extraction process for concentration of organic acids using a mixed electrolyte solvent is presented. This algorithm consists on three parts: one concerned with the analysis and definition of the problem; another with thermodynamic calculations; and the last one with the simulation and optimisation. These parts are integrated by computational simulation engineering. This algorithm uses thermodynamic insights to generate process alternatives and to obtain initial estimates for the simulation engineering, making possible the understanding of the process design. Through the analysis of the results from this algorithm, the designers will be able to get new paths to obtain the desired products. Experimental results from this work show a new process for concentration of citric acid. 1. INTRODUCTION Nowadays there is a renewed interest in fermentation processes based on renewable raw materials focused especially on the new possibilities of concentration of organic acids (Grinberg et al. 1991). Low reaction conversion and inhibition of product formation due to the accumulation of end products in the fermentation broth are characteristics of many processes. The final products have to be recovered from very diluted solutions, usually under 10 wt. % (Malinowski and Daugulis 1994). Research works have been published relating the use of liquid-liquid extraction processes for the concentration of organic acids from the fermentation broth (Wennersten 1983, Moore et al. 1989, Bizek et al. 1992). Furthermore, researchers have shown that the salting-out effect has wide application for solvent extraction processes for obtaining a better recovery of organic compounds from aqueous solutions (Marcilla et al. 1995, AI-Sahhaf and Kapetanovic 1997), by the fact that they can improve the two phase region of such systems. The use of an electrolytic liquid-liquid extraction process for recovering organic acids depends on a good selectivity and distribution coefficient of the solvent. The aim of this work is, then, to present an algorithm for analysis, simulation and optimisation of liquid-liquid extraction process for concentration of organic acids using a mixed electrolyte solvent.
344 2. BACKGROUND INFORMATION The knowledge of the thermodynamic properties is of fundamental importance to determine the appropriate solvent for the process. For this, it is necessary to determine the phase diagram for the systems, being possible to know how the solute distributes between the phases. It is important to mention that these equilibrium data are highly dependent on the temperature, being this then, an important process variable. The objective of using an electrolyte in the solution is due to the fact that it can improve the two phase region through the salting-out effect. Salt addition leads to a decrease of the solvent solubility in the aqueous phase. When an electrolyte is used, another important parameter is the electrolyte solubility in the system. Therefore, the experimental determination of the electrolyte phase diagrams, that are based on liquid-liquid and solid-liquid-liquid equilibrium behaviour of the systems, provides important insights to these questions. Zurita et al. (1998) related that mutual solubilities of partially miscible binary systems are significantly affected by the presence of a mineral salt. This phenomenon is also observed in temary systems through changes on the sizes of the partially miscible regions with respect to the condition without salt. Depending on whether this region is increased or decreased, it is called the salting-out or salting-in effect. An analysis of the addition of an electrolyte component to improve the solvent extraction can be shown in Figures 1 and 2. Since the interest is only in obtaining information on the salting-out effect, all compositions are given on a salt-free basis for the experimental tie-lines. Therefore, consider Figure 1 showing experimental phase diagrams to the quaternary system composed by Water - Propionic Acid 1-Butanol- Calcium Chloride at 302.15 K (Zurita et aL 1998).
Experimental Experimental 9 Experimental = Experimental v Experimental
.
=
9
Tie-Lines: Tie-Lines: Tte-Unes: Tie-Lines: Tie-Lines:
0 mass % CaCl= 4.3 mass % CaCl 2 9.3 mass % CaCl= 15.1 mass % CaCi 2 20.3 mass % CaCI=
o2s
*
. = v
o rs
~
"'2
1,00
=
Z:.':
0,00
.
,
0,25
.
,
#
.
0,50
Water Mass Fraction
Experimental Experimental Experimental Experimental
,
0,75
Z .
,
0,00
1,00
1,00
Tie-Lines: Tie-Unes: Tie-Lines: Tie-Unes:
/
o25r
\ oN , \
,,-
o, o/
7::" .
0,00
~
0,25
0 mass % NaCl 4.7 mass % NaCl 10 mass % NaCl 15 mass % NaCl
.
,
-~
\
.
0,50
=
0,75
.
, 0,0
1,00
Water Mass Fraction
Fig. 1. Phase Diagram to Water-Propionic Fig. 2. Phase Diagram to Water-Propionic Acid - 1-Butanol - CaCI2 at 302.2 K (Salt- Acid- 1-Butanol- NaCI at 302.2 K (SaltFree Base)- Zurita et al. 1998. Free Base)- S61imo et al. 1997.
345 Zurita et al. (1998) related that for the systems represented by Figure 1, the salt concentrations were 4.3%, 9.3%, 15.1%, and 20.3% by mass, which are within the saturation limit at 303.15 K. For this system, it is evident that the salting-out effect occurs for all the salt concentrations investigated. The w a t e r - propionic a c i d - 1-butanol- NaCI system was studied by S61imo et al. (1997). The phase diagram obtained can be represented in Figure 2. The authors related that for this system, the presence of salt decreases the solubility of the system, increasing the heterogeneous 1-butanol rich region. For both systems it was verified that the partition coefficient and selectivity increase, increasing the salt concentration. Therefore, 1-butanol in the presence of dissolved NaCI or CaCI2 appears to be a good extracting solvent for propionic acid from aqueous solutions. The authors, still, reported that all partition coefficients are far greater than unity because the propionic acid always prefers the organic phase rather than the aqueous one. For the studied systems, represented by Figures 1 and 2, the most soluble salt (Calcium Chloride) would be a better choice since this salt does not precipitate. 3. EXPERIMENTAL AND SIMULATION RESULTS The use of the proposed algorithm combines the generation of a consistent phase diagram to represent the liquid-liquid equilibrium data through the use of an appropriate thermodynamic model and an efficient computational algorithm. With this, the problems related to the analysis and design of the liquid-liquid extraction process are solved in an integrated form, providing feasible designs, operating conditions and data for analysis (mass balance). Choosing a mixture point inside the two phase region of the diagram, it is possible to obtain the mapping of the mass-balance lines. With this, an initial estimate of the operational conditions of the extraction process can be visualised, and it can be used as initial conditions for the rigorous simulation of the process. It is very important to emphasise that a correct choice of the thermodynamic model for the calculation and/or prediction of the phase equilibrium and the parameters for the selected model must be made. The phase diagrams are very sensitive to the parameters when an electrolyte system is used. An important task is, also, to consider the necessity of carrying out a preliminary treatment in the fermentation broth if solid particles are present in the broth. The main steps of the algorithm can be find in Pinto (2000). Nowadays, several studies have been reported about the use of the liquid-liquid extraction process to concentrate the citric acid from its fermentation broth and a number of liquid-liquid systems has been investigated. Grinberg et al. (1991) studied the water - citric a c i d - 2-butanol system at 298.15 K. 2-Butanol is partially soluble in water and in the presence of citric acid, so that the two-phase region is relatively narrow, reducing the practical importance of the system when the separation process is considered. With the objective of to validate the proposed algorithm, a study using an electrolyte component in the w a t e r - citric a c i d - 2-butanol system was carried out in this work. The simulation of the process was made using the ASPEN PLUS process simulator. The data bank of the simulator did not present the binary parameters for the thermodynamic model (electrolyte NRTL model) that is necessary to represent the studied system. In the open literature, it was not found any work reporting the experimental liquid-liquid equilibrium data for systems containing citric acid, being necessary to measure these data. Therefore, the phase diagram for the water - citric acid - 2-butanol - sodium chloride system was determined experimentally. The results are represented in Figures 3 to 5. It can be verified that the system presents a better two phase region when salt is used (Figure 3).
346
The use of an electrolyte component in the system does not change significantly the partition coefficients (Figure 4), but considerably increases the selectivity (Figure 5). It is necessary to emphasise that when salt concentrations from 10 mass % were used, to some initial mixture concentration, salt precipitation occurred (liquid- l i q u i d - solid equilibrium). Therefore, the operating conditions are limited by the salt solubility in the quaternary system. This can be easily determined by mapping the phase diagram. The electrolyte NRTL thermodynamic model (Renon and Prausnitz (1968), and Mock et al. (1986)), was used. The experimental tie-lines were used to obtain the model parameters.
0,0~0
Experimental Tie-Unes: 0 % mass NaCI Experimental Tie-lines: 5 % mass NaCI Experimental Tie-lines: 10 % mass NaCI
. v
0,0~5-
.
/
2-~ad
92-]~lad+Y/ol'~ 9 2-BUla~ +1
O,GD. "~ 0,015-
0,75./
0,2
\
0,00
0,25
0,50
0,75
0,O05
1,00
Water Mass Fraction
Fig. 3. Phase Diagram for the Water- Citric Acid - 2-Butanol - NaCI at 298.15 K (SaltFree Base)
2-Butanol 2-Butanol + 5% NaCi 2-Butanol + 10% NaC!
9 4,0
9 9
3,5"~
3,0-
r 2.0 -~
1.51,0-
0.000
9
I
0,005
"
I
0,010
"
I
0,015
"
I
0,020
'
I
0,025
"
Xcitric acid- mixturepoint
Fig. 5. Selectivity Profiles
I
0,030
"
I
0,035
.... 9
0.04(
O,Om 0,000
0,005 Q010 Q015
Q~
QO~5
Fig. 4. Distribution Coefficient Profiles
003
347
~.
100
100
es
"~
95
85
75
0
o
65
!
l
1,4
I 1,6
i
I 1,8
,
i 2,0
.
!
ii!ii!i! i i:
2,2
.
1
.
2,4
t
2,6
.
i
2,8
i" 80
.
i
3,0
.
~
- - - . - - Stage Number: 5 Stage Number: 10 Stage Number: 15
7o ,
3,:
0,00
Solvent I Feed Ratio
I
0.02
t
I
0,04
n
I
i
0,06
i
I
o,oe
I
0,1o
Salt Concentration in the Solvent Flow Rate
Fig. 6. Influenceof the Solvent to Feed ration in the Citric Acid Recovery
Fig. 7. Influenceof the Salt Concentration in the Citric Acid Recovery
Through the simulation and the optimisation of the key parameters of the process, it was possible to improve the distribution coefficients. The controlled parameters were the temperature of the process and the salt solubility. The optimised parameters were the solvent flow rate, the salt feed position, and the number of stages of the column. Table 1 shows the mass balance of the process. The simulation was carried out at 25~ and 1 atm. It was used 15 theoretical stages.
Table 1 - Mass Balance of the Liquid-Liquid Extraction Process Simulation
Components Water Citric Acid 2-Butanol Sodium Chloride
Feed Flow Rate (k~,/h) 9.6000 0.4000 0.0000 0.0000
Solvent Flow Rate (k~/h) 1.0000 0.0000 20.0000 0.5000
Extract Flow Rate (k~/h) 2.9130 0.3260 19.9620 0.2640
Rafinate Flow Rate(k~/h) 7.6870 0.0740 0.0380 0.2360
The effects of the stage number and of the solvent to feed ratio on the process are represented in Figure 6. It can be verified that increasing the solvent flow rate increases the citric acid recovery, but high solvent rates are necessary to obtain high recovery. Another important variable of the process is the salt concentration in the solvent flow rate. Its effect can be verified in Figure 7. It was found that increasing the salt concentration increases the citric acid recovery, being its maximum value limited by the salt solubility. 4. CONCLUDING REMARKS It can be verified that through the use of this proposed scheme, a powerful tool for studying the viability of the extraction process of organic acids are available. The algorithm is interfaced with ASPEN PLUS process commercial software. It was chosen the ASPEN PLUS software because it can simulate and optimise rigorous extraction models and it can represent
348 all thermodynamic models developed for electrolytic systems. This software is, also, able to regress experimental data. ACKNOWLEDGEMENTS The authors are grateful to CNPQ, FAPESP (Scholarship) and FAPESP (Process Number: 1997/10630-7) and for the f'mancial support which has made this work possible. REFERENCES 1. Aspen Plus Commercial Processes Simulator - Version 10.1-0, Aspen Technology- USA. 2. A1-Sahhaf, T.A. and Kapetanovic, E., J. Chem. Eng. Data, 42 (1997) 74-77. 3. Bizek, V., Horficek, J., Rericha, R. and Kousov~i, M., Ind. Eng. Chem. Res., 31 (1992) 1554-1562. 4. Grinberg, A., Povimonski, D. and Apelblat, A., Solvent extraction and ion Exchange, 9(1) (1991) 127-135. 5. Malinowski, J.J. and Daugulis, A.J., AIChE J., 40(9) (1994) 1459-1465. 6. Marcilla, A., Ruiz, F. and Olaya, M.M., Fluid Phase Equilibria, 105 (1995) 71-91. 7. Moore, R.J., Pratt, H.R.C. and Stevens, G.W., Solvent Extraction and Ion Exchange, 7(6) (1989) 1043-1062. 8. Mock, B., Evans, L.B. and Chen, C.C., AIChE Journal, 32(10) (1986) 1655. 9. Renon, H. and Prausnitz, J.M., AIChE Journal, 14(1) (1968) 135. 10. S61imo, H.N., Bonatti, C.M., Zurita, J.L. and Gramajo, M.B., Fluid Phase Equilibria, 137, (1997) 163-172. 11. Wennersten, R., J. Chem. Tech. Biotechnol., 33B (1983) 85-94. 12. Zurita, J.L., Gramajo de Doz, M.B., Bonatti, C.M. and S61imo, H.N.,. Chem. Eng. Data, 43, (1998) 1039-1042.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScience B.V. All rights reserved.
349
Estimation of the heat released by chemical reactions: Application to control of a simulated batch reactor F. Xaumier
M.-V. Le Lann b, M. Cabassud ~ and G. Casamatta ~
qnstitut National Polytechnique de Toulouse Ecole Nationale Sup6rieure d'Ing6nieurs de G6nie Chimique gaboratoire de Gdnie Chimique CNRS UMR 5503 18, Chemin de la Loge - 31078 Toulouse Cedex - France Tel.: +33 (0)5.62.25.23.00, Fax: +33 (0)5.62.25.23.18 E-mail: Florence.Xaumier @ ensigct.fr I'LAAS - C N R S UPR 8001 7, avenue du Colonel R o c h e - 31077 Toulouse Cedex 4 - France E-mail: mvlelann @laas.fr In this paper an estimation procedure for the heat released by chemical reactions in batch reactors is described. The estimation of the heat reaction rate is based on an energy balance on the reaction mixture and the reactor jacket. In order to use this estimation for control of the batch reactor, a model of the heat of reaction as a function of measurable variables is established. Different models were considered. Identification of the respective model parameters is based on a moving horizon of past estimated values of the heat reaction rate. The resulting optimisation problem is solved using a Gauss-Newton type method. This approach has been validated by simulating the thermal control (NLMPC) of a 16 litres batch reactor equipped with a multi-fluid heating/cooling system. The simulations give some promising results. 1. I N T R O D U C T I O N In fine chemical the majority of synthesises are carried out in batch reactors. The dynamic behaviour of these devices can be characterised by an intrinsic nonlinearity which renders their control difficult. To this complexity one has to add the absence of steady-state due to the discontinuous nature of the process. These complexities make the use of control techniques based on linear models less good performing. The knowledge of certain physical properties is usef\ll to make easier the control of these reactors. One of these properties, on which depends also the safety of the operation, is the heat released by a chemical reaction. Generally, the heat reaction rate is considered as a disturbance. In the present study, an estimator of the heat reaction rate in a batch reactor is implemented in a predictive controller (NLMPC) [1]. The estimation procedure is introduced to improve the control of the process and at the same time ensure greater safety. The estimation of the heat released is based on an energy balance on the reactor mixture. In order to make predictions, the heat released by chemical reactions is approximated by different models. The estimation problem is solved using an optimisation method over a moving horizon. This estimation approach has been validated by simulation of thermal control of a batch reactor equipped with a multi-fluid heating/cooling system.
350 2. DESCRIPTION OF THE METHODOLOGY FOR PREDICTIVE CONTROL OF THE BATCH REACTOR
NON-LINEAR
MODEL
2.1 Batch reactor The dynamic simulation model is composed of a set of differential equations derived from mass and energy balances on the reaction mixture and the jacket [1]. In our case, the batch reactor is equipped with the multi-fluid system implemented often in industry [2]. Either a cooling or a heating fluid, each of them available at a constant temperature, is delivered to the .jacket surrounding the reactor. On the pilot plant, four utility fluids are available: cold water, hot water, steam and a mixture of monopropylene glycol and water (50/50 weight). Moreover, to describe closely the dynamic behaviour of an industrial reactor, special attention is drawn to model the flow and the temperature profile inside the jacket. The jacket is lumped into a number of perfectly mixed tanks with individual filling-up coefficient [3]. The heat released by the chemical reaction j has been modelled by a classical Arrhenius'model:
(1)
O,- - Y-,
.1
~j,d/-/,;i
(2)
2.2 Control algorithm Temperature control of a batch reactor with a multi-fluid system requires the choice of the fluid circulating in the reactor jacket and its flowrate. This choice has to be taken on-line at every sample period in order to track the a priori defined temperature trajectory. Therefore, a supervisory algorithm is necessary to choose the utility fluid and a control algorithm is used to calculate tile manipulated variable. Non-Linear Model Predictive Control is based on an openloop constrained optimisation problem solved repeatedly on-line and calling upon a non-linear model of tile process. The model used for predictions in the NLMPC is based on the same difl:erential equations as the simulation one, except that it does not used the heat reaction rate given by (2) but the estimation given below. The dynamic model is solved using a numerical integration method supplying to the optimisation algorithm the predictions of the model output over a time horizon P. The non-linear optimisation problem ([4], [5]) is defined as the minimisation of an objective function represented by the difference between the predicted process output and the future desired set point over a time horizon P. The problem is subject to constraints on the decision variables. It is solved using a reduced projected gradient optimisation algorithm which provides the future values of the manipulated variable which allows to control the process according to the desired objective. Only the first control move is applied to the process during the reset sampling period. This procedure is repeated at each sampling period.
351
3. DESCRIPTION OF THE ESTIMATION PROBLEM Non-linear model predictive control needs predictions. These predictions will be more accurate if the heat reaction rate is taken into account by use of a model. In order to built up this model, information of the heat reaction rate must be available. These information are calculated by an observer based on an energy balance: dr,. Q(,I, = -ntrCPr ~lt + .fcCl, c( Tc - Tr ) - Ur,prAr, pr( Tr - Tpr ) - Ur,ex t Ar,ext( Tr - Tex t )
(3)
This paper presents the case where kinetics are unknown to the controller. The heat reaction rate is estimated either as a linear function of the reactive feed flowrate or by a second order polynomial function of time. This approach has been chosen because it uses variables easy to access on a real process. The use of these simple models dealt out to give a good quality of predictions as it will be shown later. Two models have been studied: a linear function of the reactive feed flowrate f~ (4) and a second order polynomial function of the process time (5): Q m o , l - al * f c + a2
(4)
Qlno,2 - bl * t2 + b2 * t + b 3
(5)
The model (4) will be used only when the reaction is instantaneous because, in this case, as there is no accumulation of reactants in the reactor, the heat reaction rate depends directly on the reactive teed flowrate. The model (5) depends on the process time only. It is intended to be used for all chemical reactions. In order to determine the parameters of these relations, a quadratic criterion is built up, comparing the observed and the estimated values of the reaction heat. The minimisation of the quadratic criterion by a Gauss-Newton method gives the desired parameters. The method of Gauss-Newton minimises a quadratic criterion, comparing the observed (or the measured) and the values calculated by the model. The variables of the optimisation problem are the parameters of the model that we want to estimate.
177 F kZ1LQob(k)-Q _
]2 --lHO,17
(k)
(6)
with Qob(k) are the observed data. In our case, this value represents the heat reaction rate calculated by the energy balance on the reaction mixture (3). Q,,~,,.,l(k) the values calculated by the model depending on a set of parameters (a if n= 1 or _b_bif n=2) The minimisation is done on a predefined number of m points belonging to a past moving identification horizon. The figure 1 presents the description of the moving identification horizon.
352
estimation horizon
_~ ~-[
Qmo, 1: model . .-
mr(t + P)Cp r
dTr(t + P) d~=Qmo,n(
t + P)
+ f cCPc( Tc( t ) - Tr( t + P )) - U r , pr(t + P)Ar,pr(t + P)(Tr(t + P ) - Tpr(t + P)) - Ur,ext Ar,ext( rr(t + P) - ~ x t ( t ))
(7) tl I11e past
t k
where Qmo,n(t+P) is given either by (4) or (5).
future
Fig. 1. Description of the moving horizon This procedure has been integrated in the non-linear predictive control algorithm. The energy balance used in the controller model to make predictions over the future horizon is the eq. (7). 4. SIMULATION RESULTS The performance of the estimation procedure has been studied for two different chemical reaction scheme. - R l: parallel reactions with different orders - R2: consecutive-competitive reactions A+B-->C (Rla) A+3B--+C (R2a) 2B-+D (Rib) A+2C-->D (R2b) Then, one temperature profile has been applied in order to examine the tracking of the mixture temperature. The initial reactor contents were heated during 1200 seconds. They were then realesed an exothermic reaction phase at a constant temperature during 1200 seconds before being cooled down during a time interval of 1200 seconds. The prediction horizon for all simulations is equal to 9 sample times. 4.1 Results for the chemical reaction R1 (Parallel reactions with different orders) For the following simulations the model used to estimate the heat released by chemical reaction, is a linear function of reactive feed flowrate (4). The equation of the reactive feed flowrate is given by: fc= 1.2E-8(t-ticoul)+ 1.E- 10. Figures 2a and 2b give the results obtained in the case where a reactive is fed from 1200 to 3600 seconds. On figure 2a, the observed (Qob) and estimated (Qmo) values of the heat reaction rate have been plotted. No distinction can be made between the two plots. So, it can be concluded that the estimation procedure performs correctly. On figure 2b, the results of temperature and manipulated variable (valve opening degree on utility fluid) have been presented. The simulation results illustrate the good performance of the temperature tracking by the control strategy including the developed estimation approach. But, during the cooling phase, one can notice that an overshoot appears when a changeover between the cold water and the mixture of monopropylene glycol and water is performed. The explanation is that during this period, the process is not controlled and then, the temperature grows. This can be explained by the fact that during the air purge the reactor jacket is empty, no cooling is carried out and therefore, the mixture temperature remains constant or increases (the increase of the temperature is due to the continuing release of heat by the ongoing chemical reaction).
353 Temperature(oC ) Qr (kcal/s) 5.00E-02
fc (kg/s) ...........................................................
0OOE+00---L~ , i , . , -5.00E-02
-
1.2
3.0E-05
Time ( s ) j
, , ~
Manipulated Variable
50
1
40
2aE-0~
0.8
1; 2.0E-05
-1.00E-01
30
0.6
...... set point
20
0.4
--
0.2
- - * ' - manipulated variab e
1.5E-05 -1.50E-01 1.0E-05
-2.00E-01 -2.50E-01
5.0E-06
-3.00E-01
9
............................................
10 0 0 0
I
I
600
1200
.....
t
t
I
1800
2400
3000
-0.2 3600
Time (s)
0.0E+00
Fig.2a. Comparison between the observed Qob and the estimated Qmo values for the heat released by the chemical reaction R1 using model (4)
Trsimul
Fig.2b. NMPC temperature control with reaction R 1
A second simulation test has been performed with a constant reactive feed flowrate from 1200 to 2400 seconds with a polynomial function (5) as the model used to estimate the heat reaction rate. Figure 3a give the comparison between estimated and observed heat reaction rate. Figure 3b shows the performance of the NLMPC. The same comments than those previously given can be made on the estimation procedure and controller performance.
Temperature(oC)
Qr (kcal/s)
Manipulated Variable
Fc=9.D-6
50
....................................................
1.2
5.00E-02
C(HI,,,I;IIII rcaclivc t;:cd O.OOE+O0 -5.00E-02
Time (s)!
1
4O
0.8
1:
-1.00E-01
30 20
~
...... set point
Trsimul
0.6
--
0.4 L_
- x - - manipulated var ab e
-1.50E-01
,o
o
-2.00E-01 -2.50E-01
0 0
{
I
600
1200
I-1800
I
I
2400
3000
--
-0.2
3600
Time (s)
Fig.3a. Comparison between the observed Qob and the estimated Qmo values for the heat released by the chemical reaction R1 using model (5)
Fig.3b. NMPC temperature control with reaction R 1
The objective to develop this estimator was to improve the prediction in the NLMPC. So, the real indication of in how far the objective is achieved is to compare the prediction with the <<true>> value observed later. For this purpose, on a figure 4, the value of Qmo.l(t+P) computed at time t (based on the parameters a], a2 estimated at time t) is compared to the <<true>>value Q,,b(t+P) given by the energy balance (3) at time (t+P).
354 Or(koa,/s~ s~176176 _5.00E_02@0 580
With regard to the quality of the prediction ~ time(si (Fig.4), one can notice that the two curves are 980 lk~ 1780 2180 2580 80 3~! close except when a changeover of fluids is -1.00E-01;~ ~%~"L .X !!4............... performed. Indeed, a changeover between hot ~S0E0~ \ ...... I-.-Opre~ I and cold water at 1600 seconds is carried out. 2s0E-01-2~17t6176 ~X~,,%~ ! ~ Oobl [ Then, the identified model gives bad predictions. -s00E-01 This phenomenon is more important at 2600 -3.50E-01 seconds when an air purge is done between the -400E-01 ............................................................................. cold water and the mixture of monopropylene O.OOE+O0
........... : ......... : ......... i ......... : ......... ~......... ~....
~
,
,
,
,
,
'2~,
'
'
':
Fig.4. Comparison between the predicted glycol. A typical change in the heat reaction rate Qpred=Qmo(t+P) and the estimated Qob values evolution can be observed at this precise time for the heat released by the chemical reaction R1 (Fig.3a). Moreover, this change in the evolution is amplified in the estimation procedure by the using model (5) parabolic form of the model. 4.2 Results for the chemical reaction R2 (consecutive-competitive reactions) Similar studies have been performed, and the same conclusions than previous ones can be given concerning the performance of the estimation and of the controller.
5. CONCLUSION An estimation of the heat reaction rate based on an optimisation over a moving horizon has been developed. This estimation approach has been included in the non-linear model predictive control algorithm. The application of such methodology has been applied in simulation for thermal control of a semi-batch reactor. The simulation results show good performance of both developed estimation strategy and the non-linear model predictive controller. Future studies will be devoted to the experimental application of the overall estimation and NLMPC strategy to the 16 litres pilot plant reactor which the simulator presented in this work was based on. REFERENCES
1. Xaumier, F., Ettedgui, B., Le Lann, M.-V., Cabassud, M. and Casamatta, G., Computer Chem. Engng., 23 (1999) $923. 2. Friedric, M., and Perne, R., Computers Chem. Engng., 19 (1995) $357. 3. Cabassud, M., Le Lann, M.-V., Ettedgui, B., and Casamatta, G., Chem. Eng. Technol., 17 (1994) 255. 4. Bequette, W.B., Ind. Eng. Chem. Res., 30 N~ (1991 ) 1391. 5. Ettedgui, B., Le Lann, M.-V., Cabassud, M., Ricker, N.L., and Casamatta, G., Proc. ADCHEM'97, (1997).
European Symposiumon ComputerAidedProcess Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
355
Modeling and Simulation of Biotechnological Processes: BIOSIM - A Package suitable for Integration in Process Engineering Tools U. Bergstedt, H.-J. K6rner, S. Kabasci and G. Deerberg Fraunhofer-Institute for Environmental, Safety, and Energy Technology Osterfelder Strasse 3, D-46047 Oberhausen, Germany
Abstract: A bioreactor can be represented as a heterogeneous system with at least three structural phases: the liquid and the gas, which together form the abiotic environment, and the biological phase, which consists of the cell population. The modeling of the bioreactor in our work is based on the zone cell network model for the stirred tank reactor containing two physical phases. In addition, devices in the reactor periphery and process control functions can be simulated. The models for process engineering and biological processes are formulated separately and implemented in a suitable program structure. The program enables to calculate different combinations of biological models and submodels in a process technology context. Different aspects of process technology (feeding strategies, control loops) can be included in the dynamic simulation. The modular integration of the biological phase models into the program structure also allows the transfer to complex network of zones modeling approaches. 1. I N T R O D U C T I O N Biotechnological processes have become increasingly important regarding the production of pharmaceutical and chemical products. Therefore, particularly large potentials are situated in the area of modeling these processes. In the chemical and process engineering industry mathematical methods are used for the calculation, interpretation, planning and optimization of these processes. But there are only a few models and programs which emphasizes both the biotechnological and the process engineering model components with the required degree of detail. 2. S T R U C T U R E O F M O D E L Microbial conversions in bioreactors involve multiphase systems with many different interactions, e.g. cellular reactions, gas-liquid mass transfer and liquid mixing. In modeling fermentation processes the microbial kinetics as well as the chemical engineering effects must be considered. This requires modeling of mass transfer effects and flow pattern in both, gas and liquid [ 1].
356 In this contribution we discuss the modeling of the bioreactor based on the zone cell network model for the stirred tank reactor containing two physical phases [2] (Fig. 1). On the level of the physical phase the reaction volume is divided into the gas phase and the liquid phase. In addition, devices in the reactor periphery and process control functions, e. g. heating and feeding procedures can be simulated. The setup of the equation system is module oriented according to the topology of the simulated process. The submodels of the process engineering components, which are defined by the user, are coupled to a differential algebraic equation system (DAE), which is solved simultaneously by numerical methods. The subsystems are coupled by the fluxes of mass, energy and information between each other. For every control volume (physical phases, temperature control system and wall) the dynamic balance equations for mass, energy as well as the phase equilibrium equations are formulated [3]. The resulting equation system additionally contains state equations for physical properties (density, viscosity, thermal conductivity, diffusion coefficients etc.) and explicit equations for the description of transport processes. The bioreactor is modeled using the three phases: liquid, gas, which together form the abiotic environment, and the biological phase, representing the cell population. The biological phase is assumed to be dispersed in the liquid phase only [4] (Fig. 1).
Fig. 1. Model of the bioreactor containing three structural phases All the reactions catalyzed by microorganisms take place in the liquid phase. The properties of these phases are characterized by time-dependent macroscopic variables such as concentrations or state variables such as temperature [5]. The physical and chemical processes in the bioreactor are described like those in multiphase chemical reactors by balances for mass and energy and the related conservation laws, additionally completed by balances for biomass and cell internal state [6].
357 Between the phases, which are modeled as ideal mixed cells, an interaction in the form of an exchange of metabolites and products takes place. The influence of the state variables of the physical phase on biological activity is taken into account. The relevant components of the biological phase are the cell mass, substrates, which are energy and nutrient suppliers, and products of metabolism. The conversion processes taking place in the biological phase are characterized by substrate consumption, growth and product formation terms. A mathematical description of the intracellular metabolic kinetics and regularization processes can be calculated by stoichiometric models and formal kinetics [7]. Each individual cell can be seen as a separate subsystem which interacts with the outside milieu. The biological models describe the rates of changes of biomass components, the metabolic rates, and their stoichiometry. The biological processes can be represented on various levels of complexity - ranging from simple formal kinetic models to complex models for regularization networks. In unstructured models, the microorganisms are viewed as a homogeneous component, whereas in structured models the microorganisms are modeled as a complex system with further sub-components. In many cases the characterization of biological activity by simply calculating the total biomass concentration is insufficient for a realistic model representation. Variations in biomass activity and composition require a more complex description of the cellular metabolism and a more structured approach to the modeling of cell kinetics. Different models for biotechnological processes are implemented in the simulation program as a FORTRAN library. The models are available in form of software packages for method and parameter values. They can be combined with each other by selection (Fig. 2). bacterial growth
I
I
base equation
I
II
L e e & Monod Rogers
I gmax
I
lag
~tll
I KS
exponent.
II ~12
11/13
I
II
I
stationary - 9 9
....
. . . . . .
~(x) = f(~ base, ~ 1, ~ / r , II/ . . . . . ) ~t
I
decline
product formation
: specific growth rate
K S : saturation constant
f (...)
Fig. 2. Strategy for modeling
:function
I I
..,
substrat consumption
...
358 The models for process engineering and biology are formulated and implemented separately. They are linked together in a suitable program structure. Temperature and concentrations of the liquid phase and the exchanged mass transfer represent the interface information for coupling the biological models with the chemical engineering processes. Calculation approaches for the process engineering aspects of the bioreactor (heat- and mass transfer etc.) are taken into consideration and are coupled with the description of biology. The base equations to compute these parameters were collected from literature and a database was built up. 3. V A L I D A T I O N OF T H E M O D E L As an industrial relevant process we used the discontinuous production of ethanol by
Zymomonas mobilis. The experimental data were compared with computer simulations of different models implemented in "BIOSIM". Most industrial bioreactors are operated under batch conditions. During the reaction period, there are changes in substrate and product concentration over time. The kinetic models we used describe inhibition kinetics with a threshold ethanol concentration to show the effect of ethanol on the growth and the product formation rate. The inhibitory effect of the biomass concentration on the specific growth rate is taken into account. Because the validity of the single models is limited the base equations are combined with different additional kinetic terms. For every growth phase different approaches are available. So by using an appropriate combination good results for the description of the whole life cycle can be obtained. The simulation could fit the experiments satisfactorily. In Figure 3 a comparison between experimentally determined values for the concentration of biomass, substrate (glucose) and product (ethanol) and the corresponding computer simulations is shown. 300 -
t6
substrate
5
250 ~biomass 200 o 150 '~ 100
3 simulation 9 experimentaldata
-
O .,,.~
2 ~ product
50-
0
.r
0
5
10
20
15
25
30
time [hi Fig. 3. Computer simulation and experimental data in batch fermentation
35
359 In fed batch operation additional substrate is fed continuously or in intervals into the bioreactor, thus giving an additional supply of nutrients to the cells [8]. A control of environmental conditions, e.g. substrate concentration can be carried out. It requires a feeding strategy to obtain the desired product concentrations.
80,0 70,0 60,0
4,5
t
4,0 9
3,5 -
50,0
3,0
-2,5
O
~, 40,0
/
--simulation
~" 9
~_~ -2,0 ~ .,..,
30,0 20,0 10,0 0,0 0
. 2
.
. 4 time [h]
.
-
1,5
-
1,0
-
0,5
0,0 6
8
Fig. 4. Computer simulation and experimental data in fed-batch fermentation 4. C O N C L U S I O N S In this contribution a strategy of including biotechnological kinetics with respect to the modeling and simulation of a complex multiphase production reactor is discussed. This is illustrated on the basis of an exemplary production process using available experimental data as well as parameters from literature for empirical-theoretical models. Since most of the values describing the system (e.g. mass transfer coefficients) are calculated scale-dependent, a scale-up is possible. The simulation package includes both phase equilibrium and mass transfer models for two phase processes, so that even commercial relevant processes can be simulated. Different aspects of process technology, e.g. feeding strategy and the calculation of optimal time profiles as well as optimal temperature control can be defined or developed using the program. The program enables to calculate different combinations of biotechnological models and submodels in a process technology context. The modular integration of the biological phase into the program enables the transfer to complex network of zones models for the representation of the reactor volume. Using regression tools, integrated into the program, a fast regression of the respective model parameters to experimental data can be achieved.
360 REFERENCES
1. J. Nielsen, J. Villadsen, Bioreactors: Description and Modelling, in: Biotechnology Volume 3, 2 nd Edition, VCH Weinheim, 1993. 2. G. Deerberg, Zur sicherheitstechnischen Beurteilung von Semibatch Prozessen mit Gas-Fltissigkeits-Systemen, Fraunhofer IRB Verlag Dissertation, 1997. 3. G. Deerberg, S. Schltiter, A. Steiff and W. Witt, Simulation of Operational Failures in Two-phase Semibatch-Processes, Chemical Engineering Science, 11 (1996) 3113. 4. Bergstedt, S. Kabasci and G. Deerberg, Mathematische Modellierung biotechnologischer Produktionsprozesse, Tagungsband DECHEMA Jahrestagung, Wiesbaden, 1999. 5. K.-H. Bellgardt, Cell models, in: Biotechnology Volume 4, 2na Edition, VCH Weinheim, 1991. 6. J. A. Roels and N. W. F. Kossen, On the modelling of microbial metabolism, Progress in Industrial Microbiology, 14 (1978) 95. 7. J. Nielsen and J. Villadsen, Bioreaction Engineering Principles, Plenum Press, New York, 1994. 8. J. Dunn, E. Heinzle, J. Ingham and J. E. Prenosil, Biological Reaction Engineering, VCH Weinheim, 1992.
European Symposiumon ComputerAided Process Engineering- l0 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
361
Simulation and Optimisation of Atmospheric and Vacuum Distillations of a Lube Plant Fernando G. Martins a*, Manuel A. N. Coelho a, Carlos A. V. da Costa a, Manuel A. S. Jer6nimo b, Carlos Martins c and Artur S. Braga c LEP/E, Departamento de Engenharia Qufmica, Faculdade de Engenharia, Universidade do Porto, Portugal
a
bUniversidade Lusfada, Famalic~o, Portugal CPetrogal, Matosinhos, Portugal Abstract The paper presents the development of complete, rigorous and integrated models including all units of the atmospheric and vacuum distillation lube oil plant, from the Petrogal Porto Refinery. The models created, with the PROII process simulator, allow the increase of the plant knowledge, the detection of the plant bottlenecks, the evaluation of the economic impact of the alteration of operation variables, as well as, the accomplishment of studies for process optimisation.
1. Introduction The use of simulation software packages of chemical engineering process (as examples, the program PROII of the Simsci and ASPEN PLUS of the Aspentech) for simulation, process design and conduction of complex industrial installations in steady-state, meets in a phase of great development and expansion. This fact is due to the recognition of its help in the design and improvement of operational conditions (Seader et al, 1999; Biegler et al, 1997): The development of models using these packages, allows actuating in areas such as: 1. Design. The construction of models, based on the study of the existing units and its simulation, will allow, in the future, establishing a base calculation form for sizing some equipment. 2. Plants flexibility. Possibility of production with different specifications of those initially considered in project design. Simulation for different conditions in terms of quality/specifications of raw materials and products; 3. Process optimisation. Improvement of the operation conditions of existing units in terms of control of the process, operationally and its economic effect in investment/cost of operation. For refinery processes, as atmospheric and vacuum distillations, the simulation models are used to translate the separation scheme and to predict the installation behaviour in different *Author to whom all correspondence should be addressed E-mail:[email protected]
362 situations. The same tools are used too in detecting new operational conditions that increase the production added value.
2. Process Description Figure 1 gives a schematic representation of the atmospheric and vacuum distillation sections of the lube oil plant. In the atmospheric distillation section, the crude oil is separated in naphtha (fuel gas, gasoline), white spirit, atmospheric gasoil and atmospheric residue. The vacuum distillation separates the atmospheric residue in vacuum gasoil, some types of vacuum distillates and vacuum residue. The atmospheric crude tower has an integrated pre-heat with bottoms, upper and lower pump-arounds and upper and lower product streams. This integrated pre-heat is represented in Figure 1 by E1 and E2. The heat is also submitted to columns through the injection of steam in the bottom of columns.
Figure 1 - Simplified flowsheet of the atmospheric and vacuum distillation sections.
3. Simulation Models The fn'st model presented here tries to simulate the plant behaviour. All columns are modelled using ideal tray towers. The heat exchangers are modelled using rigorous models and the furnaces with simple heat exchanger models. Figure 2 shows the model developed in ProlI. According to industrial experience, the specifications introduced in the simulation model were: 9 The mass yields of naphtha, white spirit, atmospheric gasoil 9 The mass yields vacuum distillates; 9 The temperature of vacuum gasoil; 9 The kinematic viscosity of D3 vacuum distillate. The performance of the simulation model was compared through the temperature profiles of distillation curves, the temperature profiles in the columns and the kinematic viscosities values of vacuum distillates.
363
Figure 2 - Simulation model for entire atmospheric and vacuum distillation section. An optimisation model was then created based on previous model. An optimiser utility model of the ProII was introduced. The objective function regarding the production add value is given by: Fob, =
P, - P r ,e - P, oe
-- P ,eom
p
(1) where Pi is the price of product i, Pcrude is the price of crude oil, PFoeq is the price of fuel consumption and P,~teamis price of steam consumption. The manipulate variables correspond to the variables specified in previous model. The constraints considered attains to the end points of distillation curves for naphtha, white spirit and atmospheric gasoil, the flash points for white spirit and atmospheric gasoil and the kinematic viscosities of vacuum distillates. 4. Simulation Results and Discussion
The profiles presented in Figure 3 shows the distillations curves for naphtha, white spirit and atmospheric gasoil. As can be seen, the differences observed between the results obtained by simulation model and the real process are insignificants.
364 A S T M D 8 6 at 7 6 0
mmHg
600 580 560 540 520 500 L = L
480
"6
3
460
~. 440 E 1-
Label Label Label Label Label Label
420 400 380 360
1 - Naphtha experimental 2 - Naphtha simulation 3 -W. Spirit experimental 4 -W. Spirit simulation 5 - Atm, Gasoil experimental 6 - At. Gasoil simulatiom
340" 320" 300 5%
10%
30%
50%
70%
90*/0 95%
5%
10%
30*/0 50% (%)
70%
90%
95%
5%
10%
30%
50%
70%
90%
95%
distillate
Figure 3 - Profiles of distillations curves for naphtha, white spirit and atmospheric gasoil. Table 1 list the values of the kinematic viscosity for vacuum distillates. The values obtained show how the simulation model has the ability to translate the process behaviour. The discrepancies between experimental results and simulation results are about 2% of the experimental value. Table 1 - Experimental and simulated values of kinematic viscosities for vacuum distillates. Kinematic viscosity at 373 K (cSt) Distillate Experimental Simulation D1 3.71 3.68 D2 6.00 5.94 D3 11.00 11.00 D4 16.66 16.99 Finally, we proceed with optimisation studies to obtain the maximum of the objective function, described in Equation (1). For the present case, assuming that there aren't commercial restrictions, it was possible to increase the production added value in 350 euros/hour, which corresponds in an increase of 12% compared with base case. However, the real situation is different. In fact, selling restrictions have to be added to the model in order to account the changing market conditions.
365
5. Conclusions This paper demonstrated how process simulators can replicate actual steady-state plant operation and how we can use them to search the production end point (maximum production added value). The simulation models could be used in many other situations not presented in this work. As examples, we can use them to test operational limit conditions, to define new heat transfer arrangements and to analyse the process behaviour for other crude oils.
References Seader ,W. D., Seader, J. D. and Lewin, D. R., Process Design Principles, John Wiley & Sons, New York, 1999. Biegler, L. T., Grossman, I. E. and Westerberg, A. W., Systematic Methods of Chemical Process Design, Prentice Hall PTR, New Jersey, 1997. Acknowledgements This work was supported by ADI, under project GALPORTO21.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
367
A coalescence and breakup module for implementation in CFD-codes Lars Hagesaether, Hugo A. Jakobsen, Kai Hjarbo and Hallvard F. Svendsen. Department of Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. Tel: +47 73 59 41 00, Email: [email protected], [email protected], svendsen @chembio.ntnu.no. [email protected] Bubble and drop coalescence phenomena observed in many industrial separation processes and in multiphase chemical reactors such as bubble columns and stirred vessels, often have a determining influence on the process performance. Even though a number of sophisticated modeling concepts have been presented in the literature over the years, the chemical and physical mechanisms involved are still not satisfactorily understood. Among the most promising methods applicable for elucidating these phenomena are the volume of fluid (VOF), level set (LS) and the direct numerical simulation (DNS) methods. On the other hand, the multi-fluid models have been found to represent a trade-off between accuracy and computational efforts for practical applications. In these multi-fluid models constitutive equations are needed to describe the coalescing and breakup processes, and due to the limited understanding of these phenomena we still have to resort to empirical correlations. In our model development we have chosen to apply a modular approach. At this stage we focus on the inclusion of elaborate models for bubble coalescence and breakup phenomena, while the flow formulation is more simplified. A population like model is developed with emphasis on the source and sink term formulations describing the birth and death rates. The model is formulated in order to facilitate the direct future inclusion into a more sophisticated flow calculation, a full multi-fluid CFD model. We therefore apply a conservative (positive definite) finite volume technique including a second order TVD scheme. The local size distribution budgets for the fluid particles are discussed.
1. THE MODEL Starting with the continuity equation for the dispersed phase (1)
~(pa__.___~)+ V . ( p f f ~ ) = 0 [kg/(m 3s)]. 8t
The dispersed phase is divided into a number of subclasses according to particle size, giving one continuity equation for each particle class. The total dispersed phase fraction and the mass averaged velocities are given as a = ~ n , n---d3 and ff=~.,(n, ff, p~9,)l~,(n, ptg,). i
O
i
(2)
i
The following balance equation for each bubble size class can be obtained in analogy to the wellknown population balance concept: ~9( pn,.______~) + V . (,off, n, )= p [B B - D B + B c - D c ~ [#. k g / ( m 6s ) ] ~t "
In this approach the individual bubble classes are assumed to have their own velocities
(3)
368
in contrast to the standard population balance where all size classes move with identical velocity to the liquid. This approach also allows for variation in gas density. Lo (1999) has developed a slightly different formulation which is implemented in the commercial code CFX. The source terms are found from breakage- and coalescence models. The breakage probability model, Luo and Svendsen (1996), is based on principles of molecular collision and isotropic turbulence. This model contains no adjustable parameters and all constants in the model are calculated from the constants of isotropic turbulence theory. The daughter bubble size distribution can be derived directly from the breakage rate model. Unlike previous work, this model does not need any prior assumption as to the distribution function for the breakage kernel. The breakup model may be written as i (1 +~)2e-ZC a ~7; d~ (4) fib (zg,' zg~fnv)=C3 (1-s ~Inin
where
Z c = 12Cs ~-1113 and We i = pLdiff~ / a
(5)
The breakup model is divided into two parts, the collision frequency and the breakup probability. A similar division for the coalescence model is written as: fic (tg,, rJi ) = coc (r3,, rJj). Pc (tg,, Oj) (6) where the collision rate, Saffman and Turner (1956), may be written as o)c (Z9~, Os) = ~ ( d ~ + ds)2nin/uis with u,:s =(~,~ + ~2),,2j
(7)
The coalescence efficiency is given as Pc =exp(-tc / ts) by Coulaloglou and Tavlarides (1977). Luo (1993) found the coalescence time and interaction time and gave the efficiency as [0.75(1 + ~)(1 ,., 1/2 Pc (Zg~,Zgj) 9 (1 +~',..j)] 9 we~ 9 = exp - C 1 ( p ~ I p L + ~/)1/2 ~0)3
'
where We~ = pLd~g~j/o
(8)
The division into classes is done based on mass, or on volume, if incompressible phases are assumed, as in our case. In accordance with Hounslow et.al. (1988), we use r),+l- 2t9, which is convenient since it simplifies the determination of which classes each broken or coalesced particle belongs to. For a particle with mass between two classes the following formula is used for splitting it into the two adjoining classes t9 = xtg~ + (1 - x),3~+~= xr3~ + 2(1 - x)tgi giving x = (2t9~ - tg) / tg~ (9) where x is the part of the particle that is put in class i and ( l - x ) is put in class i + l .
This
scheme conserves the number balance of the dispersed particles as well as the mass balance. The breakage model (4) and the coalescence model (6,7 and 8) give the source terms in (3). (9) is used for dividing all fluid particles into appropriate classes. The source terms may then be written as i-1 i-1 1 ,,5
Bc,' = E
Xnc(O,,I-'gJ)+E(1-x)nc(O,-1,~gJ)-Tnc(O,-l,l~,-i),
j=I,i~N N
BB. i -
~ j=i+l,ir
D.,, = ifiB(O,,zg, fBv)dfBv ,
j=l Oi+l
flB(Oi,Oj)+
o N-I
~ X o f 2 B ( O i + l , 0 ) d O , and De. , = ~ f ~ c ( O , , O , ) + f i c ( z 9 , , O , ) 0=0#
j=l
where i = 2..N for B(:, i and DR, i , i = 1..N for B B.i and i = 1 . . N - 1 for De. ~.
(10)"
369 2. N U M E R I C A L M E T H O D S The time discretization of the basic balance equations is performed by use of the fractional time step method that has become very popular in geophysical sciences, e.g. Berge and Jakobsen (1998). The fractional step concept is more a generic approach than a particular method. It is essentially an approximate factorization of the various numerical operators determining the transport equation. It is also possible to split the convective and diffusive terms further into their components in the various coordinate directions. Strang (1968) pointed out that the accuracy of such splitting methods depends both on the accuracy of the numerical solution methods applied to the individual operators in the equations, and on the accuracy of the time splitting procedure itself. By performing the intermediate time integrations in a prescribed order, the splitting method itself can be shown to be second order accurate in time. Therefore, when the individual operators applied are second order (or higher order) in time, the total time integration procedure will be second order accurate. The various transport, source and sink terms in the balance equations have accordingly been split into separate numerical operators that are successively solved by intermediate time integrations. The convective terms are calculated by use of an explicit second order method in space, the conservative Total Variation Diminishing (TVD) scheme. The TVD scheme applied was constructed by combining the central difference scheme and the classical upwind scheme by adopting the 'smoothness monitor' of van Leer (1974) and the Superbee limiter, Sweby (1984) and LeVeque (1990). An Euler explicit advancement is applied for the individual source terms. This approach is by definition modular, and the balance equations can easily be implemented in any consistent CFD code. 3. RESULTS AND DISCUSSION Local and global mass and bubble number budgets were obtained by integrating the convective fluxes in and out of the boundaries, and the death and birth rates within the calculation domain. The discrepancies in all balances were found to be of an order close to the machine number representation. Two examples of the behavior of the coalescence-dispersion model applied to a bubble column are shown in the following. The column used both in the simulations and for experimental comparison was 4.3 m high and with inner diameter 0.29 m. First, in order to study the breakup into various classes, it was assumed that the gas flow into the bubble column consisted of only one bubble class of diameter 15.9 mm (class 3). The whole population was divided into six classes. The energy dissipation rate was set to 0.25 m2/s3, which is a reasonable average value for a superficial gas velocity of 0.02 m/s. The bubble rise velocities for the various classes were calculated from equations given by Fan and Tsuchiya(1990). Both the energy dissipation and the relative velocities contribute to the total coalescence and breakup rates. The integration time step used in the calculations was 0.02 seconds. Figure 1 shows the development of the number of bubbles in each class with time and position. The liquid velocity is in this case assumed to be zero. Notice that the individual classes appear to have the same rise velocity. This is due to the fast coalescence and breakup. Without the source terms the difference in rise velocity is easily seen (not shown). The number of bubbles in the initial bubble class, class 3, is seen to drop rapidly. Ini-
370
tially it drops slightly below the steady state value, but rapidly stabilizes. The small bubbles in class 1 and 2 stabilize to their equilibrium value about 0.4 m above the inlet. The larger bubbles, classes 4-6, however, show an overshot right after the inlet. This is as expected as the number of class 3 bubbles is initially large. The coalescence rate to larger bubbles will then be high. As the number of class 3 bubbles drop toward the equilibrium value, the breakup processes of the larger bubbles will take over and bring the number density down toward the equilibrium value. At steady state the whole bubble distribution has reached equilibrium about 0.6 - 0.8 m from the inlet. This is in agreement with our own size distribution measurements based on the five-point conductivity method, Buchholz et. al. (1981). It thus seems that the model behaves reasonably as far as these results can be interpreted. Development of bubble size distribution 4
Class 1
x 10 4
Class 2
4000
2
2000
1
...Q
0
E
2~176
6000
3
Class 3
0
1
2
3
4
0
0
1
2
3
4
0
"1
'2 -
"4
c..-
Class 4
~=~ 9 400
Class 5
35
Class 6 2
30
~- ~ o o
1.5
25 20
1
15 I I I I
100
0 0
0.5
0 ,____~ 0
1
2
3
4
0
0
1
'2
3
4
[m]
Bubble size distribution as function of position in bubble column. The bubble diFigure 1. ameters range from 1 to 3 cm. Legend: N = 4 sec., 9. . . . 12 sec., - - = 20 sec., a n d r e = 28 sec. Secondly the model was tested against bubble size distribution data measured in the mentioned column using the conductivity technique. The bubble size distributions were measured at two positions, 0.3 m and 2 m above the inlet. The size distribution and bubble numbers at position 0.3 m were used as input to the model, and the model calculated the size distribution at position 2 m above the inlet. Measured rise velocities for the various bubble classes were used for the bubble rise velocities. This means that the liquid velocity profile was indirectly taken into account. The bubble population was split into 14 bubble classes and the results are shown in figure 2. It is seen that the model shows a change in bubble numbers in the right direction for all bubble classes. The number of small and intermediate bubbles increases whereas the number of large bubbles decreases. The change in bubble numbers, apart from the two smallest bubble classes, compares surprisingly well with the experimental data taking into account that the model only contains one empirical parameter, in the coalescence model, and that no fitting of this parameter has been done. The rise in bubble number for the two smallest bubble classes appears to be grossly over-
371 estimated. However, one should note the limitations of the experimental method. The three smallest bubble classes are for bubble diameters 0.75, 0.94 and 1.2 mm respectively. The fivepoint conductivity probe used has a lower detection limit of about l mm, and the numbers below this value are very uncertain. The numbers obtained in the simulation may therefore not be totally unreasonable, although some overprediction may be assumed. Uncertainties still exist in the underlying models. The used coalescence model can be improved and will be replaced by a coalescence probability based on a Lagrangian collision model concept, Hagesaether et.al. (1999) and Svendsen and Luo (1996). The distribution function used in the breakup model was developed and tested for liquid droplets and thus may need to be slightly redefined for gas bubbles.
6~176
Simulated bubble numbers compared to experimental data
Class 1
313 9
Class 2
2000
3000
61 9
553 9
1000
2000
800
1000
600
1000 0
Class 3
1200
1
Class 5
695 7
2
650
600
400
0
1 Class 6
822 9
[
Class 7 "851
2
620
486
600
484
~- 500 t~
580
482
400
560
480
E
tq)
0 215 1= n
1
Class 9
29 3 7
2
C l a s s 10
88 9
77.5
210 205
1
78
1
Class 13" 1
2
28
1
Class 11
18 9
2
27.5
77
1
2
27 C l a s s 14
1 9
494 9
Class 8
525 9
J 1
2
Class 4
1
2
1
2
C l a s s 12 " 4
11.5 1
2
11
0.8 i
2
0.7
i
2 [ml Figure 2. Development of bubble size distribution as function of position in a bubble column. The number on top of each subplot is experimental count at position 2 m above inlet. The bubble diameter range is 0.75 to 15 mm.
4. CONCLUSIONS A combined particle coalescence and breakup model is developed with numerical schemes specifically designed for implementation in multi-fluid CFD-codes. The model is tested on a bubble column geometry and for simplified flow situations. The transient responses obtained for a stagnant liquid phase are found to be physically reasonable. Comparison between simulated and experimental bubble size distribution show that the correct trends are obtained for all bubble classes, and that the model also predicts the quantitative changes well.
372
SYMBOLS B~ birth breakup # / ( m 3s)
Pc coalescence efficiency-
B c birth coalescence #/(m3s)
t
time
C~ constant c 3 constant, c 3 = 0.923 c f coefficient, se eq. 5
-
D8 death breakup # / ( m 3s)
D c death coalescence # / ( m 3S) d r diameter bubble class m
ni
total number of classes number in class i # / m 3
size ratio, ~ = 2 / d i
m -
tc
coalescence time
s
~!j size ratio, ~j = d, / d i -
tz
interaction time
s
p density
u We x a ?
velocity Weber number volume fraction volume fraction added mass param.
m/s
PL liquid density kg / m 3
-
eddy dissipation m 2 / s 3
f~v breakage volume fractionN
][ eddy diameter
s
t9
volume
0 surface tension N / m Zc critical breakage energy s
breakup rate
# / ( m 3s)
~ c coalescence rate # [ ( m 3S) Nc collision rate
E~ void fraction
kg ] m 3
#](m 3S)
m3
REFERENCES Berge, E. and Jakobsen, H. A., "A Regional Scale Multi-layer Model for the Calculation of Long-Term Transport and Deposition of Air Pollution in Europe", TeIlus, 50, 205-223 (1998). Buchholz, R., Zakrzewski, W. and Schugerl,K., "Techniques for determining the properties of bubbles in bubble columns", Int. Chem. Eng., 21, 180-187 (1981). Coulaloglou, C.A. and Tavlarides, L.L., "Description of interaction processes in agitated liquid-liquid dispersions",Chem. Eng. Sci., 32, 1289-1297 (1977). Fan, L.-S. and Tsuchiya, K., "Bubble Wake Dynamics in Liquids and Liquid-Solid Suspensions", ButterworthHeinemann, USA, 1990. Hagesaether, L., Jakobsen, H.A. and Svendsen, H.F., "Theoretical analysis of fluid particle collisions in turbulent flow", Chem. Eng. Sci., 54, 4749-4755 (1999). Hounslow, M.J., Ryall, R.L. and Marshall, V.R., "A Discretized Population Balance for Nucleation, Growth, and Aggregation",AIChE Journal, 34, No. 11, 1821-1832 (1988). Le Veque, R. J., "Numerical Methods for Conservative Laws", Chapter 16, Birkhauser Verlag, Basel (1990). Lo, S., "Application of population balance to CFD modelling of bubbly flows via the MUSIG model", CFX Technology, UK, Presented at GLS'99 in Delft, Netherlands (1999). Luo, H. and Svendsen, H.F., "Theoretical model for drop and bubble breakup in turbulent dispersions", AIChE Journal, 42, 1225-1233 (1996). Luo, H., "Coalescence, breakup and liquid circulation in bubble column reactors", Dr. ing. Thesis 1993:105, Dept. of Chemical Engineering, Trondheim, Norway (1993). Saffman, P.G. and Turner, J.S., "On collision of drops in turbulent clouds", J. Fluid Mech., 1, 16-30 (1956). Strang, G., "On the Construction and Comparison of Difference Schemes", SIAM J. Numer. Anal., 5, No. 3 (1968). Svendsen, H.F., and Luo, H., "Modeling of Approach Prosesses for Equal and Unequal sized Fluid Particles", Can. J. Chem. Eng., 74, 321-330 (1996). Sweby, P. K., "High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws", SlAM J. Numer.
Anal., 21, No. 5,995-1011 (1984). van Leer, B., "Towards the Ultimate Conservation Difference Scheme II. Monotonicity and Conservation Combined in a Second Order Scheme", J. Comp. Phys., 14, 361-370 (1974).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
373
Fluid Dynamics and Thermochemical Simulation of a Smelting Cyclone M. Modigell, M. Weng Institute of Chemical Engineering RWTH Aachen, Dep. Mechanical Unit Operations Turmstr. 46, 52056 Aachen, Germany The present paper discusses a new approach to describe the conversion of complexly composed solids that are dispersed in a gas flow at high temperatures. Numeric simulation of the flow field and particle trajectory is coupled with a thermodynamic equilibrium calculation. First simulation results and the comparison with experimental data are shown in this paper. 1. INTRODUCTION High temperature cyclones applied in non-ferrous metal processing and thermal environmental technology are characterised by their high specific throughput and smelting capacities [1, 2]. Smelting cyclones are operated at high swirl intensities resulting in multiple recirculation phenomena and intensive gas phase mixing (s. Fig. 1). The combustible components of the injected solid phase are oxidised, resulting in high temperatures, the inorganic ash is smelted. The liquid droplets are separated at the wall due to the centrifugal forces and form a coherent slag film which leaves the cyclone at the bottom in uniflow direction together with the off-gas. The heat and mass transfer is intensified by high slip velocities between the gas and the particle phase, increasing the rate of conversion. Hence, the cyclone's main operation feature is the combination of chemical reactor and phase separator which enables the realisation of ambitious processes. particle injection tangential The complex apparatus properties require a detailed inlet i understanding of interaction mechanisms between transport phenomena and chemical reaction. Thus, a CFD analysis has been coupled with a thermochemical simulation. The advantage of this method compared to classic combustion calculations is that it takes into account all species participating in the process. To demonstrate the application of the modelling technique, the incineration of complexly composed sewage sludge was chosen as an example.
i
I rcuaon
II/illl Figure 1
~
smelting cyclone principle
374 2. MODELLING Because of the enhanced complexity of both the flow field and the chemical composition of the input material, a calculation approach has to take into account the strong coupling between both. Since a kinetic approach seems to be inappropriate because of the high number of unknown reaction parameters, a new model technique is developed. The calculation of high temperature conversion is based on a numerical simulation of the fluid flow inside the cyclone with the commercial finite volume software CFX 4.2. Additionally, the initial temperature field and gas phase concentrations are calculated (s. Fig. 2). Since the cyclone flow is highly turbulent, closure conditions for the Reynolds-averaged NavierStokes equations are required. The commonly used standard-k-~-model (KEM) has proved to fail in the description of swirl flow turbulence. Especially at higher swirl intensities it is neeessary to use the higher order Reynolds stress model (RSM) because of turbulence anisotropy [3]. In order to limit the computation effort, the calculations were carried out in a twodimensional geometry. The high swirl stabilises the flow and causes small gradients in the circumferential direction even if the real geometry is not rotational symmetric [4]. The equations of motions for single particles are then solved explicitly according to fixed starting conditions. In the Euler-Lagrange method the heat and mass transfer between gas and solid phase are determined by the Ranz-Marshall-equation [6] Nu = 2 + 0.6- Re~
Pr 0.33
(1)
and the analogue equation Sh = 2 + 0.6. Re~
Sc0.33
(2)
Radiation is considered in this first approach as particle-wall radiation with the heat transfer coefficient T~-T~v aS=es . o - . ~
(3)
Yp - Yw
with particle temperature Tp and wall temperature Tw, respectively. The effect of gas phase turbulence on particle motion is considered by a modification of the particle drag coefficient which is dependent on local turbulent kinetic energy [6]. The discussed conversion calculation is a two step model. Firstly a certain amount of water and volatile hydrocarbons evaporates from the particle. These amounts are determined by the temperature distribution inside the particle which is calculated by solving the instationary heat conduction equation. A spontaneous evaporation of a certain fraction is assumed when the respective discrete spherical particle shell reaches temperature of 100~ or the pyrolysis temperature of 250~ respectively. According to solid phase conversion it is now assumed that the particle and the surrounding gas phase are in the state of thermodynamic equilibrium during short time steps. Consequently, the transport phenomena between gas and solid phase are responsible for the overall deviation from equilibrium, whereas kinetic inhibitions are neglected due to the high temperature level. The participating amounts of gas and solid phase are passed to the thermo
375
Figure 2
model scheme
chemical simulator ChemApp. Here, the equilibrium state of the system is determined. The gaseous conversion products represent sources for the gas phase. The locally calculated change of system enthalpy is used for the correction of the initial heat source distribution. In the following iteration the calculation of flow field, temperature and species distribution is repeated. The calculation of particle conversion is finished when the particle reaches the wall where it is separated. A cell model using the equilibrium simulation method with explicitly identified transport coefficients between the single cells of the LD process in steel making has shown good agreement with experimental data [7]. The transport terms in the here discussed method are calculated from the solution of balance equations for mass, momentum and enthalpy. The thermochemical simulation is based on a minimisation of the system Gibbs energy Gm, where the total Gibbs energy is the sum of Gibbs energy of the pure phase components G rej , the contribution of the ideal mixture entropy G~a and the excess Gibbs energy contribution
G'mx" Pure phase Gibbs energy is calculated from
/
0 -}- Cp,idT-T(Si~ef + Gref'-m Z xi H~,,,eS
!1/
cpidT)
(4)
To The ideal entropy of a mixture and excess Gibbs energy are calculated from Gim d + G ex = R T ( )--~.x~ln x i -I- ~ Xi In ~'~)
(5)
Heat capacity Cp,~ and activity coefficients i are modelled by algebraic functions using thermodynamic data from a comprehensive database. The system's enthalpy Hm is then calculated from
376
H m --Gm-T(OGml
(6)
at given pressure p and composition n. 3. SIMULATION RESULTS AND COMPARISON WITH EXPERIMENTAL DATA In Fig. 3 the profiles of tangential and axial velocities at different axial positions are compared with experimental data obtained from isothermal investigations [8]. The tangential velocity shows that the agreement of RSM is fairly good, whereas the KEM overestimates, as expected, the turbulence energy resulting in a solid body vortex that extends nearly over the complete radius. The axial velocity indicates that the places of maximum velocity and the position of the large recirculation zone (u < 0) is well determined. Deviations near the axis indicate the necessity of further experimental and numerical investigations. The input material in this investigation is industrial sewage sludge with a water content of 10% and a mean grain size of 1 mm according to experimental investigations which enable a comparison of the calculated compositions. The results of particle heating and conversion represent the initial results of a postprocessed particle trajectory without the full coupling between gas and solid phase. In the case of sewage sludge incineration the observed recirculation zones are of major importance for process operation. The oxidation enthalpy from particle conversion is carded towards the cyclone inlet causing an effective preheating of the entering particles. A supporting mechanism is the convective heat transfer intensified by the high slip velocities between particles and gas phase. In comparison to the conditions in uniflow smelting furnaces six times higher Nusselt numbers are achieved in the cyclone. Consequently, the convective heat transfer is the major mechanism for particle heating compared to radiative transport unless the pyrolysis and subsequent combustion of hydrocarbons starts. A further boundary condition for process operation is given by the fact that the particle flight time surpasses the required smelting time. Varying the particle diameter and the location of injection shows the effect on particle heating. In Fig. 4. the temperature distribution inside spherical particles is drawn along the time axis. 40
6
.,
~
4
9
30
, "-"-'--. . . . . . . . . . . . . .
2 E
"--' :=
'1o
~',, \'-.
0
,,.,
--..j
x=0.35
m' m
----
-.---
x=035
RSM
m
~ - -/ - - - ] l
I
Y,,,~
x=O.55m E x p . ~
i
-2 !
0
0,2
0,4
0,6 dR [-]
Figure 3
0,8
-4
0
0,2
0,4
0,6 dR [-]
simulated velocity distributions versus experimental data
0,8
1
377
Figure 4
particle heating at Re = 86,000, S = 5.4
Particles injected through the main air inlet reach the upper reversal of the toroidal eddy where high slip velocities and a fast heating are provided. Though, particles are separated at the wall due to the high tangential velocity before smelting temperature is achieved. Injecting the particles from the cyclone lid in axial direction enhances the flight times. The slip velocities are optimal if the place of injection is near the axis in such a way that the particles are blown into the recirculation zone. Particle smelting is limited by their size. The temperature of a 2 mm diameter particle even at optimum slip velocities is far below smelting when they are separated at the wall.
Figure 5
trajectory of a lmm particle, solid phase composition and heavy metal phase distribution, Re = 86,000, S = 5.4
378 Based on the particle tracking and interior temperature distribution the conversion of a sewage sludge particle with 1 mm diameter injected from the cyclone lid is calculated. When the pyrolysis of hydrocarbons start (after 17.2 ms), the solid phase composition widely corresponds to the initial state apart from a fractional evaporation of water. Due to its low vapor pressure, total Hg is volatilised. After 41 ms, 4% solid carbon have been oxidised and a displacement of FeO to FeC takes place. The volatilisation of Zn starts, 55% of Pb have been evaporated. At 58.2 ms the outer particle layer reaches smelting temperature. Fig. 5 shows the fractions of evaporated heavy metals, the composition of the solid phase and the interior temperature distribution after 58 ms flight time in comparison to experimental data taken from the liquid slag leaving the cyclone at the outlet [10]. The main composition of the solid phase and heavy metal volatilisation are described qualitatively good. The difference in remaining carbon shows that an additional modelling of the fixed carbon oxidation has to be included in order to describe the kinetic inhibitions of this special reaction. Simulations with increased 02 partial pressure indicate that oxidation is enhanced due to increasing flux from gas to particle phase. Thus, the efficiency of sewage sludge incineration is increased. 4. CONCLUSIONS A new modelling approach is developed to simulate the high temperature reactions in the multiphase flow of smelting cyclones used as highly efficient flash reactors in extractive metallurgy and environmental technology. A CFD simulation is coupled with a thermodynamic equilibrium calculation. This method allows the description of non equilibrium phenomena such as the reactions between gas phase and dispersed particles while assuming local equilibrium. The particular advantage of this method is demonstrated for the combustion and smelting of complexly composed industrial sewage sludge. Besides the reactions of the main components C, H and O, the evaporation and chemical reactions of minor components such as heavy metals which are of special interest in respect to environmental aspects can be determined. First results show good agreement to experimental data. REFERENCES 1. Sauert, F., Castor, L., Jones, S., Proc. of the Symposium on Recent Developments in NonFerrous Pyrometallurgy, Toronto/Kanada 1994 2. Barin I., Klefisch R., Lemperle M., Modigell M., Proc. of the Int. Conf. on New Smelting Reduction and Near Net Shape Casting Technologies for Steel, Pohang/Korea 1990 3. Modigell, M., Weng, M., Chemie-Ingenieur-Technik 71 (11), (1999) 4. Erdal, F., Shirazi, S., Papers Society of Petroleum Engineers, No. 2, (1996) 5. Ranz, W., Marshall W., Chem. Eng. Progress 48 (3), (1952) 6. Uhlherr, P., Sinclair, C.: Proc. Chemeca 1, Butterworths, Melbourne (1970) 7. Modigell, M., Traebert, A., Monheim, P., Hack, K., Proc. of 1st International Conference on Process Development in Iron and Steelmaking, (1999) 8. Lang, P., Dissertation Karlsruhe (TH) (1984) 9. Rizzon, J., Dissertation Aachen (1991)
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
379
Computational Fluid Dynamics Modelling of Multiphase Reactors Marco Bistolfi, Nicola Mancini, Fabrizio Podenzani EniTecnologie, Via Maritano 26, 20097 San Donato Milanese, Italy emai l: mb istolfi (nmancini, fpodenzanO @enitecnologie. eni. it
Computational Fluid Dynamics (CFD), originally developed for non-reacting systems and successfully applied to aerodynamic design, has been recently proposed also for process engineering, in particular for multiphase reactor simulation. In fact CFD modelling can significantly contribute to better understand the fluid dynamics of process equipment, especially when their performances have to be scaled up from a laboratory or pilot plant scale to an industrial size. In this paper two different applications of this modelling approach to multiphase reactors are presented: in the first one slurry bubble columns are considered, while the second one is related to a lab-scale stirred tank reactor (CSTR) simulation. 1. INTRODUCTION Computational Fluid Dynamics (CFD), a modelling approach based on the numerical solution of the Navier-Stokes equations in a Reynolds Averaged form (RANS), has been proposed about twenty years ago to represent and predict the fluid dynamics of turbulent nonreacting systems; nowadays a few commercial codes based on this approach are currently available. Recently many CFD applications in the field of process engineering have been proposed, and some of the most promising ones, although still under development, are related to multiphase reactor modelling. It has to be considered that in this case a complete physical description of the system (multiphase multicomponent hydrocarbon mixtures) and of the phenomena involved (turbulence, chemical kinetics, heat and mass transfer) are required; moreover, dealing with turbulent reacting flows is one of the most difficult task for CFD modelling. Therefore, to reduce the complexity of such an approach it could be useful to start with simpler conditions (ideal systems, ambient temperature and pressure, no reactions) generally obtaining a quicker although approximate response, and then gradually introduce all the other real industrial conditions once a first solution has been achieved. With this approach CFD could be used not only to describe the fluid dynamic behaviour of multiphase and multicomponent reactors (fluidized beds, slurry bubble columns, etc.), but also to validate the assumptions of simpler reactor models or to support the design of reactor components (gas distributor, internals), evaluating their performances in terms of mixing level, flow patterns, and so on. In this paper two examples of CFD applications to gas-solid-liquid reactors (CSTR slurry reactors, slurry bubble columns) are described.
380 2. SLURRY BUBBLE COLUMNS
Slurry 1 bubble columns are commonly used for industrial gas-liquid-solid catalytic reactions, due to their effective heat and mass transfer under chum turbulent regimes: in fact in these conditions the largest bubbles undergo very frequent coalescences and break-ups, inducing very significant liquid mixing and recirculation effects. Experimental observations show that, at low gas superficial velocities (Ug), a homogeneous regime of small bubbles exists, while above a specific transition value (Utrans) a churn turbulent regime of both small and large bubbles takes place. Small bubbles have sizes in the range from 3 to 6 mm, they are typically spherical in shape and their velocities only depend on liquid physical properties; large bubbles instead are in the range of 20-70 mm, their velocities do not depend only on physical properties, but also on scale parameters (typically the ratio between bubble diameter and column diameter) and wake effects between leading and trailing bubbles in the swarm. Small bubbles are substantially entrained by the batch and well-mixed liquid/slurry phase, generating the so-called dense phase; on the other hand large bubbles, the so-called dilute phase, follow a plug-flow regime, passing much more rapidly through the column. 2.1 The two phase model
To represent the hydrodynamics of such complex systems, a simplified semi-empirical model has been adopted [1 ], considering two different phases and leading to quite accurate gas hold-up calculations. In order to describe in details the fluid dynamics behavior of bubble columns, a CFD code (FLUENT) have been used, taking advantage of the same semiempirical correlations to determine some key fluid dynamic parameters. Following this approach in fact, once Utrans has been determined and the small bubble mean diameter assumed, the large bubble mean diameter is given by dtb -- O.069.(Ug-
Utrans) 0"376.
While small bubbles velocities Usb could be quite accurately calculated as:
Us b = 1.53(gcr)0.25 Pl large bubbles raise velocities Ulb are much more uncertain, because of wake effects, and could be calculated as [ 1]" Ulb = 0.71~gdlb (SF)(AF), where SF (Scale Factor) is equal to:
dlb , d lb I DT d lb SF=I, if DT dlb <0.125. SF=I.13 exp(---~), if0.125< DT <0.6; SF=0.496 ~,dlb if~DT > 0.6 with AF = 2. 73 + 4.505.(Ug-
Utrans), acceleration factor due to wake effects.
All these correlations are the same used by the simplified model [1 ], while in the case of the CFD approach also the momentum exchange between the phases has to be calculated, and in particular the drag forces contribution:
l Highly dispersed catalystpowder in liquidphase.
381 3 1 f = 4elegqglCD I U l - U g I(U 1 -Ug)--~b
In fact two different bubble diameters and drag coefficients for the two classes of bubbles have to be considered, taking into account their different behaviors (i. e. wake effects for large bubbles), and this is crucial for a good prediction of liquid velocity and gas hold-up. In order to calculate the drag coefficient the following expression is used: 4 1
2.2 Solid and p r e s s u r e effects
Slurry phase solid particles promote bubble coalescence and, as a consequence, reduce gas hold-up; this effect is taken into account modifying the bubble parameters (diameters, Co, etc.) according to the solid quantity [2], with Utra,,s = U~b"etr~,,s'(1 - etra,,~). The mean velocity of small bubbles increases because of the higher slurry density compared to the liquid one, and could be calculated with the following expression: 0.8
0.7 and:
Usb = Usb,O (1 + ..... es) Usb, 0
etrans = etrans,O ( 1 - ~ e ) etrans, 0 s
Also for dlb and A F new expressions [3] are required: A F = 2. 2 725 + 3. (Ug
-
dtb = O. 1 l" (Ug -Utrans) 0"53
Utrans)
Higher pressure values instead reduce large bubble diameters (bubbles become smaller increasing the pressure), and also in this case the correlations have been modified according to Reilly et al [4]: P g )0.48 etrans = etrans,O (
Pg,o
For the combined effects of pressure and solid the value of etrans is obtained from a combination of the previous correlations for etra,s. 2.3 S i m u l a t i o n results
Slurry bubble columns have been initially modelled as ideal gas-liquid (air-water) systems in churn turbulent regime using FLUENT; the modelling approach adopted is based on" 9 Eulerian-Eulerian formulation; 9two classes of bubbles considered; 9drag coefficient CD and bubble diameters obtained from previous experimental data correlation; 9 axial-symmetrical 2-D or 3-dimensional (3-D) time-dependent simulations; 9ReNormalization Group (RNG)/k-e turbulence model; 9SIMPLE algorithm; 92nd order numerical scheme (QUICK).
382
i' rrllIllll~trt 'II
0.6
'.q 1i1j,'J. ,r~]lllIl~rl, . . . ~ltl
0 i
~
exper.
exper. calculated
0.4
9
2D simulation
.~_ 0.4
/.,rl[llllllllllTlr,. ,P' 'ql~/
!,..'r~rTr.,,tl! lr'fl~Tt"~LII
t
0.3 ~
or}
E
"6
>
o.o
,
wall
i
-0.4
, ~,
0.2 -
!
.=,-
-0.2
p.,.,l,tas~jr;,, t
,
....
0.2
~
It'dI~t'"tllli
3DsimulaUon - ~ - ~ . ~.
\
0.1 ! simmem/axes --T--T--T-~--I 0.0
0.2
0.4
~ [ 0.6
0.8
0.0 .0
~ 0.0
1 0.2
r
I 0.4
~
~ 0.6
T
,i, !:,,
,, t
r,F,
~"t
"* Fig 1. Bubble column water velocity vectors (left) and air hold-up data [6].
T r 0.8
1.0
Although steady-state solutions are required, transient simulations are necessary for numerical reasons. Moreover, due to the periodic and non-symmetrical behavior of the flow (as shown by experiments), axial-symmetrical 2-D simulations could give reasonable results in a relatively short time only for water velocity and total air hold-up: predictions of more detailed information (like air hold-up profiles) require fully 3-D simulations. With this approach a good agreement with experimental data of water velocity and air hold-up data [6] has been initially obtained (see fig. 1); the results show that a reasonable prediction of the "total" air hold-up is possible with 2-D simulations, but the correct hold-up profile is obtained only with a 3-D calculation. Table 1 Gas hold-up values for air-water system % solid 0 15 37
measured 0.37 0.30 0.18
calculated 0.32 0.31 0.18
Table 2 Gas hold-up for an hydrocarbon system Us, m/s 0.15 0.20 0.25
measured 0.30 0.33 0.37
calculated 0.32 0.32 0.36
Using the correlations previously described also solid effects are taken into account, obtaining a good agreement between simulated and measured total air hold-up (see table 1). Table 3 Gas hold-up values for air-water system % solid 0 20 30
measured 0.30 0.18 0.17
calculated 0.32 0.15 0.13
Table 4 Gas hold-up at 20 bar and Ug = O. 15 rn/s % solid 0 20
measured 0.33 0.21
calculated 0.30 0.16
In order to validate the approach a real hydrocarbon system has been simulated and the
383 results are compared with gas-hold-up experimental data obtained with different mock-up columns (from 50 mm to 400 mm diameter). The results at atmospheric pressure and without solid are quite good, as shown in table 2 for different gas velocities. The effect of solid on total gas hold-up is a little overestimated (see table 3), and also the effects of solid and pressure at the same time are a little overestimated (see table 4). 3. STIRRED TANK R E A C T O R A laboratory-scale CSTR reactor for kinetics studies has been modelled with another commercial CFD code (CFX4), in order to predict both flow pattems and chemical reaction effects. Such reactor is made by a round-bottomed vessel with three baffles at 120 ~ and an impeller with four 45 ~ pitched blades. There are two main methods to represent the impeller effect within a CFD code: the first one directly takes into account the impeller geometry and then uses a " s l i d i n g m e s h ''2 to perform the calculation; the second one applies the body forces exerted by the impeller on the fluid through distributed momentum sources. The former method is computationally much more expensive than the latter, but is more accurate near the impeller. As the aim of this work is to describe the overall fluid dynamics in the vessel, initially the second approach was adopted; then few cases have been compared with the sliding mesh method to check the results. The body forces have been calculated from the impeller power curves for different geometrical configurations; each component of such forces was imposed as a source term in the relevant momentum equation, within the region swept by the impeller. A gas-liquid-solid three-phase system, with 50 ktm catalyst particles, was considered; a flat liquid surface (symmetry plane) has been assumed in the simulations. The work has been performed in different steps: initially the momentum source approach was validated with a literature test case [7], reporting experimental Laser Doppler Velocimetry (LDV) data in a stirred vessel; then simulations of our lab-scale reactor, considering a single liquid phase were performed; afterwards the solid was taken into account.
Fig 2. CSTR velocity vectors (left) and solid volume fraction fields 2 With the sliding mesh technique two grids are used: one describes the vessel and the fluids, while the other one is moving with the impeller along a "grid interface" in discrete steps.
384 In fig. 2 the velocity vectors flow field and the solid volume fraction distribution predicted by the code are represented. 4. CONCLUSIONS Commercial CFD codes could be used to simulate multiphase reactors with quite satisfactory results, once a preliminary validation activity has been performed. CFD capabilities could support experimental data interpretation and scale-up, predicting fluid dynamics characteristics and reactor performances. Further fundamental developments of CFD approach to real multiphase and multicomponent reacting system are required, in order to become an effective industrial tool for reactor design and process optimization.
REFERENCES 1. R. Krishna, M.I. Urseanu, J.M. van Baten, J. Ellenberger, "Risevelocity of a swarm of large bubbles in liquid", Chem. Eng. Sc. 54 (1999). 2. R. Krishna, J.W.A. De Swart, J. Ellenberger, G.B. Martina, C. Maretto, "Gas Holdup in Slurry Bubble columns: Effects of Column Diameter and Slurry Concentrations" AIChE J., Vol. 43, N.2 (1997) 3. R. Krishna, J.M. van Baten, J. Ellenberger, "Scale effects in fluidized multiphase reactors" Powder Tech. 100 (1988). 4. I.G. Reilly, D.S. Scott, T.J.W. De Bruijn, D. Maclntyre, "The role of gas phase momentum in determining gas holdup and hydrodynamic flow regimes in bubble column operation" Can. J. Chem. Engng., 72, (1994) 5. C. Maretto, R. Krishna, "Modelling of a bubble column slurry reactor for Fischer-Tropsch synthesis" Catalysis Today 52 (1999). 6. J. H. Hills "Radial non-uniformity of velocity and voidage in a bubble column" Trans. Instn. Chem. Engrs, Vol.52 (1974) 7. M. Pettersson, A. Rasmuson "Hydrodynamics of Suspensions Agitated by Pitched-Blade Turbine", AIChE J., 44, (1998).
Notation d g DT
diameter [m] gravity acceleration [m/s2] column diameter [m]
e o
f
interaction force among phases [N/m 3]
CD drag coefficient U
velocity [m/s]
volume fraction surface tension [N/m]
p
density [kg/m3]
gas large bubbles regime transition point
1 sb 0
liquid small bubbles standard conditions
Subscripts g lb trans
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
385
Simulation of silica deposition in an Atmospheric Pressure Chemical Vapour Deposition reactor, using a modified CFD software J.P. Nieto ab, B.
C a u s s a t a,
J.P. Couderc a, C. ArtufeP, S. Coletti b, L. Jeannerot b and O. Simonin c
aLGC/ENSIGC UMR CNRS 5503, 18 chemin de la loge, 31078 Toulouse Cedex 04, France E-mail : JeanPierre.Nieto@ensigct. fr bATMEL ES2, Zone industrielle, 13106 Rousset Cedex, France Clnstitut de Mrcanique des Fluides, Allre du Pr. Camille Soula, 31400 Toulouse, France The deposition of silicon dioxide in an Atmospheric Pressure Chemical Vapour Deposition reactor from TEOS (tetraethoxysilane) and ozone mixtures has been studied and modelled, with the objective to optimise the productivity of an industrial equipment. ESTET, a French commercial CFD software, has been used to solve the hydrodynamic and heat transport problems. A subroutine has been developed to treat the mass transport and chemical reactions phenomena, both in the gas phase and on surfaces. A critical point being the stiffness of the chemical system, a specific model of the boundary layer type, has been developed to treat conveniently the numerical consequences of fast surface reactions, at the immediate vicinity of the substrate. Finally, the model is able to predict gas velocity, temperature and concentration profiles, then the deposition rate variations on the substrate surface ; its systematic use has demonstrated that several regimes, corresponding to several rate limiting phenomena (species diffusion or chemical kinetics) must be distinguished. The results obtained are discussed and compared with experimental data. 1. INTRODUCTION Silicon dioxide doped or undoped films, are commonly used as premetal or interlevel dielectric layers in the microelectronic industry. They are increasingly elaborated by CVD from TEOS-ozone mixtures, because this route offers important advantages over the conventional silane-oxygen process. Moreover, Atmospheric Pressure Chemical Vapour Deposition (APCVD) processes are developing due to several interesting advantages : particularly, using continuous reactors leads to an higher productivity compared to batch reactors, and the absence of vacuum conditions decreases the technological complexity compared to Low Pressure CVD [1 ]. However, the phenomena taking place in such CVD reactors are complex, with momentum, heat and mass transfers coupled with homogeneous and heterogeneous reactions. Modelling appears then very interesting to provide a better insight into all the phenomena involved, and also as an efficient predictive tool, to optimise the process operating conditions [2][3].
386
2. THE APCVD REACTOR The reactor which has been analysed in this work is an industrial APCVD WatkinsJohnson WJ1000 system, fed with TEOS, ozone and oxygen mixtures, highly diluted in nitrogen. As illustrated in figure l a, the reactor is constituted of four deposition chambers separated by nitrogen curtains. The 200mm-in-diameter wafers are horizontally transported through the reactor by a continuously-moving belt, cleaned in a fluorhydric acid vapour bath after the downloading of the wafers. In each chamber, a gas injector delivers three different gas mixtures, organised into five thin sheaths, as shown in figure lb. The centre channel (inner port) injects TEOS diluted in nitrogen, the intermediate channels (separator ports) nitrogen, and the two outermost sheaths (outer ports) ozonated oxygen. The TEOS flow comes from a conventional temperature-controlled nitrogen bubbler maintained at 65~ The ozone flow is obtained by passing oxygen through an ozonator. exh
9
ust
i--outer p o r t - I
] l-separator-I I
vent shield injector shield
Figure la : Schematic view of the WJ1000 reactor
curtain
T
T
wafer
moving belt
Figure lb : Detail of one deposition chamber
3. THE M O D E L
Several reasonable assumptions have been made to treat this CVD problem. For symmetry reasons, only one half of a single chamber has been considered. The belt and wafers have been supposed motionless (their velocity is far lower than that of gases). Flow has been assumed to be laminar and incompressible, reacting gases are heavily diluted in nitrogen, phenomena are calculated in two dimensions, and steady state is assumed. The classical continuity, movement and heat set of equations is used to describe the system. The complete presentation of the corresponding equations has already been done elsewhere [8]. inlet : T=70~ T=400~ curtain) ~ _0u _ symmetry" u=c~ ~ 0 z 0P 0T ~ Xk=c~ .......................................... d•
9
dx
outlet: P= 1atm 0T _ 0X k 0z 0z
[llflHIllll]lllNll][ll[llHl[I]i~lllIH]l X
•M•••••••[•••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••11111••••••••••••• •••••••••••••]•]••H•H•••••••••]••]]••j•••••••H••ij•••$••••••$••i$i$•••$••••••]••••H•••i•[•H••••••• IIIIII|III ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••I•I•I•I•I•I•I•I•I•I•I•I••I•I•I•I•I•I•••••••••••••
-~.'L~jt~1]~~'~~~ OUz ; 0x k = 0 ' - ~...................................................... dx dx wafer: T=500~ ; u~= u~=0 Ux soecies flux=surface reaction rate Figure 2 : 130x26 nodes computational grid, with boundary conditions
z(3)~ Z(12~
~ w.... . . .......... ,.~:~: .,..."~C2 7
- D kn
/
Surface
/
/
C 1-Cw = surface reaction rate Z(1)/2
Figure 3 : Schematic view of the concentration profile in the boundary layer (Dkn is the diffusion coefficient for species k in the carrier gas, nitrogen in our case.)
387 A French Computational Fluid Dynamics software called ESTET has been used. This software performs the calculation of hydrodynamics coupled with heat transfer by using a combination of finite difference and finite volume methods in a half-staggered computational grid with 130• nodes as shown in figure 2 in which the boundary conditions selected are also mentioned [5][6]. A subroutine has been added to treat mass transport, with source terms corresponding to homogeneous and heterogeneous reactions. The major difficulty of this modelling work concerns chemistry. The number of reactions and species which must be considered in such a TEOS-ozone system is very high (more than one hundred reactions), which would result in very important computation times. In this analysis, chemistry has been simplified and an apparent chemical model has been used in order to keep reasonable computation times. It consists in five gas phase and two surface reactions, as described in table 1, following a proposal recently made in the literature by Zhou et al. [4]. Table 1 : Chemical TEOS/O~ mechanism [4] (Units are m, kmol, s, K) Reaction step (R1) (R2) Gas-phase (R3) reaction (R4) (R5) Surface ( R 6 ) reaction (R7)
03 + M --~ 02 -'b O +M 03 + O ~ 2 02 2 0 + M ~ 02 nt- M 03 + TEOS + M --~ INT + R + M INT ~ products TEOS+
603 ---~SiO2(s)+10H20+8CO+O 2
INT+X--~ INT-X --~SiO2(s)+X+products
Reaction rate 2.5 10'1 exp(-11430/T)[O3][M] 101~exp(-2090/T) [O31101 4 108 exp(+V20/T) [O]2[M] 4 10'Texp(-14099/T) [O3][TEOS][M] 105 exp(-5539/T) [INT] 200 [TEOSlw[O3lw~ 20 exp(-4053/Tw) [INT]w 1+ 1.14107 exp(-2578/Tw)[INT~v
In its original form, ESTET calculates the source terms corresponding to surface reactions, with the gas phase concentration of the depositing species, at the closest point near the surface (i.e. C1 in figure 3). This assumption appears convenient to treat problems in which the surface reaction rates are slow, which means that the concentration profiles are not very stiff at the surface ; thus, the error resulting from the use of C1 as the wall concentration is small. In the case considered here, the (R6) heterogeneous reaction is extremely rapid, and using the previous assumption has generated important numerical problems, due to the stiffness of the concentration profiles for 03 and TEOS at the surface. These problems have been suppressed by developing a specific numerical model of the boundary layer type which calculates accurately the wall concentration of the surface reacting species. As represented in figure 3, the surface reaction rates for the involved species (i.e. 03 and TEOS for R6, and INT for R7) are assumed to be equal to the diffusive flux in the boundary layer near the deposition surface. The resolution of the equations obtained for this first order boundary condition leads to a good estimation of the real wall concentrations Cw. In its final form, the complete model requires approximately 12 hours of CPU using a Hewlett-Packard K260/EG computer. 4. S I M U L A T I O N
RESULTS
Simulation provides a very large number of results, in terms of gas velocity, temperature, concentration profiles in the gas phase, and deposition rates, everywhere in the deposition chamber [8]. Figure 4 describes the flow field (depicted as vector arrows). Flow can be
388
described as a high speed jet impinging on the wafer surface. No recirculation region have been observed, whereas they form when the nitrogen flow through the shield is suppressed. The role of the shield appears then essential in avoiding recirculations, responsible for long residence times of gases, possible source of undesirable homogeneous nucleation and particles formation. The shield nitrogen flow also reduces strongly the risk of deposition on the upper walls of the chamber. The temperature field in figure 5 shows that the gas stream from the injector remains cool until it reaches the vicinity of the hot wafer surface, where it increases abruptly from 70~ to 500~ The thermal boundary layer underneath the injector appears to be very thin, of the order of 3mm in thickness. For the mass transfer and chemical reactions phenomena, the first simulations which have been performed, using the rate constants values proposed by Zhou et al. [4], have produced results which reproduced conveniently the major features of the experimental data. However, several discrepancies demonstrated that some recalibration of the apparent kinetic constants was necessary.
Figure 4 ' Velocities depicted as vector arrows
That part of our work has required a lot of efforts. A first observation has been made on (R6) rate equation. It rapidly appeared that, in the form proposed by Zhou et al. [4], this reaction was very rapid and played a negligible kinetic role ; in fact, the corresponding deposition rate was controlled either by the flux of ozone or by the flux of TEOS towards the surface and even large changes of the kinetic constant produced no noticeable variations of the deposition rate. Moreover, a series of experiments demonstrated clearly that the experimental results always depend on the TEOS flow rate, at least in the conditions covered in that work, as shown in figure 6 ; this experimental observation then excluded a limitation by the flux of ozone. 400 300
c "-
9~O
200
r~ ' E
O ~e" "--~
~
t-
--- 100
og .Q
0
i
0,5
i
,
1 1,5 N2 flow in the TEOS bubbler (slpm)
Figure 6 9Dependence of deposition rate on TEOS flow rate
i
2
389
As a consequence, two sets of kinetic constants have been selected by two long trial and error procedures, with which numerical simulation reproduces quite conveniently experimental data. The corresponding final results are presented on figures 7 and 8. 1
.....
~
0,8
E 8 0,8
~
0,6
0,6
8 r
0,4
~ 0,2 -I-
0,002
0,000
0,004 03 mass
0,006
0,008
0,010
0,2 o~ "1- 0 0,000
0,002
0,004
fraction
0,006
0,008
0,010
0 3 mass fraction
1
1 E
go,8 8
f
0,4
E
'0,8
0.6 0,4
~_ 0,2 o
. _
f: o
f
r= 0,6
I
$
~ 0,4 E ~ 0,2 .1= I
0,000
0,005
0,010
TEOS mass
0,015
0,020
0
0,000
0,005
fraction
0,010 TEOS mass fraction
0,015
0,020
(a)(b) Regime 2 9Limitation by diffusion of TEOS (c)(d) Regime 3 9Limitation by the kinetics of (R6) Constants recalibrated 93.5 k4;0.1 k5;0.15 k7;0.53 k8 Constantsrecalibrated: 0.00275 k6;0.1 k5;0.15 k7;0.53 k8 Figure 7 9Concentration profiles (beneath the injector) for 03 and TEOS on the deposition surface
4oo
~ r-
300
(a)
\ \ k J
(R6)+(R7)
--
400
~300
Experimental static print
(b) ~ ( R
--Experimental static - - - Model prediction
6)+(R7)
o
g200
print
200
100
....
~ 100 ""...... ~ . ( . R 6 !
0
10
: TE?S c~176 . 20
.
.
. 30
Distance from injector centre
~ .
. 40 . (mm)
"". ~ ( R 6 ) 50
0
0
10
: TEOS contribution 20
30
Distance from injector centre
(mm)
40
50
Figure 8 9Comparison of experiment and model in both regimes (a) Regime 2 9Limitation by diffusion of TEOS (b) Regime 3 9Limitation by the kinetics of (R6) Constants recalibrated 93.5 k4;0.1 ks;0.15 k7;0.53 ks Constantsrecalibrated 90.00275 k6;0.1 ks;0.15 k7;0.53 k8 Let us observe that the first mechanism, with a limitation by the flux of TEOS is not able to reproduce conveniently the deposition rate peak just beneath the gas injectors. The agreement is far better with the second mechanism, which involves a chemical limitation by reaction (R6), but that has been obtained only after a considerable change of the corresponding constant ; at the present time, we have no satisfactory explanation, except that the chemical species, reactions and kinetics are only apparent.
390 Work is presently in progress, with new series of experiments to better determine which mechanism and which set of constants will be able to represent the real behaviour in the largest possible range of operating conditions. 5. CONCLUSION Modelling of that complex CVD process in a continuous atmospheric pressure reactor is a difficult task. It is necessary to determine, at the same time, the pressure, velocity, temperature and concentration profiles, with several very rapid chemical reactions. That could be done using a conventional CFD software, but it has been necessary to add several subroutines, first to treat chemistry then to calculate exact values of surface concentrations. An important effort of recalibration of chemical constants has then been necessary ; two different sets of results have been obtained which produce quite convenient results. Discrimination between them will need additional experimental data. After this step, the model will be used first to optimise the operating conditions, then if necessary, to redesign the equipment. ACKNOWLEDGEMENT
Authors would like to thank ATMEL Corporation for financial support and experimental results. REFERENCES
[ 1] Masi M., Carra S., Vaccari G., Crippa D., "Optimization of S i O 2 atmospheric deposition in continuous belt systems", Proc.of the 14th Int. Conf. and EUROCVD 11 on CVD, Paris 5-9 september 1997, M.D. Allendorf and C. Bernard Eds. pp. 1167-1174 [2] Kim E.J., Gill W.N., "Modeling of CVD of silicon dioxide using TEOS and ozone in a single-wafer reactor", J.Electrochem.Soc. 141 [ 12] (1994) 3462-3472. [3] Dobkin D.M., Mokhtari S., Schmidt M., Pant A., Robinson L., Sherman A., "Mechanisms of deposition of SiO2 from TEOS and related organosilicon compounds and ozone", J.Electrochem.Soc. 14217] (1995) 2332-2340. [4] Zhou N., Krishnan A., Kudriavtsev V., Brichko Y., "Numerical study of TEOS/O3 CVD mechanism in an industrial reactor", 5th Int.Cons Advanced Thermal Processing of Semiconductors, New Orleans 3-5 september 1997, RTP'97, pp. 257-268 [5] Soukane S., Duverneuil P., "Hydrodynamics and mass transfer modeling for RTP reactors", Proc.ofthe 14th Int.ConEand EUROCVD11 on CVD, Paris 5-9 september 1997, M.D. Allendorf and C.Bernard Eds. pp. 238-245 [6] De Paola E., Duverneuil P., "Simulation of silicon deposition from SiHC13 in CVD barrel reactor at atmospheric pressure", Comp.and Chem.Eng. 22 (1998) suppl.S683-S686. [7] Reid R.C., Prausnitz J.M., Poling B.E., The properties of gases and liquids (McGraw-Hill, 1987) pp.581-589. [8] Nieto J.P., Caussat B., Couderc J.P., Coletti S., Jeannerot L., "Modeling of S i O 2 deposition from mixtures of tetraethoxysilane and ozone in an APCVD industrial reactor", Proc. of the 12th European Conference on Chemical Vapour Deposition, Barcelona 5-10 september 1999, pp.149-155
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
391
Validation of a CFD model of a novel recycle axial flow cyclone for droplets removal from gas streams 19. Stanbridge and R. Swanborn CDS ENGINEERING Sonsbeekweg 26 NL-6814 BC Amhem
C. P. Heijckers* and Z Olujic ~ TU Delft, Laboratory for Process Equipment Leeghwaterstraat 44 NL-2628 CA Delft
Abstract During the development of a new generation of recycle flow axial cyclones for droplet separation a computational fluid dynamics model was used to simulate the complex turbulent gas flow field downstream the swirl element. The Fluent 5 code was used in conjunction with the Reynolds Stress Model (RSM), which is generally considered to be the most accurate tool for solving rotating flow problems. Model predictions were compared with measured axial and tangential velocity profiles. Agreement proved to be reasonable for the axial profile except for locations around the centre of cyclone, where the outlet of the recycle flow tube is placed. The reason for this and other model related causes for the observed inaccuracies are discussed. Keywords: Demisters, Axial flow cyclone, Computational Fluid Dynamics, Model validation
Introduction Separation of droplets from gas/vapour streams is a widely encountered operation in process and related industries, particularly important in the offshore processing of natural gas. Traditionally inertial type demisters have been used for this purpose including mesh pads, vane (zigzag) plates and conventional cyclones 1-4. Compact recycle flow axial cyclones are a relatively new development3'5 which regarding the capacity and efficiency increases moved the technology of droplet removal devices to a higher level. In other words, in comparison to present designs these high performance devices enable both a considerable separator vessel size reduction in new designs and a significant capacity increase in retrofit situations at the same or even better separation efficiency. A detailed description of a recently introduced commercial version of an axial cyclone with recycle stream can be found elsewhere5. In the course of the development work on this cyclone, CFD modelling proved to be a valuable aid, however more to indicate the direction of work than to arrive at final designs. The latter was settled upon after a time consuming experimental effort. Fortunately CFD has recently become powerful enough to model the full extent of the complex swirling flow patterns within the cyclone. This encouraged further effort toward a thorough computational analysis and optimisation of the performance/design of the recycle axial flow cyclone. This, as demonstrated in this paper, has to be accompanied by a limited but necessary amount of experimental work arranged to validate and refine *Now with CDS Engineering w author: [email protected], Phone: +31 15 2786674, Fax: +31 15 2786975
392 properly the model employed. A CFD model must prove its value as process analysis tool before one considers its application as a predictive tool.
Physical Background Within the present oil and gas processing industry, particularly in offshore production, there is a drive to develop more compact droplet separation equipment. In this way more capacity can be processed through existing facilities and new items can be made more compact. To accomplish this more use is being made of cyclonic technology of which Figure 1 shows a side view of the so-called AXI 50 cyclone of CDS Engineering. The heart of the device is the swirl element (2) that induces a swirling motion on the mist flow entering the cyclone (5). Due to rotational flow component imposed by the swirl element liquid droplets hit the walls (3) forming a film that is discharged into liquid drainage compartment through slits in the downstream part of the cyclone walls. Droplet free gas leaves through the outlet where a ring is located (1) to prevent re-entrainment of the liquid film. The flow recycle pipe (4) connects the liquid drainage compartment with the top end (vortex finder) of the swirl element body, i.e. the centre of swirling flow. Due to the high velocities the static pressure is lower at this point than that at cyclone walls and in the liquid drainage compartment, leading to induction and maintenance of a constant recycle flow (purge gas), which in turn forces the liquid collected at cyclone walls to drain through slits. This is a special performance-enhancing feature of this device that can be utilised in an effective way only if the design of the device is fine-tuned. Among other things this implies a detailed knowledge of the complex highly turbulent gas flow field in the cyclone and this appeared to be a proper subject for utilisation of full potential of a state of the art CFD modelling.
Mathematical Modelling A velocity vector at a point in a swirling or rotating flow in cylindrical co-ordinates system can be divided into an axial, radial and tangential component. The tangential velocity determines the driving force for separation. The axial velocity determines the residence time of a particle in the cyclone. The radial velocity is usually relatively smaller than other components, however its presence may be utilised as mentioned above to enhance liquid drainage through slits in cyclone tube walls downstream the swirl element. According to the experience, free and forced vortex flows are usually encountered in a cyclone as well as a combination of these two, called the Rankine vortex. The latter one shown in Fig. 2 is defined as w ( r ) = A . r . e (-B'~)
where w t is tangential velocity and r is radius. A and B are empirical constants depending upon the geometry of cyclone. The extent of swirling in a flow is usually characterised by the so-called swirl number S, which is defined as ratio of the fluxes of angular and axial momentum. The definition of S along with those of the angular M o and axial Mx momentum are described below where R o is the radius of the cyclone wall: ,%
S
= M~ ~
Mx "Ro
Mo=2.x.pg,
u.w. r2dr o
Typically, the swirl number is less than 1.
,%
M~=2.x.pg.
u 2.rdr o
393 The rotating flow in a CFD model is usually described using fundamental mass and momentum balance equations in conjunction with cylindrical coordinates. These equations can subsequently be used in conjunction with turbulent flow by adopting the so-called "Reynolds stresses" in the Navier-Stokes equations, which incorporate the effects of turbulence. The widely used k-e turbulence model assumes that the Reynolds stresses are proportional to the mean velocity gradients and that turbulent viscosity is isotropic, i.e. same in all directions. This is not the case in swirling flows where velocities vary considerably with direction. Therefore the Reynolds Stress Model (RSM), which takes into account the anisotropy of the turbulence, was employed in this study. Grid Computational fluid dynamics modelling basically involves definition of the physical geometry, generation of volume elements within this definition and execution of the appropriate calculations on these volume elements. In view of the cyclone geometry the "unstructured", body fitted tetrahedral approach is chosen. A picture of the simulated configuration is given in Figure 3. Due to the high mesh density, required for calculation of the high swirling flow and the limited computational capacity, the simulation of cyclone operation was carried out in two subsequent parts: (i) cyclone inlet and swirl element and (ii) vortex finder, separation section, cyclone outlet, liquid drainage compartment and recycle tube. Both simulation parts are related such that the velocity profile downstream of the swirl element obtained from calculations of part (i) is applied as the inlet velocity, profile of part (ii). The grid consists of a total of 114000 volume elements for part (i) and 87000 volume elements for part (ii).
Boundary conditions Model boundaries are the cyclone inlet, walls of liquid drainage compartment and outlet box. This implies that for the inlet, effects of the entry geometry on the flow in this area are not taken into account. On the outlet side, the model has been extended to account for possible outlet flow effects on cyclone operation. _.
J tcl~
Gas Out
Ranklne V o r t e x ,
i"'ud:;~*~:~
i Radial position
Figure 1: Internal configuration of an axial cyclone with a recycle flow tube
Figure 2: Rankine- vortex flow profile
394 The inlet velocity profile of part (i) is a constant axial velocity profile. This velocity is determined from the desired gas flow rate and the cross sectional area of the cyclone tube. The inlet velocity profile for part (ii) is the velocity profile from part (i) at the cross section located 0.055 m downstream the swirl element body. This location is chosen because it is not affected by the recycle flow thereby ensuring that the cyclone model can be split into two without any special considerations being required in this regard. The inlet and outlet conditions also require definition of initial turbulence parameters. The turbulence intensity at the inlet and outlet is set to 0.05, which is a common value. The turbulence length scale is set to the equivalent radius of the inlet and outlet areas. These values are used by Fluent to derive the inlet turbulence kinetic energy and dissipation rate. Turbulence intensity will have little effect on the downstream flow profile, as the turbulence effects incorporated in the Figure 3: Views of the cyclone CFD model Reynolds Stress Model (RSM) will dominate 6. The surfaces of cyclone walls and swirl element are hydraulically smooth; therefore the absolute roughness was set to zero in all simulations.
Experimental set-up In order to validate model predictions a simple experiment was designed and carried out using a simple experimental set-up. Ambient air was used as test system, supplied at a constant flow rate from a blower. Axial and tangential gas flow velocities were measured accurately downstream the swirl body at cross sections corresponding with the beginning of the slits (HI), halfway along the slit length (H2) and close to the cyclone outlet (H3). There were 27 measuring points along each of three cyclone cross sections. A specially designed micro-pitottube was used containing only one small hole (0.3 mm) in micro tube wall. This tube can be rotated by 360 that allows determination of points of static pressure and total pressure, depending on the hole position angle. From the pressure differences measured at appropriate hole inclination angles velocity profiles were obtained and presented as a function of radial position. Results and Discussion Figure 4 shows a comparison of the CFD predicted and the measured axial and tangential velocity profiles at three cross sections downstream the swirl element. Measured tangential profiles resemble the Rankine vortex shown in Fig. 2. Tangential velocity profiles measured at H1 and H2 indicate a rather small decay of the swirl, i.e. there is relatively little dissipation as a result of internal motion and friction at walls and slits. Furthermore, for all heights, the tangential velocity is approximately zero in the centre of the cyclone. Towards the outlet, the maximum tangential velocity is located more inwards. This is a favourable situation as it results in higher centrifugal forces in the centre of the cyclone. Axial velocity, profiles at H1 and H2 indicate fairly constant axial velocities, except for the
395
central zone. Here, the axial velocities are significantly higher due to contribution of recycle flow entering the cyclone through the centrally placed vortex finder. eo
HI
Z~ 4,
~ 30
.~,~.
9
9
9 *
1 lO o -30
-20
-10
0
lO
Radial Posllon [mm]
I .......
...........
~ ~'~-~'-~- . . . . .
" ,1 20
30
-20
,~l~-m- =.,m_ w
-10
lO
0
20
P ~ d t i a l [ram]
"~
4O
H2
4
9 * * r e
* ~,**t
*
o
i
. ~9
-~o
.lO
o
lO
20
-20
Ft.adi~,l P ~ t l o n [ram]
80
-lO
o
1o
20
30
Poa/rUon [ram]
,
60 S0
1,0 H3
9
+
mrJ l P m
r
9
m-t-~-~
10
. -;~io
-20
-10
.
.
o
.
0
10
20
.30
-20
-10
0
10
I~ldlal P o t i o n [mm] i
. . . .
20
t 30
]
Figure 4." Comparison of predicted and measured axial and tangential velocity profiles
Axial profiles produced for the same conditions by the CFD model agree well with the measured one except in the central part corresponding with the location of vortex finder. The CFD model indicates an increase in the axial velocity that is much lower than that observed. The tangential velocity profile shows rapid decay of the swirl over the length of the cyclone. At H1, the trend in the predicted tangential profile is con-ect, however local velocities deviate considerably. Downstream, tangential velocities change to a constant value over the cross section H3. This is far from the actual situation. In other words, a rather rapid decay of the swirl indicates a too high turbulence dissipation rate. Namely, a high dissipation rate results in high internal shear forces, which has a dampening effect on the tangential velocity profile. The tangential velocity profile mainly determines the static pressure profile, which will therefore also appear as a rather flat profile. This, together with very high dissipation rate in the recycle flow itself explains why the recycle flow is not sufficiently visible in the axial velocity profiles. Another indication of the turbulence dissipation being incorrect and thus causing the mismatch is indicated by settings of the discretization scheme. Namely, setting the turbulence dissipation discretization scheme to second order windup for the RSM leads to divergence during solving. Divergence even occurs at under-relaxation factors as low as 0.05. Solution is
396 achieved with first order windup for turbulence dissipation, however at the cost of a decreased accuracy of the extent of turbulence dissipation. The origin of the high turbulence dissipation rate is most probably in inlet boundary conditions. Here, the turbulence intensity and length scale were set at 0.05 and 0.021, respectively, with the thought that this isotropic turbulence specification will have negligible effect on the downstream cells in the model. However, there is little turbulence production downstream of the swirl element. Therefore, the calculations which lead to a solution, i.e. equilibrium of the flow will be dominated be the inlet boundary condition. This affects the pressure profiles and the recycle flow as indicated above. Similarly, at the pressure outlet, isotropic turbulence is assumed. However, as the outlet boundary conditions are located downstream and far away from the cyclone, this will have negligible effect on the profiles. In an attempt to improve the results, the outlet side turbulence parameters from part (i) were applied to the inlet boundary of part (ii), similar to the velocities. This ensured the convergence, but did not lead to improved accuracy. Conclusions
Experiments have been carried out to validate a CFD model used to simulate the single phase gas flow field in a prototype of a new recycle flow axial cyclone. The commercially available Fluent 5 code in conjunction with the RSM turbulence model and appropriate boundary. conditions produced axial and tangential velocity profiles that partly deviate considerably from observed ones. Further model refinement work is needed to improve predictive accuracy of the model. More realistic approach will be obtained by simulating the cyclone in one piece in conjunction with a fine hexahedral mesh. This is a prerequisite for next, application oriented simulation step, the incorporation of liquid phase. Nomenclature
M r, R u v w x
momentum, kgI~s 2 radius, m axial velocity, m/s radial velocity, m/s tangential velocity, m/s distance in x direction, m
Subscripts 0 ref. to cyclone wall g gas r radial t tangential x axial 0 angular
Greek letters p density, kg/m 3 0 angle, rad
References
1. Burkholz, A., Droplet Separation, VCH Verlag, Weinheim, 1989. 2. Verlan, C.C.J., Olujic, Z. and de Graauw, J.; Performance Evaluation of Impingement Gas/Liquid Separators, Proc. of 4 th Int. Conf. on Multi-Phase Flow, Nice, 19-21 June, paper C2. 3. Swanborn, R., A New Approach to the Design of Gas-Liquid Separators for the Oil Industry, Dissertation, Delft University of Technology, 1988. 4. Verlaan, C. C. J., Performance of Novel Mist Eliminators, Dissertation, Delft University of Technology, 1991. 5. Stanbridge, D., Swanborn, R. and Olujic, Z., A Novel Recycle Axial Flow Cyclone with Strongly Improved Characteristics for High-Pressure and High-Throughput Operation, Proc. o f 9th Int. Conf. Multiphase 99, BHR Group Conference Series Publication No. 35, Professional Engineering Publishing Ltd, Bury St Edmunds and London, 1999, p. 555. 6. Fluent Manual, Release 4.3, Fluent. Inc., Lebanon, NH, USA, 1993.
European Symposiumon ComputerAided ProcessEngineering- l0 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
397
SIMULATING FLOW AND HEAT TRANSFER IN TUBES USING A FAST CFD FORMULATION Mercado, E. R. L.; Souza, V. C.; Guirardello, R. and Nunhez, J. R.* Faculdade de Engenharia Q u f m i c a - UNICAMP - CP 6066 Campinas - SP - Brazil - 13083-970 - e-mail: [email protected] A new approach to study turbulent flow and conjugate heat transfer in tubes is proposed in this work. Instead of using the conventional finite element or finite volume methods, this formulation applies a different technique that calculates both for the flow and heat transfer. It discretizes the flow in the radial direction using a 4th order finite differences method, which is more accurate than the traditional 2nd order schemes. Using this technique, a system composed of several ordinary differential equations for the temperature and a set of linear equations for the velocities and pressure gradient is obtained. The equations are then integrated in the axial direction using a 4th order Runge-Kutta method. The values of viscosity, density and thermal conductivity are dependent on temperature, which makes the model suitable for the calculation of high temperature gradients, as in the case of refinery fired heaters. The turbulence is taken into account using a zero order turbulence model. 1. INTRODUCTION The heating and cooling inside tubes has been among the most important processes in the engineering field such as petrochemical fired heaters and petrol cracking. The applications are innumerous. The modeling of these processes, based on the conservation of mass, momentum and energy associated with its boundary conditions, normally lead to a set of partial differential equations with no analytical solution. Many details of the flow cannot be captured by experimental analysis, therefore numerical procedures are needed to have a deeper understanding of these processes. Travelho and Dias (1984) have developed a model to solve the energy equation in tubes under laminar flow. The fluid is incompressible and it is assumed a parabolic profile for the axial velocity. The fluid properties are constant and the work is primarily concerned to analyze axial conduction at the tube walls. Barozzi and Pagliarini (1985) developed a method combining the finite element method to the superposition principle to solve simultaneously the momentum and energy equations in order to analyze axial conduction at the walls. The physical properties are also assumed to be constant. Martinuzzi and Pollard (1989) compared six turbulence models in tubes and interesting details about the numerical methods were explained. They arrived at the conclusion that the low Reynolds number k-r model better predicts turbulence in tubes. However, physical properties are also constant. * author to whom all correspondence should be addressed
398 In order to get more insight of how these systems behave, a new computational fluid dynamics model has been developed to calculate the velocities, pressure and temperature. The flow is assumed to be turbulent. Also, as the fluid properties are dependent on temperature, the model suitable to investigate the flow inside petrochemical fired heaters. The hypothesis of the model are: 1- The tube wall temperature is assumed to be constant, which is acceptable for the design of fired heaters (Wimpress, 1963); 2- The flow is steady-state; 3- Radial velocities are negligible in comparison to the axial velocity; 4- The flow is symmetric about the axial axis 5- Physical properties are dependant on temperature; 6- There is no phase change and the fluid is a Newtonian liquid. 2. M O D E L I N G The model equations for tubes under the hypotheses described above are well known and can be found in textbooks such as (Bird, 1982). A study of the order of magnitude was applied to the system and several terms of the equations are negligible in comparison to others. The governing equations for the turbulent axi-symmetric model are: 2.1. T U R B U L E N T F L O W 2.1.1. M O M E N T U M C O N S E R V A T I O N : Radial direction 0P
=
az
(1)
0
Axial direction 0
=
aP
az
+ -1. a r.(lz+/z,). r -~r ar )
(2)
2.1.2.ENERGY CONSERVATION
^-
p'Cp'vz
.
c~T . . c3Z
.
lc9 I (k .~t_kt ) r l r ~r r. 9
(3)
2.1.3. C O N T I N U I T Y E Q U A T I O N 1
a
7
ar
+ c3
:
(4)
2.2. B O U N D A R Y C O N D I T I O N S
Symmetry line ( r = 0 ) The axial velocity and temperature do not vary in relation to the radius:
ar)
(5)
399
(6) Tube wall ( r = R ) vz =0
(7)
-~" -~-r - - h . ( L - r )
(8)
Tube entrance ( z = 0 ) T = To
(9)
Mass conservation ( 0 _< z < L ) R
co = j p .
0
vz .
2. ~r. r dr
(10)
In order to evaluate the turbulent viscosity and thermal conductivity the mixing length hypothesis is assumed (Prandtl theory). The mixing lenght theory assumes that: Turbulent viscosity:
JUt =p'12 ] ar I
(11)
Turbulent Thermal conductivity: kt=p.Cp.l
2.
(12)
Or
In order to evaluate ~t e k t, it is necessary to know the mixing length ( 1 m ). There are several expressions for tubes. This work uses the expression by ( Rodi, 1984 ) 9
lm = 0 . 1 4 - 0.08 IR! R
2 - 0 . 0 6 (R! 4
(13)
2.3. NUMERICAL APPROXIMATION The fourth order finite differences method is applied to the radial direction. This leads to a linear set of equations for the velocities and for the pressure drop and a set of first order differential equations for the temperature. The discretization in the radial direction is applied to (m+ 1) points and (n+ 1) points in the axial direction. 2.3.1Turbulent flow From the continuity equations and the assumptions made, P = P(z), the following is true: aP
= K 0z From Equation (6); after integrating 9
(14)
400
r"r= Eri. -
,o
9
9
9Or)
+/z.
+
k, d r )
~
k, d r )
=
Or]
(15)
0
2
Therefore the velocities can be calculated by the following relation, which is the negative root of equation (22) c3v z
_
(17)
4/z~ + 2 K i r~ Pi f ( r ) i - k t i
Or r=i
--2 Pi f (r)i
Where: f(r)
= R 2-
0.14-0.08
(18)
-0.06
The energy equation is discretized as follows"
I
(~T Fi ~z . . . . - " r=i pi Cpi vzi
{/
Ill
( ki + kt ) t~ k O kt ] OT (ki + kt ) _ i + + . + . ri ~ r r=i -~r lr=i -~r r=i i
The continuity equation: Pi . r i + r i. c r ( P " Vr
(20)
= v z ( p " Vz
r=i
(19)
OZ21r=i
z=i
For the laminar case a uniform temperature profile is given at the tube entrance. From this, all physical properties are calculated. The axial velocities at different radial positions are calculated from the momentum equation in the axial direction. From the energy conservation a set of first order differential equations are obtained for the temperature which, in turn, is calculated using a fourth order Runge Kutta method. The procedure is repeated for the whole length of the tube. The radial velocities are then calculated using the continuity equation. For the turbulent case, as for the laminar case, a uniform temperature profile is given at the tube entrance. The physical properties are estimated and a set of linear equations are obtained for the velocities according to Equation (17). However, as the value of K is not known, a guess is made and the values are corrected using Equation (10). The procedure is repeated until convergence for the values of K is obtained. 3. RESULTS AND DISCUSSION All results are based on a case study of a tube with radius r = 0,1 [m]; wall temperature Tw = 90 [~ and temperature of entrance Te = 30 [~ In order to make the comparisons, a residence time of 2000 seconds was used for the two conditions (laminar and turbulent). Since the temperature varies, it was set a reference temperature equals to the average between Te and Tw, for the calculation of the Reynolds number. Figure 1 shows the variation of the temperature in the axial direction for Reynolds number of 2000, which is a laminar flow. Figure 2 shows the variation of the temperature in the axial direction for Reynolds number of 12000, which is turbulent flow. As expected, heat transfer is improved for the turbulent flow. Figures 3 and 4 shows axial velocity for the
401
laminar and turbulent cases referred above for a distance to half of the total length of the tube. As expected, the turbulent profile is flatter when compared to the profile of the laminar flow. Figures 5 and 6 show the radial velocities calculated for the laminar and turbulent flow. As expected, for these flow conditions, the radial velocities are negligible. The computational time spent for the laminar cases was about 1 second, whereas it took about 2 seconds for the turbulent case, using a Pentium II for the calculations. The new method is very fast.
Figural. Laminar profile temperature Re=2 000 Figura 2. Turbulent profile temperature Re=l 2 000
Figure 3. Axial velocity Re=2 000, z=5m
Figure 5. Radial velocity Re=2 000
Figure 4. Axial velocity Re=l 2 000, z=30m
Figure 6. Radial velocity Re=12 000
402
4. C O N C L U D I N G R E M A R K S A new and fast CFD method which is able to calculate conjugate heat transfer in tubes with physical properties depending on temperature is presented in this paper. The method is particularly suitable for the prediction of temperature and velocities in petrochemical fired heaters, where a considerable temperature gradient is present. An extension for the k-e model is under way since it is reported in the literature it is more suitable than the zero order turbulence models. 5. N O M E N C L A T U R E 6"p
- fluid thermal capacity
~
- time average fluid temperature
DP/dz
- pressure drop
Tw
- wall temperature
h k kt
- heat transfer coefficient - thermal conductivity - turbulent conductivity
v0 v Vr
-
K
- constant = - ( d P / dz ) - mixing length - tube length - pressure - time average pressure - radial position - tube radius - temperatura do fluido
vz Vz z Ar Az /~ fit P
-
1m
L p r R T
r
- angular velocity radial velocity - time average radial velocity axial velocity time average axial velocity axial position radial distance axial distance fluid viscosity turbulent viscosity fluid density
ACKNOWLEDGEMENTS The authors thank CNPq and F A P E S P for the grants received for this project. REFERENCES
1. Barozzi, G. S. , Pagliarini, G. (1985), "A method to solve conjugate heat transfer problems: The case of fully developed laminar flow in a pipe", Journal of Heat Transfer, v. 107, p.77-83. 2. Bilir, Sefik. (1994), "Laminar flow heat transfer in pipes including two-dimensional wall and fluid axial conduction", International journal of Heat and Mass Transfer, v.38, n. 11, p.1619-1625. 3. Faghri, M . , Sparrow, E. M. (1980), "Simultaneous and fluid axial conduction in laminarflow heat transfer", Journal of Heat Transfer, v. 102, p.58-63. 4. Travello, J. , Dias, L. A. (1984), "Temperature Calculation in na incompressible permanent laminar fluid flow through a circular pipe with axial conduction and viscocity", Int. J. Heat mass transfer, Great Britain, v.27, n.6, p. 1183-1187. 5. Wimpress, R. N. (1963), "Rating Fired Heaters", Hydrocarbon Processing and Petroleum Refiner, v.42, n.10, p.115-126.
European Symposiumon ComputerAided ProcessEngineering- l0 S. Pierucci (Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
403
IMPROVING THE FLOW OF STIRRED VESSELS WITH A N C H O R TYPE IMPELLERS S. M. C. P. PEDROSA, C. G. DUARTE and J. R. NUNHEZa+ aFaculdade de Engenharia Qufmica, Universidade Estadual de Campinas, C.P. 6066, 13083-970, Campinas-SP-Brazil - [email protected] [email protected] Abstract. Anchor impellers have a simple and basic configuration which is well suited for the mixing of highly viscous flow, normally in the range of viscosity from 10-100 Pa.s, typical of polymer reactions. It is widely used in chemical and food industries. The primary flow generated by this radial impeller has been reported much more than the secondary flow in the literature. The great majority of these investigations refers to experimental works. The experimental works, however, have not been able to give a detailed picture of the flow, specially for the secondary flow. This is particularly important because the secondary flow controls heat transfer in stirred tanks under laminar flow. Some computational investigations have been reported but the meshes shown are normally very coarse and some simplifications such as fiat bottom are assumed for the models. The main contribution of this paper is to present a detailed picture of the secondary flow generated by anchor impellers both for Newtonian and pseudo-plastic fluids. The case study of orange juice mixing is analyzed. 1. I N T R O D U C T I O N Anchor impellers are widely used in chemical and food industries for highly viscous flows, specially pseudo-plastic fluids, typical of polymer reactions. Its simple basic configuration of two vertical blades which follow the contour of the vessel is well suited for the mixing of viscous fluid. The reactions are normally carried out under laminar flow. The design of vessels employing anchor stirrers to date assume uniform temperature and perfect mixing, which are strong assumptions that clearly leave the designers to their own experience, especially for moderate and highly exothermic reactions and for non-Newtonian fluids. The experimental works and empirical correlations are often not suitable for many systems and also can only give a global picture of the reactor. In order to improve the design of these systems it is necessary to know a detailed picture of the flow, which would reveal details such as dead zones and other inefficiencies that could be eliminated or minimized. The high demand today for industries to comply to safety and environmental regulations as well as the need to ensure products with high quality calls for well thought and planned design. Even though experimental works have improved recently, they unfortunately have not been able to address to all the needs listed above. Improvements in these areas today call for the use of computational studies. The computational fluid dynamics (CFD) have been used in the last two decades to devise solutions and gain insight of the flow inside these systems and the CFD together with experimental validation have been able to improve the design of many reactor systems. * Author to whom all correspondence should be addressed. + The authors would like to thank FAPESP and FAEP (UNICAMP) for the grants received in this project.
404 There are surprisingly few works in the literature studying stirred tanks agitated by anchor impellers. The great majority of these works refers to experimental investigations and only a fraction of these are concerned about numerical studies. The primary flow generated by anchor impellers using a two dimensional grid have been reported by some investigators (Kaminoyama et al, 1994a; Kaminoyama et al , 1990a, Kaminoyama et al , 1990b, Kaminoyama et al , 1993, Rubart and Bohme, 1991). The grids shown are normally very coarse and many simplifications are imposed to the model. However, despite being important to know how the primary flow for these reactors behaves, it is important to acknowledge that anchor impellers are very much used for reacting viscous fluids in heating or cooling processes, and especially to avoid the stagnation of the products at the vessel walls, since the blades of the stirrer work as a scraper. Heat transfer in these systems is important and it is dominated by the secondary flow, which is the flow generated by the action of the inertial forces due to the angular movement of the blades. It is therefore necessary to gain more insight about the secondary flow of these vessels to determine ways in which these systems can be improved. A very important detail in the design that can not be simplified for the secondary flow of anchor impellers is that the bottom of the vessel should be modeled curved, and also the blades of the stirrer should follow the contour of the bottom. This work describes a model which is able to provide a detailed picture for the secondary flow of stirred vessels with anchor impellers. Several operational conditions and different geometries are tested to indicate how the detailed knowledge of the secondary flow can help to improve the design and operation of reactors with anchor type impellers.
2. MODELING AND SIMULATION The model described here calculates the three components of velocity, the pressure, temperature and non-Newtonian viscosity on a two dimensional grid for a single phase flow. Reaction is taken into account by a source term in the energy conservation equation that simulates the heat generated by an exothermic reaction inside the vessel. The set of governing equations for the axi-symmetric model is given below in cylindrical co-ordinates: 2.1 Governing equations Mass conservation OU r
Or
Ur
Olg z
r
Oz
+--+
(1)
=0
Momentum balance 9radial direction OU r
OU r
p ur ---~r + u z 0 z
UO2 (~ P . . . . +# r Or
0
/
0 Zr
_-z---
OZ2
(2)
9 angular direction /9 U
Ou o
rOr
+
u ru o
r
+u z
c3uo
-~Z
=
c3
ff-~r
c3 CrUo
-~r
(3) c3Z2
9 axial direction P Ur u r +Uz u ~ Energy conservation
= - - -OZ +#r~r
r O r J + OZUz
(4)
405
i~
(Ur--~Fnt-Uz
""7-~F k'r---~r +-~z kt
+AH
(5)
Table 1 gives the properties of the fluid and Table 2 gives the dimensions of the tank. Table 1: Fluid properties and some important parameters. Density 9 = 800 and 1320 kg.m -3 Viscosity la0 = 1, 45 and 60 Pa.s Cp = 100 J.kgl.K -1 Heat capacity k = 0.1 W.m-I.K 1 Thermal conductivity he = 500 W.m2.K -1 Heat transfer coefficient (wall) hfs = 5 W.m-2.K -1 Heat transfer coefficient (surface) Heat source (Fixed) AH = 1200 W.m -3 Table 2: Tank dimensions used in model. Tank diameter T 1,120 m Impeller diameter D 1,070 m Shaft diameter De 0,060 m Liquid height Z 1,500 m Impeller height C 0,707 m; 1,13m Blades width s 0,110 m Figure 1 shows the geometry being modeled. Since this work assumes symmetry, only a half section needs to be modeled. Figure 2 shows the mesh of 5492 control volumes.
Figure 1: Geometry of a stirred vessel with an anchor impeller.
Figure 2: Mesh of 5492 control volumes
2.2 B o u n d a r y c o n d i t i o n s Free s u r f a c e - No shear stress, therefore a fiat surface is assumed and axial velocity is null.
Bottom and walls of the vessel - There is no slip, so the velocity is null. Impeller blades - The presence of the two blades of the anchor impeller strictly calls for a time dependant, three dimensional method. However, as a first approximation, in order to enable an averaging of the effects of the blades, the approach of Kuncewicz (1992) is used. Bottom and walls of the vessel - For jacketed arrangements it is assumed that there is enough cooling liquid inside the jacket to maintain temperature constant at 283.15 K (10 ~ At the walls and the bottom of the vessel it is assumed that the heat is removed by the jacket, so the boundary condition is:
406
( /
q, : - k-~n
: h w ( T - Tw)
(6)
At the free surface heat is lost to the air:
q~,, = ht,, (T - Tr, )
(7)
The set of non linear equations describing the model are solved by the finite volumes method and the results performed using the CFX-4.2 package by AEA Technology. 3. RESULTS AND DISCUSSION A case study regarding the homogenizing of orange juice is analyzed to show how CFD can help to understand the flow of reactors agitated by anchor impellers. After orange juice is concentrated to a juice containing 35% of water, it is cooled from (40~ to (20~ and sent to a homogenizing tank, typically agitated by anchor impellers, where juice from several batches are homogenized at 363,15K (-10~ before receiving a last treatment which guarantees a juice prepared under rigid quality control. The characteristics of the product are: Temperature (~ 8.0 0.0 - 10.0 Viscosity (Pa.s) 20 45 60 A mesh independent study was carried out to guarantee that results are independent on mesh size. Figure 3 shows the velocity vector plot for the rotational speed of 136 rpm and a viscosity of 45 Pa.s. It is considered to be low speed for industrial applications for the tank diameter used in this work. The ratio between the height of the impeller blade and the height of the liquid is 0,47. It can be noticed the formation of a single recirculation zone centered near the curve separating bottom and vessel walls, a little bit above the curve of the anchor blade. Fluid is poorly mixed for this rotational speed sinize the velocities near the free surface of the liquid are very low. This suggests the velocity of the stirrer is not enough for this system. Figure 4 shows the same plot for a rotational speed of 317 rpm, which defines a medium industrial speed for this geometry. Mixing inside the vessel is improved, giving a better fluid circulation. The region of low velocity near the free surface is practically eliminated. However, power consumption is much higher. Even though it is not presented in this work, results show that mixing improves as the viscosity lessens. In order to evaluate how mixing can be improved by the use of a higher impeller blade, a geometry using a ratio between the height of the impeller blade and the height of the liquid of 0,75 was tested. Figure 5 shows the velocity vector plot for this arrangement. As expected, mixing is improved. However, power consumption is also increased so a there is a trade-off between benefits and cost. Figure 6 shows the temperature contour plot for the rotational speed of 317 rpm. Temperature is almost uniform due to better mixing for this rotational speed. Even though results are not shown in this work, the temperature distribution for the rotational speed of 137 rpm is not much uniform.
4. CONCLUDING REMARKS The model presented in this work gives a good representation for the flow and temperature fields for anchor impellers and helps to determine design features which improve the flow inside tank reactors stirred by anchor impellers. Results show that moderate agitation in industrial applications gives good fluid circulation and the use of impellers with a higher blade height also improve mixing. However, there is a trade off between the benefits of better fluid circulation and higher power consumption.
407
Figure 3: Velocity vector plot. g=45 Pa.s; 136 rpm and blade height ratio = 0,47.
Figure 5: Velocity vector plot. g = 60 Pa; 317 rpm and blade height ratio = 0,47. 5. NOMENCLATURE T tank diameter [m] C impeller height [m] D impeller diameter [m] De impeller diameter [m] Z liquid height [m]
Figure 4 Velocity vector plot ~t=45 Pa.s; 317 rpm and blade height ratio = 0,47.
Figure 6: Temperature contour plot g = 60 Pa.s; 317 rpm and blade height ratio = 0,47 s r z Uz Ur u0
impeller width [m] radial direction [m] axial direction [m] axial velocity [m.s -1] radial velocity [m.s -1] angular velocity [m.s 1]
408 n
power number which describes the non-Newtonian attributes p pressure [N.m 2] AH heat source [W.m -3]
p average reaction viscosity [kg.ml.s -1] go Newtonian viscosity [kg.ml.s -1] p density [kg.m3]
REFERENCES 1. Edwards, M. F. and Wilkinson, W. L., Heat Transfer in Agitated Vessels Part I. The Chemical Engineer, 310 - 319 (1972). 2. Foumeny, E. A., Holiday, S. 0. and Sandhu, K. S., Prediction of Flow Patterns in Polymerization Systems using CFD. Proc. 8th Int. Conf. on Num. Meth. in Laminar and Turbulent Flow, 517-528, 1993. 3. Kaminoyama, M., Saito, F. and Kamiwano, M., Numerical Analysis of Flow of a Bingham Fluid in an Anchor Impeller. Int. Chem. Eng., 34, No. 2, 263-269 (1994a). 4. Kaminoyama, M., Arai, K. and Kamiwano, M., Numerical Analysis of Power Consumption and Mixing Time for a Pseudoplastic Liquid in Geometrically Similar Stirred Vessels with Several Kinds of Plate-Type Impellers. J. Chem. Eng. Japan, 27, No 1, 17-24 (1994b). 5. Kaminoyama, M., Saito, F. and Kamiwano, M., Flow Analogy of Pseudoplastic Liquid in Geometrically Similar Stirred Vessels Based on Numerical Analysis. J. Chem. Eng. Japan, 23, No 2, 214-221 (1990). 6. Kaminoyama, M., Akabane, K., Arai, K., Saito, F. and Kamiwano, M., Numerical Analysis of Three-Dimensional Flow of a Pseudo-plastic Liquid in a Stirred Vessel with a Turbine Impeller. Int. Chem. Eng., 30, No 4, 720-728 (1990). 7. Kaminoyama, M., Saito, F. and Kamiwano, M., Numerical Analysis of Mixing Processes for High-Viscosity Pseudoplastic Liquids in Mixers with Various Plate-Type Impellers. Int. Chem. Eng., 33, No 3,506-515 (1993). 8. Kuncewicz, G., Three-Dimensional Model of Laminar Liquid Flow for Paddle Impellers and Flat-blade Turbines. Chem. Eng. Sci., 47, No 15/16, 3959-3967 (1992). 9. Nunhez, J. R. and McGreavy, C., Industrial Mixing Technology: Chemical and Biological Applications. AIChE Symposium Series, 90, 55-70 (1994). 10. Nunhez, J. R. and McGreavy, C., A Comparison of the Heat Transfer in Helical Coils and Jacketed Stirred Tank Reactors. Brazilian J. of Chem.1Enging., 12, No 1 (1995). 11. Ohta, M., kuriyama, M., Arai, K., Saito, S., A Two Dimensional Model for Heat Transfer in an Agitated Vessel with Anchor Impeller. J. Chem. Eng. Japan, 18, No 1, 81-84 (1985). 12. Patankar, S. V. and Spalding, D. B., A Calculation Procedure for Heat, Mass and Momentum Transfer in Three Dimensional Parabolic Flows. J. Heat Mass Transfer, 15, 1781-1806 (1983). 13. Peixoto, S. M a C., Escolha de Arranjos Preferenciais de Serpentinas Internas em Tanques de Mistura utilizando a Fluido-Din~mica Computacional (CFD). Ms Thesis, Universidade Estadual de Campinas (1998). 14. Peric, M., kessler, R. and Scheuerer, G., Comparison of Finite-Volume Numerical Methods with Staggered and Colocated Grids. Comp. &Fluids, 16, No 4, 389-403, 1988. 15. Rubart, L. and G. Bohme. Numerical Simulation of Shear-Thinning Flow Problems in Mixing Vessels. Theoret. Comput. Fluid Dynamics, 3, 95-115 (1991). 16. Van Doormaal, J. P. and Raithby, G. D., Enhancements of the Simple Method Predicting Incompressible Fluid Flows. Numerical Heat Transfer, 7, 147-163 (1984).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
409
Influence of turbulence modelling and grid discretization on the simulation of flow-forces on tubes in cross-flow K. Schr~Sder and H. Gelbe Institute of Process and Plant Technology, Technical University Berlin, Stral3e des 17. Juni 135, D-10623 Berlin, Germany Two-dimensional CFD-simulations of single-phase cross-flow around a single fixed tube are carried out and compared with experimental data. Several turbulence models (ke-, kin-, Reynolds stress and large eddy model) are applied in combination with different grid discretizations using the programs STAR-CD and CFX. The grid discretization together with the turbulence model has a great influence on the resulting drag and lift forces and on the frequency of vortex shedding. These values are essential in order to simulate the flow-induced vibration excitation in tube bundles, which is the object of this investigation. First results on threedimensional and unsteady simulations of the flow-induced vibration excitation of flexible tubes, tube-rows and tube bundles are published in [5]. 1.
INTRODUCTION
The CFD-analysis of the tube vibration induced by cross-flow was the subject of some investigations during the last years. Ichioka et al. [ 1] applied a Finite-Difference scheme on a bodyfitted moving grid to solve the unsteady Navier-Stokes equations without using a turbulence model. Their model is restricted to low Reynolds-numbers. Kassera et al. [2, 3] simulated the flow induced vibration for a single flexible tube and for full flexible tube bundles and Kassera [4] presented the three-dimensional simulation of the resonant vortex shedding vibration for a single tube. They used the Finite-Volume Method on a cartesian grid (see Fig. 3) and applied different turbulence models (kin-model, Large Eddy Simulation and a Reynolds stress model) to take the turbulent nature of the flow into account. Schr6der et al. [5] used for the first time a commercial CFD-program and demonstrate the problems arising in unsteady simulations of forces and tube-motions in tube bundles. The grid discretization together with the turbulence model has a great influence on the resuiting drag and lift forces and the frequency of the vortex shedding. Therefore in the investigation presented here, some turbulence models and different grid discretizations were tested for a rigid tube. 2. DISCRETIZATION OF THE F L O W FIELD Different discretizations of the geometry were compared. Five of the used grids can be seen in the Figures 1 to 3. The grid B1 in Fig. la has 216 cells in peripheral direction in the near wall region. This high resolution decreases with increasing radial distance from the wall to 54 cells by the usage of cell matching methods. The second grid B2 in Fig. lb is similar to the discretization B 1, but has 432 cells in peripheral direction near the wall. In Fig. 2a the grid
410
II . N ~ IIJNNl liar1 IIAltt~l
iir'~.4. II
!! Fig. 1. a) Grid B1 with 216 cells in peripheral direction in 2 cell layers. b) Grid B2 with 432 cells in peripheral direction in 5 cell layers.
INIIILIII IN.Ill INN_Ill ~ l l l
J~-qlll Ill
.!!!
Fig. 2. a) Grid B4 with 216 cells in peripheral direction in 50 cell layers. b) Grid B5 with 72 cells in peripheral direction in 18 cell layers.
B4 is shown. The grid has in comparison with grid B 1 also 216 cells in peripheral direction but lllll Illll [ in 50 cell layers in radial direction; this fine partition reduces to 54 cells in peripheral direction. For comparison a discretization of 72 cells in peripheral direction is used for grid B5, but with iZ a fine partition in radial direction and no cell matching in the solution domain. Fig. 3. Grid B3 with 48 cells in periphThe grid B3 shown in Fig. 3 used by eral direction. Kassera et al. [2, 3, 4] is also tested and discussed in the present work. In this case only 48 cells in the near wall region of the tubes are used for the peripheral discretization. IIIII IIlll
[ _
.
.
.
.
.
.
.
3. C O M P A R I S O N OF D I F F E R E N T T U R B U L E N C E M O D E L S Comparing different turbulence models for the computation of the flow field some two equations turbulence models, Reynolds stress models and a Large-Eddy model with different subgridscale models were tested for a single rigid tube. The computations were carried out with the CFD-programs STAR-CD and CFX.
3.1.
Computations for a Reynolds-number of 140000 The experimental data of Cantwell and Coles [6] for a Reynolds number Re = uooda/v =140000 were used for a comparison. The same parameters as in the experiments: tube diameter d = 0.10137 m, freestream velocity uoo = 21.2 m/s and the viscosity and density of air were taken in the simulation. The turbulence intensity in the upstream region for this experiment was less than 0.1%, so a laminar separation could be expected and the transition from laminar to turbulent flow lies in the boundary layer. This is an important fact for the turbulence modelling, because the transition can not be well predicted by k~-model [7]. The computations were carried out with the time step size of At = 0.0001 s, the QUICK differenc-
411 ing scheme and the grid discretization B 1 (see Fig. 1). The results for the time averaged pressure distribution obtained for three different turbulence models can be seen in Fig. 4. The pressure coefficient Cp = (p-p=)/(0.5.p.u 2) is plotted over the angle around the tube surface, beginning at the stagnation point. - - - kco-model (Wilcox) The computations with the standard ke1 "~ . . . . ke-model (standard) model and the two-layer ke-model of Norris and 0.5 k~-2-1ayer-model Reynolds [8] can not predict the pressure distri. . . . k~-model (nonlinear) bution. The computed pressure for t~ > 120 ~ in rJ~ 0 x exp. Cantwell & C o l e s the flow separation area is too high, so the re-0.5 , "" . . . . . . . . . suiting drag coefficients are too low. Franke [9] obtained much better results with a modified -1 ~ ~, ;' two-layer ke-model and a finer partition in ra-1.5 ~ dial tube direction for the pressure distribution -2 J Re=140000 and 144 cells in peripheral direction. The results da~O.10~a7m obtained with the k~o-model by Wilcox [7] can ,
I
,
I
,
I
,
,
.
describe the observed experimental pressure 0 30 60 90 120 150 180 oc distribution quite well. The pressure coefficient Fig. 4. Computed pressure distributions for the minimum is acceptable with a deviation around the tube surface for different turof less than 20 %. In opposition to the standard bulence models compared with experiand the two-layer ke-model, the pressure distrimental data by Cantwell and Coles [6]. bution computed with the quadratic nonlinear ke-model [10] is in good agreement to the experimental for a > 90 ~ This turbulence model computes the components of the Reynolds-stress tensor with algebraic equations and so takes into account the anisotropic nature of turbulence [7]. The amplitude for the lift coefficient computed with the kt~-model and the nonlinear kemodel is more than 5-times higher than the amplitudes for the standard and the two-layer kemodel. The resulting average drag coefficient CD = 1.17 is comparable to the measured value of 1.25. Using grid B 1, the k0~-model enables an accurate prediction of the Strouhal-number Sr = fvda/uoo = 0.197 for the test case with a relative error of less than 10 % compared to the measured Strouhal-number of 0.179. All ke-models fail in the prediction of the Strouhalnumber with a relative error of more than 30 %; one reason may be the inaccurate determination of the laminar separation point by the ke-models in this special test case with a high Reynolds number and a very low turbulence level in the upstream flow. The nonlinear ke-model predicts an average drag coefficient of 0.91 with an error of 27%, whereas the computed value for the standard ke-model is only 0.3. The computed shedding frequencies by Franke [9] with the modified two-layer ke-model and a differential Reynolds-stress model in conjunction with the two-layer ke-model showed also a relative error of more than 30%.
3.2. Computations for a Reynolds-number of 6450 The experimental data of Gog [11] for a Reynolds number of 6450 were used for an additional comparison. The upstream turbulence level was about 1%, so a laminar separation could not be expected. The selected time step size At = (1/100) Zvortex-shedding = 0.0007 S seems to be appropriate for an accurate time discretization of the vortex shedding excitation. The same parameters as in the experiments: tube diameter d =0.04 m, freestream velocity uoo = 2.47 m/s and the viscosity and density of air were taken in the simulation. All computations were carried out with the grid discretization B5 (see Fig. 2b). The cells in peripheral
412 direction were increased to 108 cells. ke-model (standard) 1 "'\ ___ ke-model (quad. nonlinear Fig. 5 shows the time averaged pressure 0.5 ~ ..... ke-model (cubic nonlinear distribution obtained for different ke turbulence x exp. Gog (1982) models. The MARS differencing scheme was O 0 applied for the simulations. The pressure coef-0.5 ficient is plotted over the angle around the tube surface. For this Reynolds number the standard x . . . . . . . ,.___. ke-model can predict the pressure distribution -1.5 quite well. The predicted pressure coefficients -2 Re=6450 in the separation area are nearby the measured da=0.04m values. The vortex shedding frequency could be -2.5 ' ' ' ' ' ' ' ' ' ' ' calculated with an error of less than 5%. In op0 30 60 90 120 150 180 o~ position to the standard and the quadratic nonFig. 5. Pressure d i s t r i b u t i o n s a r o u n d the linear ke-model, the pressure distribution comtube surface for different turbulence moputed with the cubic nonlinear ke-model [10] is dels computed with STAR-CD and comin good agreement to the experimental data in pared with experimental data by Gog [11 ]. the total range of o~. The computed vortex shed....... algebraic-Reynoldsstress ding frequency has an error of less than 0.5%. 1 ~ . . . . k~-model (Low-Reynolds) Additionally computations were also car0.5 [-- ~ , . differential Reynoldsstress ded out with the CFD-program CFX on the O~ 0 9 x exp. Gog (1982) same grid layout. The difference to the grid -0.5 used for the STAR-CD calculations is, that the -1 "\x x x x grid is divided in four structured blocks. For the -1.5 \\\ \ ~~.,,,,~ velocities the QUICK differencing scheme and ", -,Jfor the turbulence equations a HYBRID differ-2.5 \ encing scheme was applied. The results for the -3 \ ,-". . . . . . R'e'--"6~.'S"6"" computed pressure distribution can be seen in -3.5 ' ' '' da='0"04m Fig. 6. In comparison to the STAR-CD results 0 30 60 90 120 150 180 (z the pressure coefficients calculated with CFX Fig. 6. Pressure d i s t r i b u t i o n s a r o u n d the are lower. One reason may be the different caltube surface for different turbulence culation of the pressure field. For the CFX calmodels computed with CFX. culation a SIMPLEC algorithm was used instead of the PISO algorithm applied for the STAR-CD calculations. Comparing the results obtained with the algebraic Reynolds-stress ke-model to the differential Reynolds-stress model, greater differences can be observed for o~ > 120 in the separation area. The calculated vortex shedding frequencies of 14.8 Hz for the differential stress model and 13.4 Hz for the quadratic nonlinear ke-model are in good agreement to the measured frequency of 14 Hz. The calculated pressure coefficients for the Low-Reynolds kcomodel with a viscous damping function for the near wall cells are too low compared to the algebraic and differential Reynolds stress models. The calculated vortex shedding frequency of 20.2 Hz is much higher than the observed one. One reason for this inadequate results may be the grid spacing near the wall, so the conditions for Low-Reynolds calculation near the wall are not valid. STAR-CD offers two LES-models with different subgrid scale models (SGS), namely that of Smagorinsky and a two equations kl-model. The computed pressure coefficients in Fig. 7 are lower than the measured values for (x > 60 ~ with a relative error of up to 80% I
413 compared to the measured values. The calculated vortex shedding frequency of 13.5 Hz for the Smagorinsky and 15 Hz for the SGS N-model are acceptable. Breuer [ 12] carried out a large eddy simulation for a tube in cross-flow and a Reynolds-number of 3900. His results were in good agreement with the measurements. The reasons for the better results are twofold: the SGS-model was modified near the wall and the grid and time discretization was much finer than for the presented simulations in this paper. For acceptable results with a large eddy simulation the numerical costs will be very high.
1 " ....
0.5 O" 0 -0.5
~ ~,
-1
~\
-2 ~ -2.5
L E S ( S m a g o r i n s k y SGS) L E S (kl-model SGS) ke-model (cubic nonlinear) x exp. Gog (1982)
t 0
Re=6450 da= 0. 04m x
.'~2..-" ,
"
',
, , I , I , I , , . 30 60 90 120 150 180 (z
Fig. 7. Pressure distributions around the tube surface for LES simulations computed with STAR-CD and compared to [ 11]. 4. C O M P A R I S O N OF D I F F E R E N T GRID D I S C R E T I Z A T I O N S
The measurements of Cantwell and Coles 1 ~ I---kin-model (grid B1) I [6] for a single rigid tube were also used for a 0.5 ]- ~ I .... kin-model(grid B2) I comparison of different discretizations out[ '~ Jo----e kin-model(gridB3) J I_ 9 ] - - - - ko~-model (grid B4) r} lined in section 2. The ko)-model was applied 0 l ~ 1--- kco-model(grid B5) ]] with the QUICK differencing scheme and the -0.5 ~1~[ x exp. cantweu&colesq same time step size of At = 0.0001 s was used. o~ -1 F The results for the pressure distribution are ' -presented in Fig. 8. -1.5 The fine grid B4 with 18054 cells and a computation time of 41.10 seconds per time -2 step yields the best result for the pressure coefficient in the total range of the peripheral an0 30 60 90 120 150 180 gle. Especially the computed position of the o~ pressure minimum at c~ = 71~ and the value of Fig. 8. Computed pressure distribution for Cp =-1.4 gives a good agreement with the exdifferent grid discretizations with the experimental data. An excellent result can be perimental data by Cantwell and Coles [6]. obtained for the drag coefficient cD = 1.24. The computed Strouhal-number of Sr = 0.182 confirm, that the kco-model enables results with high accuracy. The results obtained with the grids B 1 (7686 cells and a computation time of 9.52s per time step) and B2 (9558 cells and 11,08s per time step) are good compromises obtaining satisfying results (Sr = 0.197) within an acceptable computational time. The simple discretization B3 (6950 cells and 7.63s per time step), which has the lowest calculation time, is in good agreement with the measurements between 85 and 180 degrees. The separation point is fixed by the edge of the grid at about 74 degree, so the grid cannot predict the pressure distribution in the range of 50 < ~ < 80 and the calculated Strouhal-number of 0.232 is 30% higher than the observed value of 0.179 by Cantwell and Coles. Testing the error of cell matching, grid B5 with a high resolution in radial and a low resolution in peripheral direction (see Fig. 2b) was applied with no cell matching in the solution domain; the computed pressure distribution is acceptable in comparison with the results for the grid B3, the computed Strouhal number 0.184 for B5 is quite well.
414 5. CONCLUSIONS A comparison between the simulation and experimental results for a rigid single tube in cross-flow with a Reynolds-number of 140000 show, that the best results for the pressure distribution at the tube surface, the frequency of vortex shedding and the lift and drag forces can be obtained with the implemented kor-turbulence model. The results computed with the standard ke-turbulence and the ke-2-1ayer model are not satisfying. Simulations for a Reynolds-number of 6450 with different turbulence models and different CFD-programs compared to experimental results demonstrate the strong influence of CFD-code and turbulence model. The best results can be obtained for the cubic nonlinear kemodel. The pressure coefficients calculated with the CFD-program CFX differ from the resuits calculated with STAR-CD. The differences in the simulation results for the differential stress model and the nonlinear ke-model are small. The large eddy simulations carried out with STAR-CD could not predict the measured pressure distribution. The reason for the inaccurate results are the grid and time discretization and the wall treatment for the subgrid scale implemented in STAR-CD.
REFERENCES [ 1] T. Ichioka, Y. Kawata, H. Izumi, T. Nakamura, K. Fujita, Two-dimensional flow analysis of fluid structure interaction around a cylinder and a row of cylinders, Symposium on Flow Induced Vibration, Minneapolis, 1994, ASME PVP-Vol. 273, pp. 283-288. [2] V. Kassera, L. Kacem-Hamouda, K. Strohmeier, Numerical simulation of flow induced vibration of a tube bundle in uniform cross flow, Symposium on Flow Induced Vibrations, Honolulu, 1995, ASME PVP-Vol. 298, pp. 37-43. [3] V. Kassera and K. Strohmeier, Simulation of tube bundle vibrations induced by cross flow, Journal of Fluids and Structures (1997) 11, pp. 909-928. [4] V. Kassera, Three dimensional CFD-analyses of tube vibrations induced by cross flow, ASME AD-Vol. 53-2 Vol. II, 4 th Int. Symp. on FSI, A, FIV & N, Dallas, 1997, pp. 137-143. [5] K. Schr6der and H. Gelbe, Two- and three-dimensional CFD-simulation of flow-induced vibration excitation in tube bundles, Chem. Eng. and Proc. 38 (1999), pp. 621-629. [6] B. Cantwell and D. Coles, An experimental study of entrainment and transport in the turbulent near wake of a circular cylinder, J. of Fluid Mechanics, 136 (1983),pp. 321-374. [7] D.C. Wilcox, Turbulence Modelling for CFD, DCW Industries Inc., La Canada, California, 1994. [8] L.H. Norris and W.C. Reynolds, Turbulent channel flow with a moving wavy boundary, Report No. FM-10, Department of Mechanical Engineering, Stanford University, USA, 1975. [9] R. Franke, Numerische Berechnung der instationaren Wirbelabl6sung hinter zylindrischen Kt~rpern, Dissertation, Universit~it Karlsruhe (TH), 1991. [ 10] T.H. Shih, J. Zhu and J.L. Lumley, A realizable reynolds stress algebraic equation model, NASA TM-105993, 1993. [ 11 ] W. Gog, Untersuchungen der Erregermechanismen am Einzelrohr und am querangestr6mten Rohrbtindel, Dissertation D 83, Technische Universit~it Berlin, 1982. [ 12] M. Breuer, Large Eddy Simulation of the Sub-Critical Flow Past a Circular Cylinder: Numerical and Modeling Aspects, Int. J. for Numerical Methods in Fluids, Wiley, Chichester, 1998.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
415
A CFDFinite V o l u m e M e t h o d to Generate a D e t e r m i n i s t i c Model: A p p l i c a t i o n to Stirred T a n k Reactors ~Maciel Filho, R.. ; 2Bezerra, V. M. F. 1 Laboratory of Optimization, Design and Advanced Process Control (LOPCA), College of Chemical Engineering, State University of Campinas, Email: [email protected] 2 Universidade Federal do Rio Grande do Norte - UFRN - CT -Departamento de Engenharia Quimica - Programa de p6s-gradua~o em Engenharia Q u i m i c a - PPGEQ -CEP: 59072-970 Natal - RN - Brazil- Email: [email protected]
Abstract-The objective of the present work is to analyze a deterministic model related to stirred tanks, starting from its set of partial differential equations, going forward the discretization of such set, through the Finite Volume Method and also applying a simplified procedure for obtaintion of the temperature profile of the case study considered. Comments on the discretization of the system of equations show that this particular method partitions the computational domain into a finite set of volumes or cells, assuming that the main variables are constant over each cell and this fact requires the conservation of equations being satisfied for each cell. Computational Fluid Dynamics (CFD) represents the scientific alternative to preprocess, process and post-process the fluid flow inside stirred tanks. In the core of commercial CFD packages, Finite Volume Method based discretizations for different case studies are used and the user can count with a feasible output
Keywords: stirred tanks, computational fluid dynamics, finite volume method, fluid flow.
1. I N T R O D U C T I O N Stirred tanks constitute commonly used equipments inside chemical industry. In order to study the characteristics of fluid flow inside stirred tanks, the fundamental equations of conservation (mass, momentum and energy) are used. Such approach results in deterministic models for the equipment studied. It starts from an analysis of the case study proposed. The set of fundamental relationships shows the equation of continuity (mass conservation law), the energy conservation formulae and the Navier-Stokes equations that represent the fluid flow inside the tank. In this path, the conservation laws are coupled with auxiliary equations becoming possible the subsequent numerical simulation of the system studied. It is necessary that the degree of freedom of the system be zero for possibility of solution, i.e., the number of equations is, at least, equal to the number of unknowns of the mathematical set. At this point, numerical solutions are necessary for the final output for the problem. There are diverse ways in
416 which one can choose her/his route for problem solving. Some routes can be stated as: deterministic, non-deterministic, experimental and/or non-invasive ones. In the first case, the set of partial differential equations together with initial and/or boundary conditions is discretized through the Finite Volume Method [Patankar, 1980; Maliska, 1995], p.e.. After discretization, the set of partial differential equations turns into a set of algebraic ones, to be solved through tridiagonal matrix algorithms [Patankar, 1980]. For non-deterministic models proposed for the case study, the equipment itself is considered a black box, in which the final assessment does not show conservation laws, but mathematical expressions calculating errors between inputs and outputs. At present, experimental and non-invasive techniques, as particle image velocimetry data [Fox, 1998] are used in order to compare and elucidate temperature and velocity profiles, as well as to validate data used in Computational Fluid Dynamics (CFD) simulations, p.e.. Computational Fluid Dynamics consists in studying diverse phenomena with a strong ability of post-processing the fluid flow studied in an understandable and accurate output. Examples can include stirred tank reactors and the final displays show characteristic velocity, temperature, concentartion profiles, among other, depending on the proposed problem.
2. CASE S T U D Y ar = 0 c3z exceto Ur = 0
t a_, = o
"
Parede do Tanque
&
r
li I ~j-~ Eixo (s
/
{l
4
"lamina(blade)"
i i i
l
00O=0 &
l
l
a_,= 0 Or
exceto U~,U~
l L*=0 0Z
exceto Ur
r
exceto U~ -- 0
r
Figure 1.0-Boundary conditions to a stirred tank [PIKE, 1990] Simplification of a model in mathematical terms, intends to reduce one of the among characteristics [TUCKER, 1989]: number of equations, number of terms inside the equations, degree of non-linearities, degree of coupling among the equations (mainly, stiffness degree).Taking into account our case study, it is observed that cylindrical coordinates are the pertinent representations to the deterministic model to be shaped. The main conservation equations for the system are:
417 Continuity:
1~
Op Ot
10(prUr) + (PUo)+ r Or r~--O
~z(OUz)
(1.o)
0
Simplifications:
~=0 ~0
(2.0)
ap
(3.0)
Ot
= o
that is, incompressible fluid flow is considered. In this way, Equation (1.0) has the following shape' 1 Cg(prUr)+ (pUz)=O r Or ~zz
(4.0)
Momentum equations: Component- r: OUr + Ur OUr + U 0 OU r P
&
[(
Or
63p 63 1 63 (rUt - ~Or + Ix & -r ~ r
r
-U~) -+
c~O
U z -OU ~z r ) =
r
)) q '~Ur 2~Uo~_~U]r r 2 50 2
r 2 630
CDZ2
(5.0) Disregarding angular momentum,O OU r OU r +U r 9 P Ot Or + U z - ~ z J - - O T + l t l
Component- z:
~- r~rr
2'rl
+ 63z2
(6.0)
418
P
c3Uz c3Uz UO c3Uz c3Uz / +U r + +U z & Or r 690 --~z/
@
[1 ~( O~z/
0z + g r ~ r r
r 0r/
(7.o)
1 632Uz 632Uz 1 -t 2 F r c~O2 C3Z2
J
Another simplification:
P
0t
z +Ur
&
z +U
Z&-zJ
op [lOIrOZO2z]
=_m+g
c~z
r&
+
&-J
c~z2
(8.0)
According to the a general representation of the system above, all the conservation laws can be represented through Equation [9.0] [Maliska, 1995]:
P r
(rUr~)+-(U~ r
rFr
(Uz~b) Ozz r Dr
+ -~ + r c~O
+
Ozz
Fo
(9.0)
+
3
DISCRETIZATION OF THE EQUATIONS, NUMERICAL METHOD OF FINITE VOLUMES:
USING
THE
The sequence of discretization follows: tomando-se a eq. (4.0) and multipling it by r, we have 0 c3 ( 9 r U r ) + r (9U)Or ~zz z
0
(10.0)
From eq. (10.0): enc9
~--(prU r )drdz - [(prU r) - ( p r U ) ](z e - z w ) ws & n r s
(11.0)
](2 r2 / rnZw 2
(12.0)
e
G3
i lr-~rr(PUz)dzdr-[(pU sw
) -(pU)
Ze
Integrating eq. (10.0) takes in the present 2D problem to:
IIr~(ouz ) d z d r - [ ( P U z ) n - ( 9 U z ) s ws
2/
re-r w 2
(13.0)
419
2 2
re - r w : ( r e _ r w 2
/re+rw/ - ( r e - r w ) r p
(14.0)
2
where rp is the medium radio of the control volume. Adding up Eqs.(13.0) and (14.0): (prUr)e (Zn - Z s ) - ( prur )w ( zn - Zs) + (PUz)n ( re - rw )rp
(15.0)
-(pUz)s(r e -rw)r p - 0 And eq. (16.0) represents discretization of the continuity equation (for the obtaintion of a deterministic 2D model) Analogously, these procedures can apply to the energy and momentum equations.
4. R E S U L T S
Figure 2 . 0 - Temperature distribution for axial and radial directions for a stirred tank [Excel v. 5.0, BEZERRA, 1997]
5. C O N C L U S I O N S : Stirred tanks are common and important equipments for chemical industry. In this way, their characterization becomes useful and demanding. Numerical simulations involving Finite Volume Method discretizations are present either in struturated algorithms or in commercial packages. The present work showed the main details for discretization of conservation equations to shape a deterministic model for a stirred tank. Moreover, it has shown a CFD-
420 Finite Volume procedure implicitly used in commercial packages, representing, p.e., a tank with simple movement of polymeric fluid inside it CFD tends to influence strongly in the scenario for representation of complex and/or simple problems.It is a low cost and relatively short time consuming to generate final displays for tanks or stirred tanks. It is concluded that many routes can be chosen to represent a same problem, but the non-invasive (CFD) ones emerge as important tools for designing and predicting na equipment and/or specific case studies. In the present case, it was shown the sequence of shaping a deterministic model in two ways: structured algorithm and afterwards numerical simulation and finally, a typical CFD post-processing for a 2D fluid flow problem. Both alternatives are important and feasible.
BIBLIOGRAPHY
BIRD, R. Byron et al., "Transport Phenomena", John Wiley & Sons, New York, 1960. BEZERRA, V. M. F. Metodologia de Obteng6o de Resultados em Fluido- Din6mica Computacional-Aplica96o a Reatores Tanques Agitados, Tese de Doutorado, UNICAMP, SP, 1997 Excel - Microsoft Excel Vers~.o 5.0 FOX, Rodney O . ; MENG, Hui, SHENG, Jian, Validation of CFD Simulations of a Stirred Tank Using Particle Image Velocimetry Data, The Canadian Journal of Chemical Engineering, Vol. 76, 611-625, June, 1998. MALISKA, C. R., TransferOncia de Calor e Mecdnica dos Fluidos ComputacionalFundamentos, Coordenadas Generalizadas, LTC, RJ, 1995. PATANKAR ,S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington DC, 1980.
Phoenics v 2.1.1, Cham, UK. PIKE, R.W., 1980, Ju, S.Y., Mulvahill, T.M., "Tridimensional Turbulent Flow in Agitated Vessels with a Nonisotropic Viscosity Turbulence Model", The Canadian Journal of Chemical Engineering, vol. 68, 3-16, 1990. TUCKER, Charles L. Fundamentals of Computer Modeling for Polymer Processing, Hanser Publishers, New York, 1989.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
421
Simulation of NOx formation in glass melting furnaces by an integrated computational approach: CFD+Reactor Network Analysis Davide Benedetto (a), Mariano Falcitelli (b), Sauro Pasini (a), Leonardo Tognotti (c) (a) ENEL S.p.A. Research Centre Generation Branch Via A. Pisano, 120 - 56122 Pisa, Italy. (b) Consorzio Pisa Ricerche P.zza D'Ancona, 1 - 56126 Pisa, Italy. E-mail: [email protected] (c) Universit~t degli Studi di Pisa- Dip. di Ingegneria Chimica, Chimica Industriale e Scienza dei Materiali. Via Diotisalvi 2 - 56100 Pisa, Italy. E-mail: [email protected] A procedure, called Reactor Network Analysis, has been developed for the prediction of NOx emissions by practical combustion systems. It is a postprocessor of a CFD simulation which allows to extract from CFD 3D fields an "equivalent" network of reactors, for which it is possible to use a detailed reaction kinetics. The study of two glass melting furnaces, drawn from the experience of the authors, are presented to illustrate the methodology. The furnaces were experimentally characterised, then CFD simulations were performed, setting carefully the boundary conditions for the radiative heat exchange, and adopting a simplified reaction kinetic scheme with 9 species and 10 reactions, for the chemistry. Then, from each CFD simulation, a chemical reactor network was extracted, as simplified flow model, to perform the computation of the secondary product combustion species by means of a complex kinetics mechanism. An evaluation of the models was given comparing the measurements with of both the temperature CFD field and the NOx prediction by Reactor Network Analysis. Finally, an estimate of the effect of some NOx reducing techniques was given, changing some key parameter of the reactors model. 1. INTRODUCTION Simulation of industrial combustion system using Computational Fluid Dynamics (CFD) modelling is still a challenging domain. Besides the research on fundamentals to understand the processes occurring in reactive flow systems, much work is being made for developing computational methods suitable for coupling the many important aspects of chemistry and physics in a way that is efficient enough for solving industrial problems. Research activity of ENEL, the Italian largest utility company, in collaboration with Research Centers (CPR) and University (Dept. Chem Eng.) in Pisa area, is seriously engaged in this field [1,2,3]. Up to now, the event to incorporate a detailed reaction kinetics directly in a 3D CFD code is still unfeasible, because of the exorbitant computational demands (both in terms of memory and CPU speed) it would request. Therefore approximations at some appropriate level have to be made. For the procedure showed in the present work, the choice is to perform on the first a
422 CFD simulation on a narrow grid taking into account a limited number of reactions for the fuel oxidation; then, on the basis of resulting flow fields, an "equivalent" network of ideal chemical reactors is extracted and the concentration of minor species is calculated, using a complex kinetics mechanism on the simplified flow model. The separation in two steps is possible as minor species have a neglecting influence on the flow field and heat exchange. 2. THE PROCEDURE STEP BY STEP
A basic organisation of the procedure is shown in Fig. 1. 1. CFD simulation is performed on a fine mesh computing flow, temperature and mass fraction fields of the major species involved in combustion. Then the local stoichiometry, the residence time distribution and the local concentration of some diffusive tracer are computed by post-processing tools. 2. Analysing the distribution of the local values of the CFD fields, the + § cells of the mesh are clustered by ] Fl~ } [ Temperature[ I'ConcentrationJ Species [ ranges of values of t6mperature, stoichiometry and/or mass fraction of Tracer an injected tracer. Clustering has to Concentration be made considering the correlation displayed for each system. The result of clustering is that every cell belongs to a homogeneous zone and so each zone is modelled as an ideal reactor. So far, the mesh clustering is supervised by the operator who chooses the number of reactors to extract and the criteria to define the intervals of local values. Yet, the criteria that the operator has to follow for his choices are those also adopted for mesh generation in CFD modelling, i.e. going in more detail where the conditions are critical for formation/destruction of pollutants, and then verifying the stability of the solution increasing the number of I Residencetime reactors. In short, the role of operator distribution consists of addressing the build up of the model in order to keep minimum Computationby the number of reactors and to better ComplexKineticModel exploit the computational resources. Anyway, more complex algorithms, Results Concentration) based on mixing indexes and shape factors, are under development, with the goal to make the clustering automatic in the whole. Fig. 1. Conceptual scheme of the RNA procedure.
[3D'CFD
I
~y~S NO
1
(NOx
423 3. The operative parameters are assigned to the reactors. The volume of each reactor is taken as the sum of the belonging cells volume. The reactors are considered isothermal at the operative ~- temperature computed by the enthalpy conserving expression:
Zi mi~ cp(T)dT -- 0 where mi and Ti are the mass and temperature of the i-esimal cell and the sum extends over all the cells belonging to the reactor. The clustered zones are modelled as perfectly stirred (PSR) or plug flow reactors (PFR) on the basis of velocity vector distribution: PSR better represents the zones where the directions of velocity vectors are randomly distributed, as in the injection and recycling zones; PFR correctly characterises a one-directional flow. 4. The mass exchange between all the reactors and the feeding streams are computed using the CFD flow field as the sum of the mass flows between cells belonging to different reactors. In this way all the mass exchanges are considered and the network is designed including recycling streams. 5. A check of the reactor network can be performed comparing the residence time distributions, obtained from a CFD post-processing tool, with the dynamic response of the network to the introduction of a finite pulse of flow. The single reactor types (PSR or PFR) are changed until a satisfactory agreement is obtained. Anyway the application of the procedure to the cases studied so far have shown that, for NOx prediction, the reactor type (PSR or PFR) is not a critical parameter with respect to the kinetics computation performed on the network, when the network includes a relatively large number of reactors. 6. The kinetics computation is carried out on the reactor network using a detailed kinetic model for chemical species involved in combustion. The mechanism used so far is that elaborated by Ranzi and co-workers [4]. The hydrocarbon combustion mechanism involves about 200 species and more then 3000 reactions, the nitrogen sub-mechanism involves about 200 reactions and 40 species. 3. APPLICATION OF THE PROCEDURE TO A GLASS FURNACE. The application of the procedure on a practical combustion system is shown: an "end-port" regenerative glass melting furnace. This kind of furnace is essentially a large tank, covered by an arch ceiling. The walls and the ceiling are refractory lined, in order to ensure a high thermal inertia and to allow an uniform irradiation of the melting glass. The batch enters from the "dog-house" and the melted glass flows towards the end-wall, which is opposite to the inlet-outlet front-wall. On this side, there are two firing ports, each one equipped with three under port barrel bumers, fed by natural gas. Besides, regenerative heat exchangers are located before each port, and the fumace is fired altemately from either port (with a cycle time of about 20 minutes); so that, in a cycle, preheated air of 1200 ~ is fed through the inlet port, while exhaust from the outlet port allows the other heat exchanger to regenerate. The study was performed in three steps: experimental characterisation, CFD simulation, Reactor Network Analysis. Two furnaces of different size (5 and 10 MW of thermal power) were fully characterised by means of "in situ" measurements of temperature and chemical species [5]. For the 3D CFD calculation the IPSE code was used. It is an original code, that belongs to ENEL, with a very flexible architecture in which many different models can be arranged in order to have the best phenomena simulation; a detailed description of IPSE can be found elsewhere [6]. The finest grid adopted was a cartesian mesh with 24192 nodes (36x42 for the
424 base, 16 for the height). For the turbulence, the Jet Model with a constant kinematic viscosity was adopted. Sources terms from heat release of chemical reactions and thermal radiative transport were considered. The Discrete Transfer Radiation Model [7] was employed, with $4 approximation and one grey gas (0.27 of emissivity). The solving algorithm proceeds as explicit, for the fluid dynamic and heat transfer, and semi-implicit for chemical kinetics and species transport (STIK method) [3]. 4. RESULTS AND DISCUSSION The goal of the investigation was to tune the computational tools, already tested on the utility boilers[1 ], in this kind of industrial furnace, in order to allow their use for addressing the design. Many CFD simulations were performed, adopting different boundary conditions for the radiative heat exchange. For each simulation a chemical reactor network was extracted by RNA, as simplified flow model, and the chemical species concentrations were recalculated by means of the detailed kinetic mechanism. The effects of the models and conditions were evaluated comparing both the flow and temperature CFD fields and NOx concentration calculated by RNA, with the measurements. Then on the reactor network model a sensitivity analysis was performed in order to single out the response of the system to critical parameters which could be controlled at the planning stage' The simulations performed have shown that the resulting CFD temperature field is strongly dependent on wall conditions and the local oxygen concentration field is strongly correlated to the chemical reactions sub-model in the CFD code; further; the prediction of the RNA procedure was found to be fairly sensitive to these changes in the CFD fields. The best agreement with the measurements was obtained using the following conditions. The boundary conditions for the calculation of the radiative heat transfer, was set by the wall equation:
,y.(qi-crTint4)-k(Tint-Text) the emissivity ~, the conductivity k and the external temperature Text were specified on the basis of manufactures data and temperature measuremems, thus, during the CFD computation run time, the imemal temperature T~,,tof the walls was recalculated at each iteration using the incident radiation q~. The heat release by chemical reactions was calculated adopting a simplified reaction kinetic scheme which includes one irreversible reaction for the demolition of the fuel in CO and H2, and a CO/H2 oxidation mechanism with 8 (CO, CO2, H2, H20, 02, OH, O, H) species and 9 reactions derived by Westbrook et al.[8]. For both the furnaces the resulting flow field was "U" shaped with two main recirculating zones: one cemral, the other placed near the end comer opposite to the outlet port. The comparison of the temperature fields with the measurements, performed on 34 probing points, showed a good agreemem: for the 5 MW furnace the mean shifting (IAT]/T) was 0.03, with standard deviation (sd) 0.01; for the 10 MW furnace it was: mean IAT[/T = 0.03, sd=0.02. The reactors network "extracted" by the postprocessing procedure for the 5 MW furnace is shown in Fig. 2: it consists of two reactors receiving the main feeds (R1, R2), two reactors, one reducing, the other oxidising, for the zone with highest temperature of flame (R3 , R12), one series of reactors with excess of oxygen (R4, R9, R8), one series of sub-stoichiometric reactors (R5, R10), one series of reactors with unitary stoichiometry (R6, R7, R11), as imerface between the other two, which follows the formation of exhausts. The scheme includes the main streams and feeds; infiltrating air stream, evaporation from the melted bath and minor recycling streams are not drawn to not complicate the scheme itself. Anyway, all the feeds
425
R4
~ i:,.'~I-.-,* I
3~
I!iii!ii!~ii!~i~!i!il
R9 1143
R87~-~
,LJ. 29ooi .IX? sl
/"
R2
,~TT4d., 7,52
R5 3
R10
Fig. 2. Reactors Network produced by the procedure for the 5 MW furnace. The numbers near each reactor show the concentration of NOx (ppm vol.) resulting by the kinetic computation. and mass flows of the system were considered when the complex kinetic calculation was performed. Reaction progress within the individual reactors was calculated using a CHEMKIN like software with a special way for solving the reactions with radical extraction, on a Personal Computer with a Pentium II processor. CPU time demands for a plug flow reactor was about 25 seconds, while for a perfectly stirred reactor it was about 5 seconds and thus 1 hour and quarter for the present network looped 25 times. The final NOx concentration calculated by RNA was in good agreement with measurements for both the furnaces. In the exhaust coming from the 5 MW furnace, a concentration of 1459 mg/Nm3@8%O2 dry of nitrogen oxides, a s NO2, was measured, while the simulated value was 1500 (3 % greater).
5O
c .o
--..--Internal
40 30
Recycle R5-->
2200J
R2
- - z x - - F G R f r o m o u t l e t into
._~ E
2O
J~•
10
~ooo+o ,
0
1400-I
z ~o r.9
-
........ ..... 9 v a r i a t i o n of p e a k t e m p e r a t u r e X . . . . . . ,',..... o n e o n l y P e r f e c t l y S h r r e d R e a c t o r , , . " .
4 2000-1
R2 (2/3) e R1 (1/3)
zE
-10
, oo I
..~ ,
/'
:"
......- " ..........................
"..... "
I I
9.....
N -20
~ >
-30
o~
-40 -50
~ ~oo~....~ 6
i'o 2'o 3'o 4'o so
% Recycled
Flow
Fraction
Fig. 3. Effect of external or selfinduced Flue Gas Recirculation (FGR) on NOx emissions, simulated by RNA.
~
600,~'" . , x . . . . . . . . . . . . . I 1600 1650 1700 1750 1800 1850 1900 1950 2000 Temperature (~
Fig. 4. Effect of changes of peak flame temperature and extreme case "flameless combustion", represented by one only Perfectly Stirred Reactor. The cross labels show: on the abscissa, the mean temperature of the furnace (1660 ~ on the ordinate, the NOx predicted by the unchanged model.
426 For the 10 MW furnace the measured value was 1600 against the simulated value of 1572 mg/Nm3@8%O2 as NO2 (less than 2%). A sensitivity analysis performed on the reactor network showed that the increase of number and the change of typology of the reactors affected the result by 3 %, while the oxygen concentration and the temperatures are crucial parameters. Furthermore an estimate of the effect of some NOx reduction techniques was given performing small variations of some key parameters of the reactor network model. As shown in Fig. 3 both the external and self-induced flue gas recirculation can contribute to lower the emissions. Fig. 4 shows how they could be reduced lowering the peak flame temperature; Fig. 4 shows also the NOx produced vs. temperature by one only Perfectly Stirred Reactor with the volume as the furnace, receiving all the feeds. This to represent the extreme case "flameless", when there are not peak temperatures, as the fuel oxidize without burning, because of the full mixing of high preheated air with exhaust. The two graphs cross at the value of the mean temperature of the furnace (1660 ~ The slope of the graph relative to the only PSR is very high, showing that slight changes in the mean operating temperature of the furnace can produce strong changes in the NOx emissions. 5. CONCLUSIONS A procedure, called RNA, have been developed. It allows to extract from the CFD fields a Reactors Network model, on which detailed kinetic mechanisms can be employed for the calculation of the pollutant formation/destruction. This approach would be a reasonable tradeoff between the complexities of the phenomena occurring in reactive flow fields and the engineering demands for addressing the design of practical combustion systems. The concept is not restricted to the NOx calculation, it can be used also for any other species for which a detailed reaction scheme is available. This procedure well performed for glass melting furnaces: a 12 reactor network suitably describes the complex flow in these systems, resulting in a very good agreement between measured and predicted NOx concentration in the exhaust. Furthermore, it have been shown how changing some key parameter of the reactors model it is possible to estimate the effect of some NOx reducing techniques. REFERENCES
1. Benedetto, D., Pasini, S., Falcitelli, M., La Marca, C. and Tognotti, L., Mediterranean Combustion Symposium- 99, Antalya, Turkey, 1999, pp. 432-443. 2. De Michele, G.,. Ligasacchi, S., Pasini S., Tognotti,L.,-"Designing low-NOx combustion systems" Int. Symp. of AFRC, Baltimore, Oct.1996. 3. De Michele, G., Pasini S., Tozzi, A., "Simulation of Heat Transfer and Combustion in Gas and Oil-Fired Furnaces". The Combustion Institute, Meeting between Italian and Soviet Sections. Pisa, Italy, November 5-7, 1990. 4. Ranzi, E., Faravelli, T., Gaffuri, P., Sogaro, A., D'Anna, A., and Ciajolo, A., Combust. Flame, 108:24-42 (1997). 5. Benedetto, D., Gelmini, A., Mola, A., Pasini, S., Santero, A., XVA.T./. V. Meeeting-Glass Industry Towards 2000- Parma, 15-17 September 1999. 6. Tozzi, A., Merlini, S., Bellanca, R., "Manuale all'uso di IPSE". Matec srl., Milan, 1997. 7. Carvalho, M.G., Farias, T. and Fontes, P. In: Fiveland, W.A. (Ed.) Fundamentals of Radiation Heat Transfer, ASME HTD, 1991, 160:16-26. 8. Westbrook C.K. et al., Prog. Energy Combust. Sci. 10:1-57 (1984).
European Symposium on Computer Aided Process Engineering - l0 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
427
C F D - A n a l y s i s o f Heat Transfer and Initiator M i x i n g Performance in L D P E H i g h Pressure Tubular Reactors F.-O. M~ihlinga, A.
D a i B b,
N. Kolhapure c and R.O. Foxr
Elenac GmbH, Technology Center TER, M510, 67056 Ludwigshafen, Germany bBASF AG, ZAT/EA, L544, 67056 Ludwigshafen, Germany c Iowa State University, 2114 Sweeney Hall, Ames, Iowa 50011-2230, USA a
Computational Fluid Dynamics combined with accurate physical property data and reaction kinetics provides the opportunity to study important process engineering problems in detail. In this study CFD is applied to both heat transfer and mixing in high pressure tubular reactors for LDPE manufacture. Two very important physico-chemical processes occurring in these reactors are a) the rapid mixing of initiators with the bulk ethylene/polyethylene mixture, and b) the efficient removal of heat from the reaction mixture. Simulation results are compared with standard values for the heat transfer coefficients. Significant differences were found for the polymerization section along the tubular reactor. In this part of the reactor mixing plays an important role for initiator efficiency and the formation of product characteristics. By use of the four environment CFD micro-mixing model the initiator efficiencies for a common peroxide initiator and appropriate reactor stability curves were calculated. 1. Introduction and Motivation
The high pressure tubular reactor process is used for the manufacture of low density Polyethylene (LDPE). The Polymer properties, characterized by the molecular weight distribution and the number of short and long chain branches are mainly determined by the temperature, pressure and by modifier concentration during polymerization. Monomer conversion in tubular LDPE reactors can reach values up to 40 % and is strongly influenced by the ability to remove heat from the reaction mixture. It is widely believed that there exists a polymer film at the inside reactor wall which should be responsible for the observed large heat transfer resistance. On the other hand, thermodynamic investigations of phase behavior exclude phase separation in an equilibrated system under the typically applied polymerization conditions. Therefore fluid dynamic investigations were made to understand these phenomena from first principles. The goal is to improve the ratio of heat transfer to pressure loss in tubular reactors. Other criteria for optimal operating conditions in LDPE reactors have to consider fast initiator mixing. If mixing time is much shorter than the initiator half life, high initiator efficiencies and good product homogeneity can be achieved. Mixing is also crucial in avoiding hot spots and thus preventing transient and global ethylene decompositions. As tubular reactor capacities increased over the last 20 years from 60.000 t/y up to 320.000 t/y per line, there became a strong demand for simulation tools for plant design, scale-up and process optimization. Therefor, reaction kinetic models combined with fluid dynamics and accurate physical data can be applied. The presented simulation results were done for Elenac's LDPE process LUPOTECH T.
428 2. The Tubular Reactor Process
LDPE tubular reactors consist of a large number of jacketed tubes with a total length to diameter ratio between 10.000 and 40.000. The inner diameter of the high pressure tubes range between 10 and 80 mm having a monomer throughput between 10 and 140 t/h. Polymerization pressure is typically between 2000 and 3500 bar and maximum temperatures are well below 350 ~ In the cooling jacket pressurized hot water in co- or countercurrent flow absorbs approximately 50 % of the polymerization heat, which must be removed to achieve high monomer conversions, to meet the desired product specification and to avoid thermal decompositions of the reaction mixture. Initiators like peroxides or oxygen are introduced at several injection points starting the free radical polymerization. Polymer molecular weight is controlled by means of chain transfer agents such as hydrocarbons, ketones or aldehydes. Monomer conversions can reach values up to 40 % per pass depending on the product grade. Under polymerization conditions the polymer is dissolved in the reaction mixture. The separation of polymer and unreacted monomer occurs typically in two stages at pressures of about 300 bar and 2 bar, respectively. The unreacted ethylene is recycled to the appropriate compressor stage and the polymer is extruded, pelletized and conveyed to degassing silos. For a detailed description of Elenac's Tubular Reactor Process LUPOTECH T see [ 1]. 3. Modeling 3 . 1 . 2 D Reactor Model
For the description of the heat removal from the reaction mixture a reactor model has been developed and implemented in the commercial software package CFX4 | [2]. The model describes the fluid dynamics of the reactor in detail taking into account the variation of density, specific heat, thermal conductivity and viscosity as a function of temperature, pressure and mixture composition [4,5,10]. However, the model neglects micro-mixing effects as it supposes that the initiator is instantaneously perfectly mixed over the whole reactor cross section. A detailed reaction kinetic scheme is used to describe the polymerization [10]. In order to describe the evolution of the molecular weight distribution function the method of moments is used. Partial differential equations of the form
a---7
x-T= ax--7
+
9
are solved for the zeroth, first and second moment of the molecular weight distribution function and the other chemical species present in the system. Herein, ~b represents the respective moments or species mass fractions, respectively. The model results in a set of thirteen coupled partial differential equations which are solved for a simplified two dimensional axi-symmetric reactor geometry. For the third moment the closure assumption of Hulburt and Katz is applied [3]. Since turbulence has a very important influence on mixing and heat removal from the reactor much care has been taken about the choice of a suitable turbulence model. In order to be able to resolve the laminar sub-layer at the wall of the reactor, which is supposed to be a limiting factor for heat removal, the Wilcox k-co model [6] is used for the description of turbulence effects in the flow. This model offers also the advantage that a strongly varying viscosity can be easily taken into account.
429
Figure l : 2 D Reactor Model The heat transfer between the inside wall of the reaction tube and the cooling water flow is modeled by heat transfer resistance factors for the reactor steel tube, the fouling layer at the walls of the cooling water flow and the heat transfer resistance factor due to the boundary layer in the cooling water flow [7,8]. Figure 1 shows the geometric model that underlies the calculations. Calculations were done for different flow velocities, tube diameters and monomer conversions (viscosity). Strong deviations from the standard correlation are revealed directly behind the initiator injection points. The quality of the initiator injection itself determines the probability of local hot spots in the reaction mixture. This topic will be tackled using the 4 environment CFD-model described in section 3.3. In figure 2 the calculated polymer mass fraction is shown for a reactor section of 100m length which starts directly in front of a initiator injection point. The flow velocity is selected for calculations to be 8 and 16 m/s, respectively. In figure 2 the radial co-ordinate is scaled by a factor of 1000. It can be seen that immediately behind the initiator feed a region with a high polymer mass fraction is formed at the wall. There the polymer mass fraction reaches values as high as 60 percent. This region can be interpreted as a polymer-rich flow layer with a very high viscosity, even it is not a thermodynamically caused phase separation. This behavior can be understood from the longer residence time of the fluid near the wall and the turbulent transport of initiator to this layer. Further downstream the reactor axis this layer is dissipated by turbulent mixing and disappears completely about 40 to 100 m downstream the initiator injection point depending on the considered mean flow velocity. For high mean flow velocities the mixing is stronger and thus the layer is more rapidly dissipated than for low flow velocities.
Figure 2: 2D distribution of polymer mass fraction calculated from CFD modeling
Figure 3: Heat transfer coefficients along the reactor axis
430 Figure 3 shows the effect of layer formation on the heat transfer coefficient, which is plotted as a function of the axial position. Heat removal breaks almost completely down where the "polymer layer" is formed. It decreases down to a value of almost zero, since the polymer layer possesses a very high viscosity and a very small flow velocity which leads to laminar flow conditions in that layer. Downstream, as the polymer layer is dissipated, the heat removal re-increases and reaches a constant value where the polymer layer is totally disappeared. Due to the higher polymer mass fraction and thus higher viscosity of the reaction mixture the heat transfer coefficient is smaller than in front of the initiator feed position. A comparison between the simulated heat transfer coefficient along the axial position and coefficients taken from a standard correlation [7,8] is also given in figure 3. As expected, the simulated values and those taken from the correlation compare rather well in front of the initiator feed and far behind it. Between these two boundaries simulation predicts much lower values for the heat transfer coefficient leading to a much lower overall heat removal in this section. Once again, this effect is due to the polymer layer formation which is not taken into account in the standard heat transfer correlation.
3.2.3D Reactor Model Without Polymerization Kinetics For the optimization of the peroxide initiator injection and mixing device 3D calculations were performed for several geometries and operating conditions. In these calculations the geometry of the reactor tube and the injection device was modeled in detail. However, chemical reactions were not taken into account. Also, the transport and material properties were assumed to be constant. The initiator injection was treated as a pure mixing problem. Turbulence was taken into account by the RNG-k-e-model. The three dimensional calculations were done with the commercial CFD code CFX4 |
Figure 4: 3D Simulation of Initiator Mixing
431 Figure 4 shows the mixing of the initiator into the ethylene stream for two different geometries. In the first geometric case the tube diameter is constant throughout the whole initiator injection section, whereas in the second geometric example the tube diameter is reduced to 2/3 of the original tube diameter around the injection point. About 1.9 m behind the initiator injection point the initiator is much better mixed in the geometry with reduced diameter. By looking at the time scale it becomes even more obvious that the diameter reduction leads to a much better (faster) mixing. After about 0.057s (the time the fluid needs to reach the position of 1.9 m in geometry 2) the fluid is already well mixed in the geometry with diameter reduction, whereas it is badly mixed in the case without diameter reduction. By use of this 3D reactor model a detailed design of initiator injection devices becomes feasible. 3.3. Four Environment CFD Micro-Mixing Model Hot spots in LDPE High Pressure Tubular Reactors should be avoided by means of improved mixing and use of appropriate initiator mixtures. Therefor the mixing of initiator is investigated with the help of a so called four environment CFD micro-mixing model [9], which takes into account that for fast reactions the reaction rate is limited by transport processes on a molecular level. The impact of the initiator injector/nozzle design on initiator efficiencies and hot spot formation can be predicted. The micro-mixing model is used together with the same detailed kinetic scheme for polymerization as mentioned above extended by decomposition reactions. The model is used for a two dimensional analysis of the influence of the initiator injection mode on hot spot formation and inhomogeneous reaction conditions. Since hot spot formation is supposed to be the source of local or global ethylene decomposition the influence of the injection mode on the product quality and the stability of the reactor operation can be studied [11]. As an example of the results figure 5 shows a stability diagram for the reactor operation using TBPP (tert.-Butylperoxypivalate) as initiator. It can be seen that for a given mean temperature and pressure a stable reactor operation can only be assured for a limited range of injection modes.
Figure 5: Reactor Stability Plot for the Initiator TBPP (tert.-Butylperoxypivalate)
Figure 6: Initiator Efficiency of the Peroxide TBPP (tert.-Butylperoxypivalate)
432 For a given initiator concentration and centre-mode injection of the initiator TBPP, the safe operation of the reactor is ensured if the monomer feed temperature is less than ca. 290 ~ As the extent of premixing between initiator and monomer increases due to ring- and uniform-mode injections, the possibility of reactor runaway decreases. Moreover, the large unshaded area (see fig. 5) indicates minimal possibility of the decomposition in case of low temperature initiators such as TBPP, and the reactor can be operated safely even at very high feed temperature when perfect mixing can be achieved. The imperfect mixing of initiator can lead to its inefficient consumption and can have a direct impact on the product quality. Each type of initiator has his special range of operating temperatures depending on the quality of mixing. As an example figure 6 shows the efficiencies of TBPP depending on temperature and the extend of premixing (represented by the fractional volume Pl in the 4-environment model [9]). Initiator efficiency curves were calculated for a variety of peroxides and can serve as a basis for the development of initiator mixtures depending on the desired T and p range. 4. Conclusion
There are process and operational relevant impacts based on the results from fluid dynamic and kinetic investigations. Mixing inside the reaction fluid has considerable influence on cooling and on the quality of the polymer product, especially on the occurrence of high molecular weight material. By use o f the model presented, an optimization of reactor performance and product characteristics becomes feasible. Coolant operating conditions as well as the ethylene flow velocity, initiator mixture compositions and initiator injection nozzles can be optimized using the combination of reaction kinetics and fluid dynamics in computation. References
[1]
Elenac GmbH, Licensing Department, Brochure: L U P O T E C H T High Pressure Tubular Reactor Process for LDPE, Frankfurt (2000).
[2]
CFX 4.2 Users Manual: AEA Technology, Oxfordshire, U.K. (1997).
[3]
Hulburt, H.M., Katz, S., Chemical Engineering Science, 19 (1964) 555.
[4]
H. Benzler, A. v. Koch, Chem.-Ing.-Tech., 27 (1955) 71. Raft, R.A.V.; Allison, J.B.: Polyethylene, Interscience Publishers, New York (1956).
[5] [6]
Wilcox, D.C.: Reassessment of the Scale-Determining Equation for Advanced Turbulence Models, AIAA Journal, 26 (11), (1988).
[7] [8]
VDI-W~irmeatlas, 5. erweiterte Auflage, VDI-Verlag, Lb2. (1998). VDI-W~irmeatlas, 5. erweiterte Auflage, VDI-Verlag, Abschnitte Eb 1-Eb6, Gb 1-Gb6. (1998).
[91 R.O. Fox, Chem. Engng. Proc., 37 (1998) 521. [10] J. Schweer, Dissertation Thesis, G6ttingen (1988). B. Tilger, Dissertation Thesis, Darmstadt (1989). F.-O. M~.hling, R. Klimesch, M. Schwibach, M. Buback, M. Busch, Chem.-Ing.-Tech., 71 (1999) 1301. [ 11] K. Tsai, R.O. Fox, AIChE J., 42 (1996) 2926. S.X. Zhang, N.K. Read, W.H. Ray, AIChE J., 42 (1996) 2911.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
433
Dynamic Simulation of Complex Reaction Schemes and Biochemical Applications in Stirred Tank Reactors with Respect to Imperfect Mixing Dipl.-Ing. U. Boltersdorf, Dr.-Ing. G. Deerberg, Dr.-Ing. S. Schltiter Fraunhofer-Institute for Environmental, Safety, and Energy Technology Osterfelder Strasse 3, 46047 Oberhausen, Germany
Abstract: This paper presents a model of stirred tank reactors, which is able to solve the dynamic mass and energy balances on the basis of certain fluiddynamic simplifications. One possibility is to give a macroscopic flow field as an input, but it is more suitable to fit local distributions of velocities to the reactor model. To close the model, empirical knowledge for the required model parameters is included in form of empirical equations. 1. INTRODUCTION Stirred tank reactors are widely used in process industries for their flexible mode of operation. These reactors can be used in batch or semi-batch processes (i. e. production of fine chemicals) or they run continuously as usually done in large scale production (i. e. polymerization processes). The design of these reactors is often based on simple models or empirical equations for single design parameters such as for example heat or mass transfer coefficient. In contrast to this there are specialists of Computational Fluid Dynamics (CFD), who compute the flow field of stirred tank reactors. It is nevertheless a non standard problem to compute flow field, chemical reactions and in case of multiphase flow mass transfer simultaneously as necessary for reactor design. Therefore we developed a reactor model which is able to solve the dynamic mass and energy balances on the basis of certain fluiddynamic simplifications. One possibility is to give a macroscopic flow field as an input, but it is more suitable to fit local distributions of velocities to the reactor model. The velocities can either be measured by L D A - o r PIV techniques, calculated by means of CFD or using empirical equations published by Platzer and Noll. To close the model, we integrated empirical knowledge for the required parameters in form of the above mentioned empirical equations.
2. STRUCTURE OF THE MODEL It has been proved that the network-of-zone model [ 1] is able to describe the imperfect mixing in stirred tank reactors and the dynamic performance of the reactor simultaneously. Using a network-of-zones approach to model the mixing characteristics in stirred tank reactors, the whole reactor is divided into smaller perfectly mixed cells which are connected corresponding to typical flow patterns. The model as presented by Mann and Knysh envisages the flow as an axisymmetric, two-dimensional flow of liquid through a fixed network of cells ("numerical grid") forming concentric circulation loops. Their models describe the convective flow in the
434
concentric loops by a volumetric flow rate, related to circulation flow number Cz of the impeller: 3
n 9dimp Vloop = C z
"limp "Ncells.radial
(1)
Additionally the mass exchange due to turbulent motion is described by lateral equal and opposite exchange between two concentric loops, which is modeled by a turbulent exchange coefficient O:
%,, =O'V,oo.
(2)
Because of lateral equal and opposite exchange, turbulent mass exchange does not affect the total mass balance of each cell, but the mass balance of each component. Although the results of M a n n el. al. are very impressive, some limitations can be observed: 9 The model assumes that the total volume can be represented by the circulation patterns. It is well known, that for some geometric conditions there are regions of negligible convection and turbulence. These regions usually appear at the bottom of the tank or close to the free liquid surface. 9 The model of convective fluxes using a volumetric flow rate can not describe local flow phenomena (i. e. secondary vortices). 9 There is no proposal to estimate the model parameter for turbulent mass exchange ~. This parameter is defined as a constant and therefore neither a function of time nor of process parameters. In contrast to this, it is well known that there is a distribution of turbulent kinetic energy and its dissipation rate. Based on the previous studies we developed an improved model compensating the limitations illustrated above. The basic idea is not only to divide the reactor in smaller cells, but in regions which are characterized by transport phenomena. This includes the identification of areas in the continuous liquid phase below the free liquid surface (zone 2) with negligible convection and turbulence, which are therefore dominated by diffusion and natural convection (called "stagnant zones" or "dead zones"). The algorithms and criteria for this identification are reported by Kipke and Platzer [2]. Additionally we consider a continuous gas phase at the top of the apparatus. This region, named zone 1, is again dominated by diffusion and natural convection. It is not allowed that dispersed phases appear in zone 1, whereas dispersed solid and gas may appear in zone 2. The calculation of diffusional mass exchange is based on Fick's law, whereas natural convection is calculated by means of transport coefficients reported by Rayleigh or Grashof. Most parts of the reactor volume are of course dominated by the circulation flow induced by the impeller. Therefore the most important modeling effort refers to convection and turbulence in these regions. As we pointed out above, we want to model flow characteristics as detailed as possible. Therefore we established the possibility to use local velocities and local turbulent properties to compute the coupling energy and mass fluxes between adjacent cells. These data can either be obtained from measurements of the flow field (LDA-or PIV-measurements), calculations by means of CFD or using empirical
435 correlations. The distribution of local velocities can directly be used to compute the mass exchange between two cells, equation (3) and (4): I V ( ( i , j ) = c ( ( i , j ) . Ara d (i, j ) . e,i q (i, j ) . Wrad (i, j )
(3)
l~l~(i, j ) = cr
(4)
j ) . Aax (i, j ) " Eli q (i, j ) " Wax (i, j )
Equation (3) and (4) describe the mass flow leaving cell (i,j) in radial respectively in axial direction. Knowledge of local velocities includes information about secondary flow patterns and gives a more precise picture of mixing characteristics. The description of turbulent mass exchange is given in different ways. One possibility to describe this transport mechanism is to use the turbulent exchange coefficient, but in contrast to M a n n ' s model this coefficient is allowed to vary with the position in the vessel. To get a dimensionless parameter we relate the local turbulent dissipation rate to its maximum value in the reactor: O(i, j ) - e(i, j )
~
_R~ turb(i,j)-- fb(i,j)"_IV ....(i,j)
Emax
(5)
Other approaches to model turbulent dispersion can be titled as follows: 1. Dispersion models 2. Using fluctuating velocities to compute mass fluxes referring to turbulence 3. Stochastic calculation of mass fluxes. To evaluate dispersion models additional mass fluxes are computed on the analogy of F i c k ' s law of diffusion. The turbulent dispersion coefficient can be calculated when the turbulent kinetic energy k and the dissipation rate e are known using the assumption of isotropic turbulence: Dt . v,. Sc
.0.09. . k2
Sc. e
w i t h k - -3- . w '2 f o r i s o t r o p i c t u r b u l e n c e 2
(6)
To use this approach the fluctuating velocities have to be known. As illustrated above this is possible using P l a t z e r ' s approach as well as CFD calculations or measurements. The fluctuating velocities can furthermore directly be used to calculate the turbulent dispersion by equation (3) and (4) replacing the fluid velocities. The assumption of isotropic turbulence moreover holds. Stochastic approaches are implemented as follows (Eq. 7), but solving the material balances is difficult because of stochastic elements in the Jacobian matrix. The upper limit of one third is set with regard to experimental data. iv (,urb =
RAN(O;~). ]v ~onv
(7)
3. EXAMPLE OF CHEMICAL REACTION As an example for chemical reaction we present the esterfication of acetic anhydride with methanol catalyzed by sulfuric acid in the liquid phase. The reaction system consists of
436
parallel and consecutive reactions. This reaction recently was object of research for various reasons: 9 The reaction system is highly exothermic and is thus suitable for safety studies. 9 The kinetic is well understood, especially the effect of catalysis can be included into the kinetic model. Experiments can be done in standard apparatus and all species are simple to handle, easy to dispose and not too expensive. So it is obviously an appropriate model reaction system for reactions in stirred tank reactors. The reaction network consists of the following three reactions: Acetic Anhydride + Methanol
~
Acetic Acid + Methyl Acetate
(1)
Acetic Acid
~
Water
(2)
~
2 Acetic Acid
+ Methanol
Acetic Anhydride + Water
+ Methyl Acetate
(3)
Reaction (2) is in fact a reversible reaction, but will be treated as irreversible with a modified kinetic expression. The dominating reaction is reaction (1), whereas reaction (3) is insignificant to reactor dynamics in most cases (for volume fraction of water less than 6%). Based on the kinetic data measured by Neumann [3], including the influence of the sulfuric acids (catalyst) concentration, our model is able to predict spatial distribution of any substance and temperature as well. Figure l a shows the average concentrations of substances as function of time, when methanol and water is added in semi-batch operation. The semi-batch feed time was 600 s. The added water causes a dramatic increase of formation of acetic acid due to reaction 3. Figure l b shows the distribution of temperature at t = 22,5s. You can see that almost no radial gradients exist, although a cooling jacket is installed. But in axial direction temperature is not distributed homogeneously (high temperature at the free liquid surface, about 335 K and lower ones, about 320 K, at the bottom). These simulations are done using a velocity distribution for the convective transport and fluctuating velocities to model turbulent dispersion, as described by Platzer.
Figure 1: (a) Concentration of substances of example process
(b) temperature distribution
437
4. EXAMPLE OF BIOCHEMICAL REACTION As an example of biochemical processes the fermentation of lactose under anaerobic conditions using Lactobacillus plantarum is presented. This process is characterized by product inhibition, because growth of biomass decreases with lower pH. Fu and Mathews [4] presented a kinetic model including the required information about pH dependency of growth and yield coefficient. Our calculation of biochemical conversion kinetics is done by using the program BIOSIM [5]. The results (Fig. 2) presented refer to a batch fermentation process (Simulation 1) and a fedbatch process (Simulation 2) in a 1.8m 3 vessel stirred with a pitched blade turbine. The pH decreases with production of lactic acid, but pH = 4,5 is set as lower limit. This can be achieved by a controller system. Further decrease in pH otherwise will limit production of lactic acid significantly. For the fed-batch process additional lactose is added at the bottom of the tank directly below the impeller during the 10th and 16 th hour. Due to the long process time no remarkable spatial deviations in state variables are observable. The reactor in single phase operation can be modeled as ideally mixed except for the regions close to feed position. The feeding strategy enables an increase in yield of lactic acid, but this effect is even more impressive when the additional lactose is given at t = 60 h, when concentration of substrate is low. 60
...........................................................................................................................................................................................................................
50 40 .o ..a
= 0
30 20 10 0
'
0
~
~
10
20
30
40
50
60
70
80
90
100
time [h] --~ simulation 1, lactic acid
--~ simulation 1, biomass
--~ simulation 1, lactose
--~ simulation 2, lactic acid
--~ simulation 2, biomass
- 4 - s i m u a l t i o n 2, lactose
Figure 2: Dynamic performance of the reactor for both processes 5. C O N C L U S I O N S We developed a model which is able to describe imperfect mixing and dynamic behavior of stirred tank reactors simultaneously. This model bases on certain fluiddynamic simplifications, but allows the representation of macroscopic flow structures by using characteristic networks of ideally mixed cells. It is furthermore possible to fit a distribution of velocities to the network of cells. These data can either be obtained from measurements, semi-empirical equations or CFD calculations. The model provides the possibility to use
438 different methods for modeling convective and turbulent mass transport in the reactor. The combination of this approaches has to be defined by the user of our software. The approach is suitable for the design of stirred tank reactors with regard to complex reaction schemes and biotechnological processes. These processes can be represented by special kinetic data. Reaction rates can be calculated as a function of local state variables such as temperature, concentration or in case of biotechnological processes, pH. The model makes a compromise on accuracy and complexity. It is as detailed as necessary to detect the problems of mixing and chemical reactions but as simple as possible to be solved in adequate time. 6. NOMENCLATURE 17 A c C Cp Cz d Dt i
volumetric flow rate area molar concentration constant pumping capacity circulation flow number diameter turbulent dispersion coefficient cell number in axial direction
[m3/s] [m z] [mol/m 3] [-] [-] [-] [m] [m2/s] [-]
Indices axial or ax cony max radial or rad turb
j k n N NB V w w' Z
cell number in rad. direction turbulent kinetic energy impeller speed number of cells Number of blades volume velocity fluctuating velocity number of impeller stages
[-] [m2/s 2] [rev/s] [-] [-] [m 3] [m/s] [m/s] [-]
Greek Symbols axial direction referring to convection referring to maximum value radial direction referring to turbulence
e e 0 ~
void fraction turbulent dissipation rate turb. exchange oefficient component
[-] [m2/s 3] [-]
7. REFERENCES
[ 1]
R. Mann, L. A. Hackett, Fundamentals of Gas-Liquid Mixing in a Stirred Vessel: An Analysis Using Networks of Backmixed Cells, 6th European Conference on Mixing 1988, Pavia [2] B. Platzer, G. NoU, Modelling of Local Distributions of Velocity Components and Turbulence Parameters in Agitated Vessels - Methods and Results, Chem. Eng. Progr., 23 (1988) pp.13-31 [3] J. Neumann, Zur Frtiherkennung sicherheitsrelevanter Betriebszustande in Chemieanlagen mit neuronalen Netzen, Ph.D. Thesis University Dortmund 1998 [4] W. Fu, A.P. Mathews, Lactic Acid Production from Lactose by Lactobacillus plantarum: Kinetic Model and Effects of pH, Substrate and Oxygen, Biochemical Engineering Journal 3 (1999), pp. 163-170 [5] U. Bergstedt, Mathematische Modellierung biotechnologischer Produktionsprozesse DECHEMA Jahrestagung 1999 Wiesbaden
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
The steady state analysis
of the twin
439
helix heat
exchanger
Elena Daniela Lavric and Vasile Lavric University POLITEHNICA of Bucharest Chemical Engineering Department RO-78126, Polizul-5, Bucharest, Romania Enhancement of the heat transfer is done, in many industrial applications, by p e r m a n e n t change of the fluid flow, as in spiral, compact or coiled tube heat exchangers. The Dean vortices, which appear as a result of the secondary flow, are responsible for the increase of local turbulence, and thus, the decrease of probability of stagnant zone development, which, in turn, lower the chance for the solids to sediment. A new patented heat exchanger was studied whose both cold and hot fluids flow along paired helical path. The ratio heat transfer aria~equipment volume is sufficiently high to classify it as compact. The experiments proved its capacity to deal with important thermal duties even for small driving forces, due to the high partial heat transfer coefficients obtained for low Reynolds numbers. Also, good values for the exergetic coefficient were acquired. A mathematical model for this heat exchanger was developed, its solutions permitting a better understanding of the impact that the design parameters like spiral step or wall thickness have upon its performance. This model consists of a system of ODEs, resulted from the spatial periodicity of the helical channels. The technique used to solve it has an iterative nature, because the temperature map must be assumed. The convergence is obtained if two successive maps are close enough. A fairly good agreement between the experiments and the model was observed. 1. I N T R O D U C T I O N One of the most popular methods to enhance the property transport is the continuous change of the flow direction, either using various obstacles arranged in many different ways, or using curved flow spaces. The main advantage of the later method is the smoothness of the surface. When a fluid flows through a curved space (in most practical cases, a duct) a secondary flow occurs, determined by the existing difference between two adjacent elements flowing with non-equal axial velocities [1+3]. The element flowing in the core region is pushed to the exterior wall by the centrifugal force acting upon it and, then, forced to move toward interior, along the wall, due to a pressure gradient. The secondary flow emerged appears like two twin vortexes, rotating in the opposite directions, which improve the momentum and heat transport (Dean vortices) [1+3]. The flow
440
Fig. 1 Twin s t r e a m s
field is divided in two regions: a core zone, where the centrifugal forces are balanced
Fig. 2 The overall h e a t t r a n s f e r coefficient dependency upon the hot fluid velocity
by the gradient pressure and a boundary layer where the pressure forces are variable in cross section. When a fluid flows t h r o u g h a helical duct, the secondary flow field induces a t r a n s p o r t of the fluid across the axial section of the duct, d e t e r m i n i n g t h a t the axial velocity profile to be curbed to the exterior side of the coil. Depending on the Reynolds number, there is a s u p p l e m e n t a l vortex, which appears and vanishes, eventually [1+3]. Based on these findings, a new type of h e a t exchanger was proposed, tested and p a t e n t e d [4]. Basically, it consists of two adjacent helical channels such as a l t e r n a t e hot and cold fluid s t r e a m s will exchange h e a t through common walls (Figure 1). Thus, in the axial direction of the exchanger there is a periodicity of these two fluids. The m a i n advantages of the twin helix heat exchanger are its compactness, the increased h e a t t r a n s f e r area and the e n h a n c e m e n t of the h e a t t r a n s f e r coefficients due to the curved flow [5]. 2. T H E E X P E R I M E N T A L W O R K
The e x p e r i m e n t a l work was conducted, on a semi-pilot scale, in a classic setup, with both hot and cold fluids being degassed and t e m p e r a t u r e controlled. The t e m p e r a t u r e was m e a s u r e d with six thermocouples (made of NiCr and NiSi wires if 0.3 m m diameter) placed at inlet, outlet and the middle of exchanger for each stream. The twin helix h e a t exchanger has the following geometrical characteristics: exterior radius = 0.028 m, interior radius = 0.0135 m, step (exterior radius) = 0.015 m, step (interior radius) = 0.012 m, m e a n thickness of the h e a t transfer wall = 0.005 m and total length of the active p a r t = 0.67 m. The e x p e r i m e n t was carried out at low flow rates and relatively small driving forces, to emphasize the good behavior of twin helix heat exchanger [5, 6]. As can be seen from Figure 2, there is an asymptotic increase of the overall h e a t t r a n s f e r coefficient with the hot fluid Reynolds n u m b e r since, due to the
441 small curvature radius, the secondary flow is expected to be well developed even for small axial velocities. The partial heat transfer coefficients (PHTCs) were computed by regression, using an original algorithm, the iterative ratio method, presented elsewhere [7] together with a thorough analysis of the results. It should be pointed out that, for the same Reynolds Fig. 3 The channels cross section and number, the values of the cold a r r a n g e m e n t (h-hot, c-cold, w-wall, u-up, PHTCs are slightly larger than the d-down, 1-1eft, r-right). corresponding hot ones, the departure decreasing with the increase of velocity. Here, we present, in Table 1, the final results, the parameters of the equation Nu = a Re b Prc (Pr/Prw)d , obtained by minimization of the square sum of residuals model-experiment. Table 1 Regression coefficients for the two regimes of circulation (Recrit-2100) regime a b c d laminar 0.759 0.447 0.45 0.106 turbulent 0.0794 0.611 0.84 0.156 3. THE M A T H E M A T I C A L M O D E L
To develop the mathematical model for the twin helix heat exchanger, a set of simplifying assumptions must be taken under consideration [5, 6]: plug flow for both fluids; the heat transfer through the lower and upper walls are due to the immediate neighbors; the cross section of the channels can be approximated by a trapeze. Keeping those in mind, one can depict the physical model as in Figure 3, which shows the cross section for two adjacent stream channels for hot and cold fluids (see Figure I for the overall picture). Observing the physical model, one is able to write the heat balance equation for the hot fluid (the steady state case): dT._ = - % {hiTh -[h:Thw~ + h3Thwd + h, (Thw, + Th,~)~} dz
442 where the notations are as follows: h 1 = h 2 +h 3 +2h o "h 2
h k e ;h 3
h L i ;h 4
h ~/(R e
R,) 2
(k e
El) 2 ;h
2 /1;DeqDhCphW h
For the cold fluid, the heat balance equation is:
dL dz
= --(~'c {[C2Tcwu + C3Tcwd -[-c4 (TcwI + Tcwr)]- ClTc}
for which the notations are: C1 . C 2 + C. 3 + 2 C .4 ; C 2. C k.e ; C 3 . C k i.; c 4 . c ~f(R e
Ri) 2 + ( L e - L i)2 ; c =
2 71;D eqPcCpcWc
In the last differential equation, the minus sign appears because the circulation is counter-current and both differential equations are coupled, as a result of the h e a t transfer. Since the dz is oriented and the hot fluid moves from zero to z, it is obvious that, for the cold fluid, it m u s t be a minus in front of dz. To compute the wall t e m p e r a t u r e , one has to write the conservation law for the heat fluxes across the walls. It is readily observed that, for each wall, there are two possibilities: the hot s t r e a m is between two colds ones, which represent the beginning and the end of a spiral, or the opposite situation. Remembering one of the simplifying assumptions, we can write the heat transfer balance for the upper wall: hot fluid central (see Figure 3 for details) (:I,h (Th -- Thwu ) = (I,c,left (Tcwu,left - To,left ) + O~r
(Tcwu,right -- To,right)
Since the wall is isolated three sides, there are no t e m p e r a t u r e gradients in it, so the upper wall t e m p e r a t u r e is Twu, given by:
Twtl
(J'hTh -t- {~r
+ 1~c,rightTc,fight
(~h "4-(~c,left q" O{'c,right
cold fluid central (same as in Figure 3, with reversed notations) The derived relationship is essentially the same as above, except t h a t indices c and h should be interchanged. At entrance and exit, we assumed t h a t the heat exchanger begins and ends with the cold fluid channel, so in the last equation ~h,right or {~h,left is, conveniently, dropped.
443 For the lower wall, the equations are the same, the only difference being the replacement of u with d as index. For the t e m p e r a t u r e of the lateral walls, the central fluid concept is used again. 9 hot fluid central 9 cold fluid central 9 right (index 1 for the cold 9 right (index 1 for the hot fluid) / left (index 2 for the cold fluid) / left (index 2 for the hot fluid) wall fluid) wall
where the overall heat transfer coefficient, kc0), is: 1
1
b --c(1)
O~h
---
5
+-+~ k
1
where the overall heat transfer coefficient, k~(l~, is: 1
1
k~,(1)
C~r
__-(~r
5
1
+__~_~ ~ ahl(2)
At both sides of the twin helix heat exchanger, the wall temperature formulae are as above, with the appropriate use of indices. 4. S O L V I N G
THE MATHEMATICAL
MODEL
The t e m p e r a t u r e profiles for both hot and cold fluid are the result of the integration, from one edge to the other, of the system of differential equations, which constitute the mathematical model of the twin helix heat exchanger. Due to the counter-current flow, imposed to maintain a relatively constant driving force along the exchanger, one of the equations has to be integrated backwards. The entrance of the hot fluid is considered the beginning of the integration process. Solving the mathematical model implies the knowledge of initial temperature fields for both fluids; if not, it is impossible to compute the wall temperatures. As the process of integration develops, new temperature fields are computed, replacing the old ones. When two consecutive fields agreed in the limit of an admissible error, the iteration process stops. The particularity of the solution of the mathematical model consists in the division of the whole channel through which a fluid flows into N spirals, each one from zero to 2n, viewed as distinct cells. The exit of each cell is the entrance for the following one. In this manner, we have to solve a system of 2N-1 equations, describing the behavior of the heat exchanger. A Runge-Kutta method was chose for the integration of the system of differential equations, with adjustable steps and h 3 precision.
444 1[~.
,
,
i
I\
I
I
hot ~uid
i
........ c o l d f l u i d 0.95
.~ 0 9
i
0.86
O8
0
02
I
04
06
08
Dimensionless l e n g t h
Fig. 4 Axial temperature profiles (countercurrent)
1
To design such an exchanger, the only adjustable variable is its length, since the technological process imposes the operating conditions. Thus, there are two iterative processes: o:o the inner one, in which are found the temperature fields; o:~ the outer, in which is found the length of the heat exchanger. A typical solution of the mathematical model is depicted in Figure 4.
5. CONCLUSIONS A new type of equipment, the twin helix heat exchanger, was designed, tested and mathematically modeled. The laboratory experiments permitted to find out, by regression analysis, the equation giving the partial heat transfer coefficients. These were used into the mathematical model to better understand the behavior of this new heat exchanger type. The results, both theoretical and experimental, showed that the twin helix heat exchanger could be better equipment than the classical heat exchangers. The exergetic yield is sufficiently high, its mean value being 0.637, thus proving the efficiency of the twin helix heat exchanger in performing heat transfer, even for low values of driving forces or velocities [5+7]. 6. R E F E R E N C E S
1. T.W. Gyves, T.F. Irvine and M.H.N. Naraghi, Int. J. Heat Mass Transfer, 4 2 (1999) 2015-2029. 2. L.J. Li, C.X. Lin and M.A. Ebadian, Int. J. Heat Mass Transfer, 42, (1999) 4147-3158. 3. A. Mokrani, C. Castelain and H. Peerhossaini, Int. J. Heat Mass Transfer, 40, (1997) 3089-3104. 4. V. Lavric, E. D. Lavric and D. F. Florea, Heat Exchanger, RO Patent No. 111131 (1995). 5. E.D. Lavric, Ph. D. Thesis, Univ. POLITEHNICA of Bucharest, RO (1998). 5. E.D. Lavric, Gh. Jinescu and D. F. Florea, The double helical heat exchanger: a theoretical and experimental approach, CHISA'96, 25-30 Aug., Prague, Czech Republic (1996). 7. Gh. Jinescu and E.D. Lavric, A new non-linear regression technique to compute the partial heat transfer coefficients, Chemistry and Chemical Engineering Conference, 16-18 Oct., Bucharest, RO, vol. I, (1997) 109-114.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
445
S i m u l a t i o n o f the b u b b l e f o r m a t i o n d y n a m i c s in r h e o l o g i c a l l y c o m p l e x fluids Huai Z. Li and Youssef Mouline Centre de G6nie Chimique des Milieux Complexes, CNRS-ENSIC-INPL, 1 rue Grandville, BP 451, 54001 Nancy Cedex, France E-mail : [email protected]
ABSTRACT - A new theoretical model was developed for describing the bubble formation at an orifice submerged in rheologically complex fluids. The equations of motion for the gasliquid interface were combined with the thermodynamic equations for the gas in the bubble and the chamber below the orifice as well as the fluid rheological equation. The present model is able to calculate the instantaneous shape of the bubble during its formation and determine the final size of detachment. The results predicted by this model compare satisfactorily with the experimental investigation. 1. INTRODUCTION The bubble behaviour in rheologically complex fluids is of key importance in such diverse fields as polymer devolatilisation, composites processing, boiling, bubble column, fermentation, cavitation, plastic foam processing and bubble absorption. In all such industrial processes, bubbles set the initial conditions for the heat and mass transfer from a dispersed gaseous phase to the liquid phase. Moreover, rheological properties control to large extent the final bubble size, shape and formation frequency in rheologically complex fluids. Due to the inherent complex nature of bubble phenomena, a complete theoretical analysis is still impossible at present. A somewhat simplified starting point in this field has been the study of bubble formation from a single submerged orifice. The literature on bubble formation from a single submerged orifice is large in Newtonian fluids. Despite the numerous theoretical and experimental investigations, the mechanism of bubble growth and detachment remains far from fully understood as pointed out by Ponter and Surati [ 1]. Especially, the study of bubble formation in rheologically complex fluids is relatively limited [2-3]. This is the topic for consideration in the present paper. 2. EXPERIMENTAL STUDIES The main features of the experimental set-up consist of a Plexiglas cylindrical tank surrounded by a square duct. The diameter of the tank was 0.30 m and its height was 0.50 m. Bubble generation was through an orifice of varying diameters (1 - 5xl 0 -3 m), submerged in the liquid on the centre at the bottom section of the tank. The air entered into the fluid and formed a set of bubbles rising in line. The bubble volume and shape were evaluated by means of camera visualisation and image analysis. It was also observed that under a stationary flowrate, the bubbles formed in line had the same shape and identical volume.
446 The three rheologically complex fluids used in this work were 1% (wt) polyacrylamide (PAAm) in 99% water, 1.5% (wt) PAAm in 49.25% (wt) water - 49.25% (wt) glycerol and 1.7% (wt) carboxymethylcellulose (CMC) in 44.6% (wt) water- 53.7% (wt) glycerol. A Rheometrics Fluid Spectrometer RFS II (Rheometrics Inc. USA) was employed to measure the rheological properties of these solutions that behaved as shear-thinning fluids. In the range of shear rates corresponding to the bubble formation and moving in this study, the rheological behaviour of these fluids can be fitted by the power-law model: z-:K~"
(1)
where r is the stress, 2> shear rate, K consistency and n flow index. A sequence of the bubble formation in 1.5% PAAm solution is shown in Figure 1.
Figure 1. Bubble formation through an orifice of internal diameter lxl 0 3 m in 1.5% PAAm solution at different stages: (a) beginning; (b) intermediate; (c) detachment.
3. MATHEMATICAL FORMULATION AND NUMERICAL SIMULATION We present a non-spherical bubble formation model by revising the model developed in Newtonian fluids to take into account the fluid rheological properties. The bubble surface is divided into a lot of small elements as shown in Figure 2. The modelling of bubble formation is based on the following main assumptions: the bubble grows symmetrically about the vertical axis on the orifice centre; the liquid around the bubble is incompressible and of infinite extent; the gas flow is adiabatic; the gas pressure inside the bubble is uniform. Gas enters the reservoir at a constant flowrate Q~. When the increase of the gas pressure in the bubble Pe is great enough to overcome the sum of resistances due to the hydrostatic pressure and surface tension, gas flows across the orifice and a bubble begins its growing procedure. The modelling consists essentially of two motion equations, which describe respectively the radial expansion and the vertical ascension of the bubble.
447
Pc
QG~
Gas reservoir
Figure 2. Schematic presentation of the non-spherical bubble formation. As in any event the flow around the bubble is incompressible and irrotational, the radial velocity ur at distance r from the centre of the bubble and at time t after the initiation of the flow, follows from the continuity equation for incompressible fluids:
R2k u~ -
r
2
(2)
R and /? are the equivalent bubble radius (2/R = 1/R' + 1/R", see Fig. 2) and growth rate at time t. Substituting Eqn. (2) into the radial component of the equation of momentum conservation and integrating from the bubble wall R to infinity gives:
3
Rk +-R
2
2 =
p,-po PL
lmiw PL
d
(3)
where PL, P~ and PL are respectively the liquid pressure at bubble wall, ambient pressure and liquid density. The pressure PL is related to the readily measured gas pressure in the bubble PB through the normal stress condition at the interface: 2o" p PL + rrr[r=R +----~-= B o" is the surface tension.
(4)
448 Combining Eqns (3) and (4) leads to the equation that governs the radial expansion: (3k2) P, =P~+PL RR+
2
20+--+ R
4K(2x/-3~<(RR__")" n
(5)
It is worth noting that the above-developed formulation is only applicable in the case of the formation of a single bubble. When a train of bubbles is formed at the orifice under constant gas flow conditions, the pressure field in the wake of preceding bubbles will affect the formation of the next one. The modified equation taking into account this effect exerted at 0 = n at the mean height of the forming bubble is then given by: 23R2 ) + 2 0 -+4 K 2 ~ ~ ( ~ ] " P, =P~ + PL ( RR +-=R n ~,R)
pw
(6)
where Pw denotes the pressure in the wake [4-5]. The sign minus means decompression due to fluid's viscoelasticity. The motion equation governing the vertical bubble ascension is described by a balance of different forces: inertial, buoyancy and viscous drag forces, the vertical component of surface tension acting on bubble surface as well as gas momentum rate through the orifice: 11
d (VBU,)=(pL_p~)gVB
PG "at''-~PL ~
rcDZaxcz~pLUZ_4rcc r
--T
Req
cosOsinOdO+~
)TO2
(7) where VBand UB are the bubble volume and rise velocity, Dm~xmaximum bubble diameter, Do orifice diameter, C~ drag coefficient and 0 angle between the normal to the bubble surface at any element and the vertical axis. The pressure change due to the flow in the reservoir is adiabatic:
aPc
~:Pc
The pressure in the bubble PB is related to that in the reservoir Pc by the orifice equation with a constant ko [3]:
(to) 2
Pc - P8 = Oo
(9)
Equations (6) - (9) are then solved simultaneously at each element using a proper constitutive relation along with the following boundary conditions to get the bubble shape at each time interval by means of a finite-difference method. At the start of the computation (t = 0), the bubble is assumed to be hemisphere of radius equal to the orifice radius. As the bubble grows,
449 new points should be added by interpolation to assume the coherent increasing gas-liquid interface. The numerical calculation is finished when the bubble neck is closed: in this case, the bubble detaches from the orifice. The interface grid evolution is illustrated in Figure 3 in 1% PAAm solution.
Figure 3. Grid evolution of gas-liquid interface in 1% PAAm: Orifice diameter Do-- 1x l 0 -3 m, gas flowrate Qc -0.15x10 6 m3.s"1, time interval between two contours At = 0.05 s.
4. RESULTS AND C O N C L U S I O N An example of the theoretical bubble shape evolution in time is shown in Figure 4.
a
b
-3 Figure 4. Shape evolution in 1.7% CMC: Orifice diameter Do = 5xl 0 m, time interval between two contours At = 0.05 s except the last one. (a) Qc =0.4xlO -6 m3.s -1 ; (b) Qc =2x10 -6 m 3. s -1 .
450 A comparison of the bubble volume at detachment from the orifice is shown in Figure 5 between the experimental measurements and the predicted values by the model. The good agreement is observed for a wide range of experimental conditions of orifices, liquids, gas flowrates. 0.8 ~0.7
f
-
O
>.
I
0.6 0.5 -
o ;> 0 . 4 -
,.Q ..~ 0 . 3 -
Theoretical 9
Experimental
m 0.2 0.0
I
I
I
0.5
1.0
1.5
I 6
3
2.0
_]
2.5
Gas flowrate Qc. xl0 (m .s ) Figure 5. Agreement between the experiments and the model in 1.7% CMC. Orifice diameter Do - - 5xlO -3 m . A rigorous model, taking especially into account the in-line interactions, was elaborated to describe the bubble formation at a submerged orifice. This model can predict the instantaneous shape of the bubble during its formation and determine the final size of detachment of bubble formation. Good agreement of the simulation results with experimental data has been achieved. It is hopefully possible that the present bubble formation model and the knowledge on the bubble interactions and coalescence after their detachment [6-7] can lead to a complete understanding and modelisation of bubble behaviours in industrial installations. ACKNOWLEDGEMENTS" The support by the French Minist6re de l'Education Nationale, de la Recherche et de la Technologie (MNRT) is gratefully acknowledged. REFERENCES
1. A.B. Ponter and A.I. Surati, Chem. Eng. Tech., 20 (1997) 85. 2. A.K. Ghosh and J.J. Ulbrecht, Chem. Eng. Sci. 44 (1989) 957. 3. K. Yerasaka and H. Tsuge, Chem. Eng. Sci., 46 (1991) 85. 4. H.Z. Li, Y. Mouline, L. Choplin and N. Midoux, AIChE J. 43 (1997) 265. 5. H.Z. Li, Y. Mouline, L. Choplin and N. Midoux, C.R. Acad. Sci. Paris 324 (1997) 491. 6. H.Z. Li, Y. Mouline, L. Choplin and N. Midoux, Int. J. Multiphase Flow 23 (1997) 713. 7. H.Z. Li, Y. Mouline, D. Funfschilling, P. Marchal, L. Choplin and N. Midoux, Chem. Eng. Sci., 53 (1998) 2219.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
451
Coarse-grained formulation for the time evolution of intermaterial contact area density in mixing systems Alessandra Adrover, Marcello Fidaleo and Massimiliano Giona Dipartimento di Ingegneria Chimica, Universith di Roma "La Sapienza", via Eudossiana 18, 00184 Roma ITALY E-mail: [email protected] This article develops a simple coarse-grained time-continuous formulation for the evolution of intermaterial contact area density in laminar chaotic flows that can be implemented in CFD codes starting from knowledge of the velocity field. The Eulerian approach towards statistical, geometrical and measure-theoretical characterization of chaotic flows is highlighted. 1. INTRODUCTION In recent years great attention has been focused on analysis of the geometrical properties of partially mixed structures in fluid-flow systems [1-3]. In phenomenological terms, geometric invariant patterns can be clearly detected in twodimensional periodically forced flows, by injecting a blob of dye into a chaotic mixing system and by analyzing the evolution of interfaces between the dyed and undyed regions at periodic intervals of time corresponding to the characteristic period of the velocity field. This produces a very complex but invariant geometric structure formed by thousands of striations, the number of which grows exponentially in time, as observed in numerical simulations, laboratory-scale mixing systems and mixing devices of industrial interest [4-6]. The dynamic explanation of this result is that a chaotic mixing system behaves within a chaotic region as a hyperbolic system as defined by Anosov and Pesin [7]. As a result, the geometry of partially mixed structures is controlled by an invariant vector field (invariant subbundle) {eU(x)}, which at each point x gives the orientation of a generic material line passing through that point and evolved for sufficiently long time. In practical applications, it is of the utmost importance to obtain a statistical description of the lamellar system generated as a result of stirring, stretching and folding. Such a global characterization can be achieved by considering the intermaterial contact area density (or for two-dimensional systems the intermaterial contact length density), which expresses the fraction of contact area (length) between two fluid elements (dyed/undyed) per unit volume of the mixing system. [4]).
452 Analytical expressions for the intermaterial contact interface density have been obtained in [8-10] for two-dimensional time-periodic flows and three-dimensional model systems [11], but the approach envisaged in these articles requires a significant computational effort. In this work, we propose a very simple computational approach to the pointwise coarse-gained evolution of the intermaterial contact interface density, which is suitable for practical implementation in engineering mixing problems. For the sake of simplicity, a two-dimensional model flow (sine-flow) is considered, although the method can be straightforwardly applied to more realistic flows both two- and threedimensional. 2. LAMINAR CHAOTIC FLOWS AND INTERMATERIAL INTERFACE DENSITY Let us consider two-dimensional periodically forced systems (such as drivencavity flow) and a Poincar6 map 9 associated with the kinematics of a fluid particle advected by the velocity field. Given the particle position Xn at time t = nT (T is the period of the forcing), ~(Xn+l) returns the particle position at time t = (n+l)T. In this article we consider a simpler model flow on the two-dimensional torus that possesses the same qualitative properties as physically realizable two-dimensional time-periodic flows and is characterized by a Poincar6 map, O(x), x = (x~,x2) O(x)
x ~ + Tpsin(2 7tx 2)
mod. 1
x 2 + Tpsin(2rc(x 1 + Tpsin(27rx2)))
mod. 1.
(1)
This is referred to as sine-flow [12]. The parameter Tp is half of the period of the external forcing T = 2Tp. We shall use ~ , ( x ) = c3~(y)/c3y[r=x to indicate the Jacobian matrix of~(x). Since det(O*(x))= 1, sine-flow is a model system for incompressible flows. The phase portrait associated with Eq. (1) for Tp = 0.6 (considered in the simulations) is characterized by a main chaotic region C and much smaller islands of quasiperiodicity. This behavior is typical of many mixing systems, including stirred reactors [5]. As extensively discussed elsewhere [2,3], for each point x~C an asymptotic unstable direction eU(x) may be defined that spans the invariant subspace EU(x) at that point, and each tangent vector to a curve passing through x and evolved for a sufficiently long time will be attracted towards this unstable orientation. By enforcing this geometric property, it has been possible to relate the intermaterial contact interface measure to the properties of the stretching field. More precisely, let us use IXL(A,t) to denote the normalized interface measure at time t returning the fraction of the total length of the curve advected at time t starting from an initial curve ~/, and falling within the subset A of the mixing space. Normalization implies that the total length falling within the set A at time t is the product of ~tL(A,t) times
453 the overall length of the advected curve achieved up to that time instant. It has been shown both analytically (for some model systems [8,11]) and numerically [9,10] that IxL(A,t) for t = nT (i.e. sampled at the period), and for large enough n, converges towards an invariant measure IxL*(A) independently of the location of the initial interface. Such an invariant measure can be ~obtained by means of a sequence of approximants IxL(n), defined in terms of the densities pL(n)(x), dixL(n)(x) = pL(n)(X)dx and IxL~")(A)= ~AdIxL(n)dx converging in measure to IxL*(A) and given by p("' L,i : A a,i IIVr
i : 1,2
~
(2)
where (I)i"n is the i-th component of the n-th iterative of the inverse map ~-1, "11 is the vector norm and A~,i are normalization constants so that the resulting measure over the whole mixing space equals 1. Both the approximations for i = 1,2 yield equivalent results in the limit of n ~ ~. The spatial structure of the intermaterial interface measure is extremely singular. To give an example, Fig. 1 A shows the behavior of the box-sectional measure along the line x2 = 1/2. The box-sectional measure Ix *(xi,e) is the normalized intermaterial contact interface measure falling at fixed x2 and referred to an interval I=(xi e/2,xi+e/2) of length e and centered at xi, normalized in such a way that Y.iIx*(xi,8)8 = 1. For details, readers are referred to [ 10]. 6
,
,
,
,Z I
/
4
'
'
,
log < ~(~1 > /
09
0.2
A
0.4
Xl
0.6
A
0.8
1
Ol
,
,
,d"
I
2
,
I
3
I
4
n
I
5
~'
I
6
7
Fig. 1 : A) Box-sectional measure along x2=1/2 for the sine flow map at Tp = 0.6. B) Comparison of log < Z,~n) > (curve a and dots .) and < log Zr(n) > (curve b and clots o ) v s n obtained from the ergodic average along chaotic trajectory within C (solid lines) and from the coarse-grained approach Eqs. (7)-(8) for the sine-flow at Tp=0.6.
454 3. F I N I T E - V O L U M E
ALGORITHMS
FOR INTERFACE DENSITIES
The approach discussed in section 2 is particularly suitable for analysis of the statistical properties of the intermaterial interface measure but it cannot be directly applied to engineering problems - such as the modeling of reaction/diffusion kinetics - due to computer-time limitations and the need of a continuous-time formulation. Both these problems can be overcome by adopting a spatially discretized approach towards the spatiotemporal dynamics of the material interface density. From continuous mechanics [4], it follows that the interface density, henceforth pL(x,t), for a two-dimensional system satisfies the Lagrangian equation dPL = ( D ~,g)p L , dt
(3)
where d/dt=8/&+v.V, D= (Dij) is the deformation tensor
--
Dij
+
-2 C~Xj
.
6~ i
(4)
The symbol : in Eq. (3) indicates dot tensor product, and Z, is the unit vector tangent to the interface at point x. Eq. (3) can be used as a starting point to develop coarse-grained models encompassing the geometric properties discussed above. Let us consider a spatial discretization of the mixing space into cells (or,13), ~t, 13= 1, .., N, and let PL = (Pt~13) be the vector of discretized interface densities referred to this discretization. By enforcing the invariant geometric properties characterizing chaotic flows, it is possible to express the factor (D:ZZ) within each cell as the average of this quantity with respect to the invariant unstable directions (where they exist) for points belonging to a chaotic region, and to randomly oriented vectors for points belonging to islands of quasiperiodic motion. This approach is extensively discussed elsewhere. In point of fact, it is possible to develop the formal simplifications in a coarsegrained approach still further by skipping the averaging procedure discussed above and leaving the dynamics to perform it automatically. In a discretized formulation, Eq. (3) can be expressed e.g. by means of a finite-volume algorithm of the form dp~ + Fp~ = ( D ~,X)p~, (5) dt where FpL is the finite-volume representation of the convective term v'VpL. Eq. (5) should be coupled with the equation describing the convection of the field of tangent directions, which reads as
455 d~ + FE = A~,. dt
(6)
where AX is the finite volume representation of the term (Vv)Z,. Eqs. (5)-(6) are the coupled systems of Eulerian equations to be solved simultaneously in order to obtain a spatially discretized description of the intermaterial interface density and to compute all the statistical quantities of physical and practical interest. For example, the Liapunov exponent A associated with the flow can be obtained from the scaling of the quantity g(t)= t 1 ~(D'~,X,)
t',
(7)
a,p=l
and in particular g(nT) = nA. Analogously, the scaling exponent ,~ of the length of a generic material line can be computed from the quantity G(t) = log
1
~PL~,~(t) , ot,13=l
(8)
since G(nT) = n,~, as obtained from the solution of Eqs. (5)-(6). It is to be observed that the scaling exponent 8 is always greater for physically realizable flows than the Liapunov exponent of the system, as extensively discussed in [13]. Figure 1B shows the comparison of the scaling of g(nT) and G(nT) with the corresponding quantities < log ~,~(")>, and log < ~,~(")> obtained from ergodic averages. The quantity ~(")(x) is the stretching rate referred to the invariant unstable unit vector basis { e~(x)}, ~(")(x) = II~n*(x)eU(x)ll. The simulations refer to a 100• 100 discretization of the unit toms. A good level of agreement can be observed between the coarse-grained results and the ergodic averages. Let us now consider the evolution of the intermaterial interface measure e.g. starting from a horizontal initial interface located at x~ = 1/2. Figure 2 shows the comparison of the coarse-grained results obtained by solving Eqs. (5)-(6) and the corresponding quantity obtained through Eq. (2) by using the same spatial discretization e = 10.2 i.e. N = 100. It should be observed that the coarse-grained formulation is able to capture all the main non-uniformity (both qualitatively and quantitatively) in the spatial distribution of the interface length, although (as expected) the fine structure of the singularities associated with this measure is lost. In any case, the results deriving from the integration of Eqs. (5)-(6) are highly satisfactory for practical purposes since they make it possible to obtain the pointwise characterization of spatial interface distribution induced by chaotic flows in an easy and computationally economical way.
456 4
i
i
I
4
I
A
I
I
0.2
0.4
I
I B
PL
2
~
0.2
0.4
0.6 Xl
0.8
1
0.0
Xl
0.6
0.8
1
Fig. 2 ' Comparison of the sectional box-measure along x: = 1/2 (Figure A) obtained from Eq. (2), (e=102) and the coarse-grained density (Figure B) obtained from Eqs. (5) and (6) at t = 10T on a 100xl00 lattice.
REFERENCES [1 ] D. Beigie, A. Leonard, S. Wiggins, Chaos Solitons & Fractals 4, 749 (1994). [2] M. Giona, A. Adrover, F. Muzzio, S. Cerbelli, Chem. Eng. Sci. 55, 381 (2000), [3] M. Giona, A. Adrover, F.J. Muzzio, S. Cerbelli, M. M. Alvarez, Physica D 132, 298 (1999). [4] J.M. Ottino, The Kinematics of Mixing, Stretching and Chaos (Cambridge Univ. Press, Cambridge, 1989). [5] G.O. Fountain, D.V. Khakhar, J.M. Ottino, Science 281,683 (1998). [6] D.M. Hobbs, P.D. Swanson, F. J. Muzzio, Chem. Eng. Sci. 53, 1565 (1998). [7] A. Katok, B. Hasselblatt, Introduction to the Modem Theory of Dynamical Systems, (Cambridge Univ. Press, Cambridge, 1995). [8] M. Giona, S. Cerbelli, F.J. Muzzio, A. Adrover, Physica A 254, 451 (1998). [9] M. Giona, A. Adrover, Phys. Rev. Lett. 81, 3864 (1998). [10] A. Adrover, M. Giona, Phys. Rev. E 60, 357 (1999). [11] M. Giona, A. Adrover, Invariant geometric properties of a class of 3D chaotic flows, Physica D (1999), accepted for publication. [12] M. Liu, F.J. Muzzio, R.L. Peskin, Chaos, Solitons & Fractals 4, 869 (1994). [ 13] A. Adrover, M. Giona, F.J. Muzzio, F.J., S. Cerbelli, M.M. Alvarez, Phys. Rev. E. 58, (1998).
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rightsreserved.
457
Dynamic Optimization of Semicontinuous Emulsion Copolymerization Reactions : Composition and Molecular Weight Distribution C. Sayera'b*, G. Arzamendi ~ J.M. Asua b, E.L. Limaa and J.C. Pintoa aprograma de Engenharia Quimica / COPPE, Universidade Federal do Rio de Janeiro, CP: 68502, CEP 21945-970, Rio de Janeiro, Brazil. e-mail: [email protected]; pint o@p eq. coppe, ufrj. br. blnstitute for Polymer Materials POLYMAT and Grupo de Ingenieria Quimica- Universidad del Pals Vasco, Apdo. 1072, 20080 San Sebasti/m, Spain. e-mail: [email protected] ~Departamento de Quimica - Universidad Pfiblica de Navarra - Campus de Arrosadia, 31006 Pamplona, Spain. e-mail: [email protected] Iterative dynamic programming is used to compute optimal monomer and CTA feed profiles to produce polymer with pre-specified copolymer composition and MWD. This approach can deal with constrained optimizations of systems described by complex mathematical models, as those needed for the emulsion copolymerization kinetics, especially when the computation of the whole MWD is included. The proposed approach is applied to the semicontinuous MMMBuA emulsion copolymerization, using dodecanethiol as CTA, allowing the effective computation of feed policies for the production of constant composition copolymer with well-defined MWDs. 1. INTRODUCTION Many polymer latex applications, such as paints, adhesives and paper coatings, require well-defined molecular weight distributions (MWD). Therefore, there is a strong incentive to develop strategies to control the complete MWD, and not only the molecular weight averages/1-3]. Molecular weight averages can be misleading when the MWD presents bimodalities or high molecular weight tails, common in monomer systems that undergo transfer reactions to polymer chains or other reactions that lead to chain branching. The closed-loop control of the MWD of emulsion polymers is an unsolved issue because the on-line measurement of the MWD by GPC is rather time consuming and virtually impossible in emulsion polymerization reactions (due to the problems associated with sample preparation). In addition, for the general case, the MWD is not observable from measurements of other variables. Therefore, open-loop strategies should be used. In this work, the method of iterative dynamic programming t4'51 is used to compute optimal monomer and chain transfer agent feed profiles in order to produce polymer with prespecified copolymer composition and MWD. *Present address : Departamento de Engenharia Quimica - Universidade de S~o Paulo, Av. Prof. Luciano Gualberto, travessa 3, n 380, CEP 05508-900, S~o Paulo, Brazil. e-mail: [email protected]
458 This approach presents the advantage of being able to handle constrained optimizations of systems described by complex mathematical models, as those needed for the emulsion copolymerization kinetics. The proposed approach is applied to the semicontinuous MMA / BuA emulsion copolymerization, using dodecanethiol as chain transfer agent (CTA). 2. OPTIMIZATION PROBLEM FORMULATION The optimization procedure of iterative dynamic programming divides that the process time into several intervals and the integration of the process model is performed only in the intervals that are affected by the change of the manipulated variables under consideration. For instance, consider a process described by the following system of differential algebraic equations: dx - : f(x,u) dt
(1)
y : g(x,u)
(2)
where x ~ R n is a vector of state variables, y is a vector of variables computed by algebraic equations and u ~ R n is a vector of manipulated variables to be optimized and limited by:
(~i--
(3)
State variables and other output variables are limited by the following equality and inequality constraints; ~j =Tj,
j = l ..... s
and
Wj <_~,j, j = l ..... t
(4)
where ~ may be either x or y. Process constraints may be treated as penalty functions, resulting in an augmented objective function[6]:
F=)--'~Pk[_k:, Ok
+ : Pl
y~
where Ok are the objectives (desired values of ~k) and pk (k : 1, ..., N O ) and pl (1 = 1..... NC) are the penalty factors of objectives and constraints; for inequality constraints pr=O if q6<_y~. The problem then consists of the determination of constant profiles u / O in the time interval ilk-z, td, in order to minimize the objective function. 3. EMULSION POLYMERIZATION MODELING The mathematical model of the semicontinuous MMA/BuA emulsion polymerizations used in this optimization procedure was developed in a previous w o r k [7'17] and validated with experimental results [81. Polymerization mechanisms considered by the model include transfer to monomer, CTA and polymer, and termination by disproportionation. Using standard kinetic assumptions, the mass balance of initiator, monomer, CTA, radicals and polymer chains with length n for a semicontinuous seeded emulsion polymerization system, at constant temperature can be expressed as:
459
dI=i qe_i k dt e i
(6)
du = qe Uwe dt
(7)
fiNp dMi =Mie q o k [M~]p dt NA P dMT ~=MTe dt dR n - -
dt
-_
qe-
fiYp NA
(8) (9)
kfr [T]p
0 = ~+k~n[Mlp~-~Rn +k [T]pZR n IT
n=l
+
r-ptMapRn_l,Sn>l
+
nMn
Rn
- kp[M]pRn - kfm[M]pRn - kfr [T]pRn - (mk;p n~=lnMn) Rn - }I-[ 2kta N a k ) p~._. en Rn
dMn =kfM[M]pRn+kt~[T]pRn+I kfp n~__ln~}Rn kfp oo +I 2ktd ~ } R dt [NADp -- NADp nMnn_-l~ [NADp n=l
(10) n
(11)
where ~ is the average number of radicals per polymer particle, calculated as proposed by Hansen and Ugelstad[91; and [MJp and [T]p are, respectively, the concentrations of monomer i and of CTA in the polymer particles, calculated by the method presented by Eehevarria et al. [21. To consider copolymerization reactions, average rate coefficients were used ~~ In order to compute the MWD and avoid the huge computational effort needed to solve eqs. (1 0-1 1) for n = 1,...,5e5, an adaptive orthogonal collocation technique was apllied. This technique was developed by Nele et al. [121,based on a previous work of Pinto and Biscaia [~31. The basic assumption of the method is that the chain length distribution (CLD) ui(O of a polymer at any time t can be expanded in a series with the form: n
u i (t) = 0(i, p)~-~ a k (t,p)1 k(i)
(12)
k=0
where O(i,p) is a strictly positive integrable reference function, which may depend on a timedependent parameter p ; n is the number of collocation points, ak(t,p) are the expansion coefficients of the MWD; and lk(i) are Lagrange interpolating polynomials: n i-sj 1k (i) = l - I - j=oSk -sj j~k
(13)
where si are the (n+l) roots of (n+l) th degree polynomial of a family of orthogonal polynomials. The resulting system of algebraic and differential equations is solved with the numerical integrator DASSL [141.
460 4. R E S U L T S AND DISCUSSION The production of copolymers with homogeneous compositions and with well-defined MWDs (narrow unimodal, broad, as well as bimodal MWD) was considered for the dynamic optimization. In order to carry out the optimization, the process was divided into 11 intervals. Ten intervals for the feed period, plus one interval to achieve final conversion. To allow the independent control of composition and MWD, three feed streams were used: Feed 1) less reactive monomer (in this case, BuA), which determinates the reaction time; Feed 2) preemulsified monomer feed with CTA, emulsifier, water and a relatively small amount of MMA, which determinates the MWD; Feed 3) pre-emulsified monomer feed containing amounts of water, emulsifier and MMA, which determinates the copolymer composition and the solids content during the polymerization. The formulation considered in this optimization in presented in Table 1 and the model parameters are given in Table 2. Table 1. Formulation of seeded emulsion copolymerization. (80~ feed time = 70 min) Reactants (g)
Initial charge
Feed 1
Feed 2
Feed 3
36.30 146.5 BuA 234.3 AA 2.34 0.36 1.47 CTA 2.98* Water 650.0 28.0 113.0 Emulsifier 0.4 0.30 1.30 Initiator 1.50 Buffer 1.50 Seed 150.0 "5 g CTA in optimization presented in Figures 1 and 2. MMA
Table 2. Kinetic Constants. Parameter Value 0.2e-4 ki 4.6e7 kpAA 1.16e6 kpBB 0.711*kpBB kfn3 kfrB*kpAA/kpBB kfrA 1.06e0 km 1.56e3 kfA 0.573el kfpB 5.25e4 kfpA 0.203ell ktdB~ 0.36e7 ktdA~ rA, rB 0.414,2.24
Units 1/s cm3/mol s cm3/mol s cmS/mols cm3/mol s cmS/mols cm3/mol s cm3/mol s cm3/mol s cm3/mol s cmS/mol s -
Ref. [11] [15] [11] [16] [17] [18] [17] [10] [17] [11] [11] [16]
The optimization goals were included in the following objective function:
F=pl
f f 2 QB~ fQB~d Q~I +p2
f
d 2
~.,[y(i)__y(i)d
+p3 i_-1L
Y(i)d
+ k~..lp4 [Fk --Fkd --
L
(1
4)
F~fd
where the first two terms of the right-hand side member compare the amounts of monomers actually fed into the reactor with those in the formulation; the third term accounts for deviations in the copolymer composition along the process; and the fourth for the differences between selected points of the experimental (Fk=dWf/dLog(M)) and desired (Fkd) MWDs. Penalty functions were represented by weights (pi) with the following values: p~ = p2 = 40, p 3 - 8 and p4= 2. The values assigned to the penalty functions indicate that more importance was given to the control of copolymer composition than to the control of MWD. In order to initialize the iterative dynamic programming procedure, initial values must be assigned to the feed streams. Feed streams used to initialize the procedure were set constant and equal to the total mass of each feed stream of the formulation divided by the total feed time. Such feed conditions (constant feed streams) would lead to the production of a rather narrow unimodal MWD quite different from the desired MWD.
461 I
8 0.8
0.8
0 _1
0.6
0~
0.6 ..... ~.. ..... o . . . . . ~ ..... .~. ..... ~
"0
,,-.~ 0.4
.... z'~ ..... ~ . . . . . ~
..... ~ . . . .
~ .... -o
m 0.4 r
._o ,Q>
0.2
0.2
>
t-
0
2
0
O
....
3
4
5 6 Log(MW)
7
8
o
9
Fig. 1. Comparison between the desired ~ with Fk (-o- - -o-) and the obtained M W D ( - - )
0
10
20
30 40 50 time (min)
60
70
80
Fig. 2. Evolution of conversion and molar copolymer composition. (o) conversion; ( - - - ) desired composition; (o) obtained composition.
Figure 1 presents the MWD for the optimal process in which a unimodal MWD and a 50/50 molar copolymer composition was desired. Figures 1 and 2 show that the desired MWD and copolymer composition were obtained using the monomer and CTA feed profiles calculated through the optimization approach. 250
. . . D . . . . . o . . . . . . o . . . . . . ~.
200 .,..,,
.-
.o"
.....~ " .
8 0.8
"
./"
E
150
o
o.
"|o
...(~
I00
. ~ : : :: : ~ "
G9
EL
50
"
4?"
.~ .....~ ....
""
........
~
0.6
c
mr-
0.4
,
0.2
.EJ'"
.i
. . . . Er"
";" 0
1 ._
~z3
"" . . . . E2 . . . . .
0
..-.is] . . . . . -[3 . . . . . . i " ~
2
4
m. ,
i
6
8
Interval
c 0
,
o
10
Fig. 3. Feed profiles, (o) Feed 1, (o) Feed 2 and (o) ]:eed 3.
.... --0 ...... ,-~----0
0
.... ~
..... ~........................................ o
o
o
o
o
o
0
10
20
30 40 50 time (min)
60
70
80
Fig. 4. Evolution of conversion and molar copolymer composition. (,) conversion; ( - - - ) desired composition; (o) obtained composition.
Figure 3 presents the feed profiles for the optimal process in which a bimodal MWD and a 50/50 molar copolymer composition was desired. Figure 3 shows that to assure the production of a constant copolymer composition, Feed 1 which contained the less reactive monomer in the copolymerization (BuA), was fed very slowly at the end of the feed period. CTA is barely fed during the first half of the feed period to initially produce the high molecular weight. 0.8
0.5
0.7
0.4
0.6
0.5 ~
_1
0.3
0.4
aD
:,
0.3
._1
o.
N 0.2
0.2
"o
0.1 0
3
4
5 6 Log(MW)
7
8
9
Fig. 5. Comparison between the desired MWD with Fk (-o- - -o-) and the obtained M W D ( - - ) .
0.1 0
2
.
3
.
4
.
5
.
6
7
t ....
8
9
Loo(MW) Fig. 6. Simulated results" evolution of MWD.
462 Figures 4 and 5 show that bimodal MWDs of copolymers with controlled composition were also achieved. Figure 6 presents the simulated evolution of the MWD. It can be seen a high molecular weight polymer was initially produced, whereas the low molecular weight was mainly formed at the end of the process. 5. CONCLUSIONS Iterative dynamic programming was used to compute optimal monomer and CTA feed profiles to assure the production of polymer with pre-specified copolymer composition and MWD. The optimization approach uses a detailed mathematical model for emulsion copolymerization in which the MWD is computed by an adaptive orthogonal collocation technique. The proposed approach was applied to the emulsion copolymerization of MMA/BuA and allowed the production of copolymers with homogeneous compositions and with well-defined MWDs (narrow unimodal, broad, as well as bimodal MWD). ACKNOWLEDGEMENTS
The financial support from CNPq - Conselho Nacional de Desenvolvimento Cientifico e Tecnol6gico, FAPESP - Fundagao de Amparo/l Pesquisa do Estado de Sao Paulo and CICYT (project TAP95-1020) is gratefully appreciated. REFERENCES
1. Charmot, D., in Polymeric Dispersions. Principles and Applications. J.M. Asua Ed. Kluwer Academic Publishers. 1996. p. 79. 2. Echevarria, A., Leiza, J.R., de la Cal, J.C. and Asua, J.M.,AIChEJ., 44, p. 1667, 1998. 3. Congalidis, J.P. and Richards, J.R., Polym. Reac. Eng., 6, p. 71, 1998. 4. Bojkov, B. and Luus, R., lnd. Eng. Chem. Res. 31, p. 1308, 1992. 5. Oliveira, A.T.M., Biscaia, E.C. and Pinto, J.C., J. Appl. Polym. Sci. 69, p. 1137-1152, 1998. 6. Luus, R. and Rosen, 0., Ind. Eng. Chem. Res. 30, p. 1525, 1991. 7. Sayer, C., Arzamendi, G., Asua, J.M., Lima, E.L. and Pinto, J.C., VI Simp6sio Latino Americano de Polimeros, Vifia del Mar, October 1998, Chile. 8. Sayer, C., Lima, E.L., Pinto, J.C., Arzamendi, G. and Asua, J.M., accepted for publication at J. Polym. Sci. Part A: Polym. Chem., 1999. 9. Hansen, F.K. and Ugelstad, J., J. Appl. Polym. ScL 16, p. 1953, 1978. 10. Arzamendi, G., Forcada, J. and Asua, J.M., Macromolecules 27, p. 6068, 1994 11. Barandiaran, M.J., Arbina, L.L., de la Cal, J.C., Gugliotta, L.M. and Asua, J.M., J. Appl. Polym. Sci. 55, p. 1231, 1995. 12. Nele, M., Sayer, C. and Pinto, J.C., Macromol. Theor. Simulat., 8,, p. 199, 1999. 13. Pinto, J.C. and Biscaia, E.C.,Latin Amer. Appl. Reseach. 26, p. 1, 1996. 14. Petzold, L.R., SAND82-8637, Sandia National Laboratories, 1982. 15. Van Herk, A.M., in Polymeric Dispersions. Principles and Applications. J.M. Asua Ed. Kluwer Academic Publishers. 1996. p. 17. 16. Hutchinson, R.A., Paquet Jr., D.A. and McMinn, J.H., Macromolecules 28, p. 5655, 1995. 17. Sayer, C., DSc. Thesis, COPPEAJFRJ, Rio de Janeiro, RJ, Brasil, 1999. 18. Tefera, N., Weickert, G. and Westerterp, K.R., J. Appl. Polym. Sci. 63, p. 1663, 1997.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
Optimizing Separation
463
the Operation of a Sequential-Simulated Process Using MINLP
Moving-Bed
Stefan Karlsson, Frank Pettersson*, Hans Skrifvars, Tapio Westerlund Process Design Laboratory at/~.bo Akademi University Biskopsgatan 8, FIN-20500 Abo, Finland * Heat Engineering Laboratory at ~bo Akademi University Biskopsgatan 8, FIN-20500/~bo, Finland email: [email protected], [email protected] email: [email protected], [email protected]
FAX: +358-2-2154791
Abstract In the chemical processing industry today there is a need for efficient methods to separate multicomponent mixtures within reasonable costs. Chromatographic separation, and especially the simulated moving-bed (SMB) chromatographic separation, is a good alternative to this need. It is not only important to find the best separation process, but also to be able to run it in the best possible way. Optimization of these complex dynamic systems is a challenging task and the schedules are often a result of a very long development process performed by experienced schedulers and production engineers.
K e y w o r d s : Optimization, Chromatography, Dynamic Mixed Integer Non-Linear Programming, Simulated Moving-Bed, Production Planning, SMB, MINLP. 1. I N T R O D U C T I O N Two main types of simulated moving-bed methods can be identified, when considering preparative scale chromatography. The first one is the continuous process, also known as the Sorbex process, where all flows (feed, eluent and outtake of products) are continuous. The second type is the sequential process where the flows are sequential but the concentration gradients move continuously in the columns. The main advantage of the sequential process is that more than two components can be separated with high purity and yield. Operating the separation in a sequential manner also results in a highly flexible process. The optimization of the continuous process has been addressed in the pertient literature, eg. Strube et al., (1997), and Diinnebier and Klatt (1999). Optimization of the sequential process is more difficult, partly due to the great flexibility of the system. The problem has been formulated earlier as a mixed integer linear programming (MILP) problem by Karlsson et al., (1999). Although the linearized model is interesting, the obtained results are not completely satisfying, due to the somewhat restricted model and the long calculation times during the optimization process. In this paper the planning of optimal sequencing of the sequential SMB is formulated as a mixed integer non-linear programming (MINLP) problem, including a dynamic model
464 of the separation process. It is worthy of note that every function evaluation in the optimization of the MINLP problem requires a solution of the partial differential equation (PDE) system. Due to this time consuming procedure, it is preferable to select an MINLP optimizer requiring as few function evaluations as possible in the iteration procedure. The Extended Cutting Plane (ECP) method (Westerlund and Pettersson (1995) and Westerlund et al., (1998)) has been shown to meet this requirement quite well as illustrated e.g. in Skrifvars et al., (1998). The model presented in this paper also covers axial dispersion, and will be used in an optimization of the sequential SMB separation of fructose and glucose. This chromatographic separation process has earlier been studied, e.g. Saska et al., (1991) and Ching et al., (1987). The optimization is done with the m• program package (Skrifvars (1998)), using the a-ECP method as presented in Westerlund et al., (1998). 2. M O D E L I N G A model that take into account all the important effects of non-idealities, is the general rate model (Guiochon et al., 1994). Axial dispersion and mass transfer resistence are mainly responsible for these non-ideal conditions. Let ci represent the concentration of the solute i in the fluid phase. Di is the dispersion coefficient for component i, and c is the void fraction. The mass-transfer coefficients k influence on concentration profiles, depending on the concentration for the components in the system. Assuming that no radial distribution of flow rate u and ci, the following sets of PDEs are obtained for one column:
Ot q-
~
9 ki - ~ -~- E/=Ikil " c i - ~ -+- Cl--~
-+- U-~z -- Di Ot 2
The PDE system is solved by a Forward Difference Equation method. 3. P A R A M E T E R
ESTIMATION
The estimation of the parameter values for the dispersion and mass-transfer coefficients (D:s and k:s) was done by the Nelder-Mead algorithm (Downhill Simplex Method) with data from a laboratory test of a two-component glucose-fructose system. The fit for the fructose is excellent, and the fit for glucose was good.
c o m p o n e nt 1
IXll c~X'2/ "~.'>,.x,, prou \,-,
T x~
pro~
I
prod
x~,
XOX3 1;1 q;2
Figure
1. Column configuration
Figure
~4
~5
~6
Vr
component 2 2. Concentration and time zones
465 4. C O L U M N
SYSTEM
The number of separation columns can vary from one, that can be operated as a semibatch, to several multi-connected columns. A three column system with all possible connections between the columns can be seen in Figure 1. The possible inputs to the columns are feed of solution, eluent or partly separated fractions from another column. At the bottom of a column it is possible to remove a product from the system, to recycle the fractions to another column, or to dilute the feed with unpure fractions. 5. O P T I M I Z A T I O N
MODEL
The goal of production planning is to denote different actions to distinct times. Every action at a certain time can be described with a discrete formulation, either it happens or not, and can easily be implemented with binary variables. Therefore, the rest of the model depends on the decisions at these times, where a change in an action takes place. The time representation used in the sequential SMB in this paper is continuous, and the maximum number of actions that can occur in the time horizon of interest is denoted by T. The concentration profiles are calculated using the PDE model, depending on the operation parameters, i.e. the times for feed of solution into the system, the times for recycling from one column into another, and the times for collecting each product. For each time interval, the total mass for every component can be calculated as an integral. The concentration profiles with time zones are shown in Figure 2. The actual time horizon can be significantly reduced if the patterns for the occurences are repeated in cycles, with the time for the cycle length denoted as TT . 5.1 O b j e c t i v e f u n c t i o n The main goal is to collect as large a quantity of sufficiently separated products as possible from the outcoming streams. This can be formulated as a sum of the total mass, over all products, columns, and time intervals in the product streams, denoted s tk,i" Pi is sales price for component i. In order to have comparable results, it is of interest to know how much is obtained per time unit, therefore the sum is normalized by the cycle time, resluting in a pseudo-convex objective function, 1
K
C
T
ax-ZEE 7"
.
(2)
k,i
T k = l i=1 t=0
5.2 L i n e a r c o n s t r a i n t s The total purity of the product streams are of greatest interest when dealing with separation processes. The mass of the pure component in the product stream divided by the total mass of the product should be bigger than a value Ri for the respective product component i,
EE t=O k = l
--
k,1 t=O k - 1
"
(1
-- Yk,i)
Vi
(3)
I-1
The m:s are masses at the bottom of a column in a time interval, a n d y:s are binary decision variables defining a possible outtake of a component. The s:s in the objective
466 function are identical to the masses m at outtake, otherwise they are defined to be zeroes according to
s tk,i -< m tk,i
Vk, i, t
(4.1)
8 tk,i < Vk, i t (4.2) -- M " ytk, i The feed into a column consists either of the solution, the eluent or the recycle from another column and can be formulatedas K
Xj, k ~ 1
V k, t
(5)
j=l
where the left hand side, equal to zero, is the feed of eluent. Similarly the stream out from a column can only be handled as a product stream, a recycle stream or a dilution of the feed (LHS=0) during a single interval: c K E y ~ , i + E Xk,jt _< 1 i=1
V k, t
(6)
j=l
5.3 N o n - l i n e a r constraints The only not linearly rewritten non-linear constraints occuring in this formulation are found in the calculation of the masses, m tk,i~ of the components. The concentrations c are simulated by the PDE system for all the times and the mass components can be calculated as the time integrals of the different concentrations multiplied by the flow rate Tt+l f~
mtk,i -- I
~/z " ctk,i (m)dm
V k, i, t C [0, T - 1]
(7)
~'t
5.4 A d d i t i o n a l constraints Some constraints can be added to the model in order to enhance the solving, without restricting the solution space. It is of interest to guarantee that whenever a discrete time point occurs, a change in the column configuration also takes place. If this cannot be guaranteed, then the time points can occur everywhere and there exist multiple solutions. The constraints avoiding this phenomena are written in two different ways here. The first one is a version of the integer cut model, which will force at least one of the binary variables to change from a time point to next time point. This formulation is eq(8). The alternative formulation described here is: if the column configuration is the same for consecutive time points, then the time points should be equal eq(9.1). In the same way, if the configuration for two consecutive time points differ, then the time points should also differ eq(9.2). K
E
(f~ ft-,
(1-t
(1-t-,)
k=l K +
K
K
. t-1
(1_
t
(1
t-,
j=l k=l C
t yt--1 k,, + ( 1 E E (Yk,i" k=l i=1
-
(1 - yt-1 k,,)) Y~,i) -
__
C) - 1 Vt E [1,m I
(8)
467
( ~
K
k=l
(Wt+l --
K
I~,~-t+e
t -t- E -- Xj,k[
j=l k=l
K --~"
K
+Z Z
If~+1 -
Tt+l -- "It -- TC
K
7"t) -Jr-E I f ~
)
< 0 Vt C
[0, T - 1] (9.1)
k = l i=1
K
+1 -- f ~ [ + E EIX}; j=l k=l
k=l
C
t + l -- Yk,i Z ] oYk,i t [ K
1 --
X},kl if" Z
C
ZIY~ +1
Ytk,i] < 0 Vt C [0, T - 1] (9.2)
-
k = l i=1
The formulations for the absolute values can be rewritten linearily. The difference between times are zero or at least l/K, where ~ is an arbitrary chosen parameter, preferable as small as possible. In order to avoid multiple solutions with respect to time events, the following constraint can be included.
<_ TC" (Tt+l --
Tt -- T t - 1
"It)
Mt e [1, T -
1]
(10)
6. E X A M P L E A binary solution (glucose/fructose) has been used as an example to illustrate the feature of the model. An estimation of the component parameters has been done with the method discussed earlier. The column system consists of two columns with the restrictions that no recycle to itself and no feed of solution to the second column is allowed. The solution time for one MILP problem is approximately one second to some minutes on 700MHz pentium, and 25 PDE simulations are needed to produce the cutting planes (including all gradient calculations) for the next MILP-problem in this particular example. One PDE simulations takes about 8 seconds with a discretization of 0.8 rain in time and 10 cm in space. The concentration profiles can be seen in Figures 3 and 4. 7. D I S C U S S I O N A method for optimizing a sequential SMB using solving of PDEs in combination with MINLP is presented in this paper. The a-ECP method for optimizing the MINLP problem is used to avoid a great number of function evaluations. This MINLP formulation is an improvement from the MILP formulation, even if global convergence cannot generally be guaranteed, because of the non-linear constraint (7), which is the case for the linear formulation in Karlsson et al. (1999). The advantages of this model are that the continuous time representation minimizes the number of binary variables, and that way also the number of simulations of the PDE system. Furthermore, this MINLP formulation better describes the separation process than the linear formulation does. Another big advantage of this model is that it is suitable for multi-component systems, and for simultaneous solving the scheduling and the synthesis problem. 2o~ . . . . /Y"\
,5~,"
, ....
,
[ -- glucose column .... fructose column
",,,
2(1 . . . .
i ....
i .... I .... i , , I glucose column 2 frucose column 2
~,5 1o ~
o
.,,
i
"
'5',
11,o
,5o
2,x,
Time [minl
Figure 3. Column 1 cone. profile
o() . . . .
50 ~
,
,
,
ioo
15(1
2(x)
Time [min]
Figure 4. Column 2 cone. profile
468
The inaccuracies in the parameter estimation for glucose can cause the concentration profiles to differ slightly from reality. Furthermore, the P D E model used, can, in some cases, cause higher concentration levels than can be observed. The used P D E model describes, however, the process well enough. Though, further improvements can be done, especially the possibilties to also optimize the flow rates in the different intervals. NOTATION i 1 k j t
Ri 7t Pi K C
Tc T c
= = = = = = = = = = = =
component index component index index for into or in which column index for from which column index for event point purity demand for component i the start of time at period t sales price for component i number of columns number of components maximal cycle time the maximum number of event points concentration
c u
Di ki kil
= = = = =
8 tk,i = m tk,i =
f~
-
x},k ytk,i -II
=
void fraction in the column linear flow rate dispersion coefficient mass-transfer coefficient mass-transfer coefficient for component i dependent on component l mass of component i in product stream mass of i at the end of column yk feed of solution into column k, binary recycle from column j to k, binary outtake of component i from column k at event t, binary volume flow rate
REFERENCES
1. Ching C.B., Hidajat K., Ruthven D.M. (1987) Experimental study of a Simulated C o u n t e r - C u r r e n t adsorption system-V. Comparision of resin and zeolite adsorbents for fructose-glucose separation at high concentration. Chem. Eng. Sci., 40, pp. 2547-2555. 2. Diinnebier G., Klatt K.-U. (1999). Optimal Operation of Simulated Moving Bed C h r o m a t o g r a p h i c Processes. Computers Chem. Engng Suppl, 23, pp. $195-$198. 3. Guiochon G., Shirazi S.G., Katti A.M. (1994). Fundamentals of Preparative and Nonlinear Chromatography. Academic Press, San Diego, CA. 4. Karlsson S., Pettersson F. and Westerlund T. (1999). A M I L P - m e t h o d for optimizing a preparative simulated moving bed chromatographic separation process. Computers Chem. Engng Suppl., 23, pp. $487-$490. 5. Saska M., Clarke S.J., Mei Di Wu, Khalid Iqbal (1991), Application of continuous chromatographic separation in the sugar industry. Int. Sugar JNL., 93, pp. 223-228. 6. Skrifvars H., m i t t l p - A Program Package for Solving MINLP Problems. Process
Design Laboratory, ~bo Akademi University report 98-158-A, ISBN 952-12-0205-X 7. Skrifvars H., Leyffer S., Westerlund T. (1998), Comparision of certain MINLP algorithms when applied to a model structure determination and p a r a m e t e r estimation problem. Computers Chem. Engng, 22, pp. 1829-1835. 8. Strube J., AltenhSner U., Meurer M., Schmidt-Traub H., Optimierung kontinuerlicher Simulated-Moving-Bed Chromatographie-Prozesse durch dynamische Simulation. Chemie Ingenieur Technik, 69, pp. 328-331. 9. Westerlund T., Pettersson F. (1995), An Extended Cutting Plane Method for Solving Convex M I N L P Problems. Computers Chem. Engng Suppl., 19, pp. S131-S136. 10. Westerlund T., Skrifvars H., Harjunkoski I., and P5rn R. (1998). An E x t e n d e d C u t t i n g Plane Method for a Class of Non-Convex MINLP Problems. Computers Chem. Engng, Vol 22, pp. 357-365.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
469
Multiperiod Planning for a Utility System Considering Emergency Situation by New Approach Jeong Hwan Kiml, Sangaun Ju2, Chonghun Han 3, Sang Hyun You4 1'2'3Automation Research Center, Department of Chemical Engineering, Pohang University of Science and Technology, San 31, Hyoja, Pohang, Kyungbuk, 790-784, Korea 4Hyundai Petrochemical Co. Ltd, 679 Daisanri, Seosan, Chungnam, Korea
Abstract In this paper, multiperiod planning considering emergency situation (MPCE) has been performed. To avoid the whole plant shutdown due to major equipment's unexpected failure, multiperiod planning framework for handling the emergency situation is proposed. To solve the MPCE within an allowable computation time, three-step approach which uses the dynamci programming and heuristics has been applied. In the case study, MPCE has been solved using proposed approach, and the result shows that MPCE is indispensable to operate the plant with the lower operating cost avoiding the whole plant shutdown.
1. Introduction Optimal multiperiod planning has been a hot issue in chemical industries to maximize the profit under varying process circumstances. The operating decision choices for different periods can have large economic impact on operation profit. Without proper operational planning based on forecasting, company cannot avoid paying high transition cost resulted from drastic production change. Multiperiod planning of a utility system is a specific concem in chemical industry, because utility costs often a major portion of the total operating cost in the chemical plant. Various approaches have been suggested for the optimal operation of the process. Papoulias and Grossmann (1991) proposed a structural optimization approach using MILP, Ito et al. (1987) suggested an operational planning model for gas turbine cogeneration plants considering startup and shutdown costs. Kalitventzeff(1991) solved the utility plant network management problem by MINLP. Papalexandri et a1.(1998) suggested multiperiod optimization method where the variable energy demands and uncertainties are considered. Iyer and Grossmann (1997) proposed the two-stage algorithm using bilevel decomposition and modified shortest path algorithm to solve the MILP problem within the reduced computation time. Kim et al. (1999) proposed a two-level approach to solve the multiperiod planning problem. However, there have been concerns from the industry that the optimized operation may bring the shutdown of the whole plant due to too aggressive operation. Because the shutdown *Author to whom all correspondence should be addressed Tel: +82-562-279-2279
E-mail: [email protected]
470 of the process bring about a big loss, operators in the industry are conservative in applying optimization result to the plant. Although it is a significant problem, not many researches have been made on the multiperiod planning considering emergency situation (MPCE). MPCE is indispensable to maximize the profit while preventing the shutdown of whole process in case of emergency situation. In this paper MPCE has been performed using three-step approach. This paper is presented as follows. In section 2, MPCE formulation and three-step approach is explained, and case study is performed in section 3.
2. MPCE using three-step approach 2.1. MPCE Emergency situation can be resulted from various reasons, and can cause the shutdown of whole process. An Act of god such as earthquake, heavy rain, storm, thunder can cause the shutdown of the plant, but these things are beyond our ability to control. In this paper, we define the emergency situation as the one resulted from the failure of the equipment. Especially for the utility plant, the failure of the boiler may cause the shutdown of whole process, because it fails to produce required amount of steam demand. The down stream plant can endure the shortage of steam supply for the limited time, and we define it as the buffer time, which is usually within a few minutes. When one of the operating boilers experiences unexpected failure, the steam production rate of the other boilers should be increased to their maximum to compensate for the deficient amount of steam. However, due to the mechanical limitation, the steam production rate can not be increased drastically to its desired value. Therefore, the operation which can guarantee avoiding the whole plant shutdown by compensating for the deficient steam amount with the other boilers is required. The formulation of MPCE is as follows; Objective function Min [Total Cost] = M i n I ~ ~ ( Hout, m, , * Fo ...... t - I~ ..... * F, .... t I * CF, + E, * CEt + W~ * C W + SCt + TCt + P S D C t l ) J xo [ "T" "ff ~, rlk, , * C P k, ,
(1)
Constraints
Z F n , k,t--ZFout, k,t--O ZH,,,~,,F,o,~,,-ZHo,,,,~,,Fo,,,,~,,-O
Vk, t
(3)
Vk, t
C2 L,~,t <- U k,t ( F , T , P ) <_ ~ v ,~,t ~
(2) Vk, t
(4)
Vk, t
(5)
Vr,k,t
(6)
PG k,t + PP~ >- P D , k
_G
r,k
,t
--
r
k
qk,, - a
* Fk,, 2 --]-b
* F~,t + c
Vk, t
(7)
471 N
(8) i;~f
f~j,
i
- F B~, j , i (1 + vj)t~ _ F0,j
,i
Vj, i
(9)
The objective is to minimize the total cost over the planning horizon. Total cost is composed of operating cost, switching cost, transition cost and plant shutdown loss. Operating cost is composed of fuel cost, electricity cost and water cost. The fuel cost shows the nonlinear function form. To consider the nonlinearity of the problem, the problem is formulated as MINLP form. The decision variables are boiler flowrate and turbine flowrate at each period and integer variables which specify the configurations of equipment at each period. Switching cost is assumed to be constant and transition cost is given to reflect the penalty for drastic operation change. Plant shutdown loss is charge when required amount of steam can not be supplied within the buffer time. Constraints are material balance, energy balance, operation region of the equipment, steam and electricity satisfaction for the demand, boiler efficiency relationship, and emergency handling constraints. Constraint (8) represents the emergency handling constraints. To avoid the whole process shutdown when one of operating boiler experiences shutdown, the other operating boilers should produce more than the amount produced within the buffer time (tb). Due to the mechanical limitation, the boiler can increase the steam production by vj, and constraint (9) shows the steam amount to be produced at its maximum within the buffer time by the j,h boiler at period i. To solve the large size problem efficiently, three-step approach has been applied.
2.2. Three-step approach Three-step approach is proposed to solve the multiperiod palanning problem for the utility plant. At the first step, alternatives for optimum configuration at each period are generated, and nonlinear problem which minimizes the operating cost at each period considering emergency situation is solved. Consideration of emergency situation is made in the nonlinear problem by incorporating the constraints of equipment's capacity to compensate the loss of equipment to avoid the shutdown of whole process. By introducing heuristics into the alternative generation step, infeasible and non-optimal configurations are excluded and this leads to the reduction of the computation time. At the second step, the optimum mode for boilers are determined using dynamic programming. Transition cost for startup of equipment is considered in determining the optimal planning for given horizon. At the third step, fine search for the optimum is made using iterative search. More reliable and accurate solution that takes into account nonlinear characteristics of the system is obtained and the computation time is greatly reduced by the combination of nonlinear programming and heuristics combined dynamic programming.
3. Case study The proposed approach has been applied to the simulated model for an industrial utility plant of Hyundai Petrochemical in Korea. Industrial data has been used for the simulation study. In the case study, MPCE for 2 days planning horizon which is composed of 6 periods
472 have been performed. Table 1 shows the steam and electricity demand change and unit electricity cost change. According to the time of the day, the unit electricity cost is differently charged, and the demand pattern is different, which gives the chance of multiperiod optimization. Table 2 shows the emergency handling capability of traditional equal load allocation method, and multiperiod planning with/without considering emergency situation under the given demand scenario. The shutdown frequency and buffer time is assumed to be constant. The result shows that multiperiod planning without considering the emergency situation can not supply the required steam within the given buffer time, and experiences the whole plant shutdown. For the steam and electricity demand of 4 th period, simple multiperiod planning fails to handle emergency situation when the number 2 boiler experiences sudden failure. The total cost including the expected plant shutdown loss shows that MPCE gives the lowest total cost. The result shows that MPCE can handle the emergency situation efficiently with the lower cost. If we consider the related unfavorable side effect resulted from the whole plant shutdown, MPCE is indispensable for profitable and safe operation. Equal load allocation can handle every emergency scenario well, but it costs more than MPCE. According to the buffer time, the optimal operation changes heavily. The check for the buffer time is required to establish the proper emergency managing framework. Figure 1 shows the relationship between buffer time and optimum steam flow rate. As shown in the figure, the optimum value converge to the specified value after marginal buffer time, and this value is equal to that of simple multiperiod planning result. Figure 2 shows the optimal operation plan result using MPCE strategy, and the optimization result avoid the whole plant shutdown with the lowest possible cost. t-
160 ---e--
.,..,
.9 9
140
~ "~
--~.
="~-'~"
120
9
--=.-
Boiler#1 Boiler#2 Boiler#3
-9 B o i l e r # 4
Boiler#5
o E
100
E Q.
80
o
4
5
6
7
8
buffer time(tb) [rain]
Fig.1.
Optimum steam production for the varying buffer time --e--
,..-, .E l0
9 140
--?-4-
Boiler#1 Boiler#2 Boiler-#3 Boiler#-4 Boiler#5
t._
~
120
=_-F-~
E 1~
100
E
80 0
1
-
i9 9
2
3
= I_ 4
5
6
7
period
Fig.2.
Optimum steam production rate of each boiler
473
4. Conclusion In this paper, multiperiod planning considering emergency situation (MPCE) has been performed. To avoid the whole plant shutdown, emergency handling constraints are incorporated to the multiperiod planning problem formulation. To solve the MPCE, three-step approach has been proposed. By using the three-step approach, search space is greatly reduced and the optimum plan is found with an efficient manner. In the case study, MPCE is solved for the utility plant, and the comparison of the result with that of simple multiperiod planning and equal load allocation shows that MPCE is indispensable to operate the plant with the lowest cost avoiding the whole plant shutdown. For the safe and economic optimization of the process, the systematic emergency handling framework is proposed by combining the MPCE with the simple multiperiod planning.
Acknowledgement The authors deeply appreciate the financial support of the Korea Science and Engineering Foundation through the Automation Research Center at POSTECH.
Nomenclature ~.. tolerance -OLk,,: lower operation bound for equipment k at period t .(2v,k,, : upper operation bound for equipment k at period t v/maximum steam increasing rate at jth boiler at tth period r/k,,: efficiency of boiler k at time t CE, : unit electricity cost at tth period CF, : unit fuel cost at tth period CPk,,: Heating vale of fuel at kth boiler at period t. CW, : unit water cost at tth period E, : electiricity usage at tth period FBj 0: present steam production atjth boiler at period t FBji,'b:Steam production atjth boiler at period t during buffer time F~,k,,. flowrate at kth boiler at time t Fi,.....,: inlet stream flowrate for unit m at time t Fou,,,,,,: outlet stream flowrate for unit m at time t F,: fuel usage at tth period Hi,.....,: inlet stream enthalpy for unit m at time t Ho,,,,,,,,: outlet steam enthalpy for unit m at time t PD,: power demand at tth period PGk,,." power generation at kth turbine at tth period PP,: power purchase at tth period QD,,k.,: steam demand for type r using kth equipment at period t QGr,~.,,: steam generation for type r using kth equipment at period t SC,: start up and shutdown cost at period t TC, : transition cost at tth period Uk,, (F, T,P): operation range of equipment k for flowrate, temperature, and pressure
474 W,. water usage at tth period h: planning horizon tb: buffer time References
B.Kalitventzeff, Mixed integer non-linear programming and its application to the management of utility networks, Engineering Optimization, Engne Optim. Vol 18, pp183-pp207,1991. Jeong Hwan Kim, Moo Ho Lee, Chonghun Han, Seong Hwan Kim, Sang Hyun You, multiperiod planning for utility systems using dynamic programming, Computers Chem. Engng Vol 23, $519-$522, 1999. K.Ito, R.Yokoyama, S.Horii, Y.Matsumoto, and S.Adagi. An optimal operational planning model for gas turbine cogeneration plants. In Tokyo International Gas Turbine Congress, 1987. K.P.Papalexandri, E.N.Pistikopolous, and B.Kalitventzeff, Modelling and optimization aspects in energy management and plant operation with variable energy demands-applicaation to industrial problems, Computers Chem.Engng Vol 22, No9.pp. 1319-1333,1998. R.R.Iyer and I.E.Grossmann, Optimal multiperiod operational planning for utility systems, Computers Chem.Engng Vol 21, No8.pp.787-800,1997. S.A.Papoulias and I.E.Grossmann, A structural optimization approach in process synthesis-I, Computers Chem Engng, Vol 7, pp695-pp706,1991. Table 1. Process demand change during the planning horizon Period HPS [ton/h] MPS [ton/h] LPS[ton/h] Electricity[MW] Unit electricity cost[won/kw]
1
2
3
4
5
6
220 100 70 65 55
180 70 65 70 80
140 5 40 60 25
240 120 75 72 55
165 85 60 65 80
160 55 45 55 25
Table 2. The emergency handling capability and total cost comparison
Equal Load Allocation Multiperiod Planning MPCE
1
2
3
4
5
6
T.O.C. [106Won]
Expected S.D.C. [106Won]
Y
Y
Y
Y
Y
Y
766.9
766.9
y y
y y
y y
n(b2) y
y y
y y
765.4
766.4
765.9
765.9
'y' denotes the emergency handling capability 'n' denotes it fails to produce the required steam within the buffer time
Y.C.
[106Won]
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
475
M I N I M I Z A T I O N OF N A T U R A L G A S A N D W A T E R C O N S U M P T I O N IN THE O P E R A T I O N OF U T I L I T Y P L A N T S Sergio M. CORVALAN and Ana M. ELICECHE PLAPIQUI- Chem. Engng. Dept.- Universidad Nacional del Sur CONICET, Camino La Carrindanga km 7, 8000 Bahia Blanca, Argentina. Email: [email protected] ABSTRACT The main objective of this paper is to report reductions in water, natural gas consumption and operating cost obtained by choosing optimally the operating conditions of a real utility plant. There are continuos and discrete optimisation variables, therefore a Mixed Integer Non Linear Programming problem is solved with the code GAMS. The main continuous operating variables to be selected are temperature and pressure of the high, medium and low-pressure steam headers, de-aerator tank pressure and flow rates such us letdowns and vents. Binary variables are introduced for the selection of boilers, pumps and drivers. A rigorous simulation of the steam and power generation plant was developed interfacing a subroutine for water property prediction. The numerical results quantify the improvements that can be expected in the utility sector by selecting the continuos and discrete operating variables. KEYWORDS: Utility plant, operation, optimisation, natural gas, water.
INTRODUCTION The utility system provides the required power to drive the process units, high, medium and low-pressure steam for the chemical plant. The main units shown in Figure 1 are: boilers, high, medium and low pressure steam headers, steam turbines (back-pressure, condensation), electrical motors, pumps, heat exchangers, valves, flash drums, de-aerator, let down valves, vents and other equipment associated with steam system. The utility system of an ethylene plant is studied in this work. A strong heat integration and mass recycle exist between the ethylene plant and the utility sector. 50% to 60% of the high-pressure steam is generated in the pirolysis furnaces. The residual gas of the demethanizer column is burnt in boilers and furnaces. Petracci et al. (1991) selected the driver's configuration and the de-aerator pressure of the utility plant. In that work, temperature and pressure of the steam headers were not included as optimisation variables. Thus, non linear equations appear only in the enthalpy balances of the de-aerator unit. Most of the model equations were linear and only a few equations were nonlinear. The
476 MINLP problem was decomposed in Nonlinear Programming (NLP) and Mixed-Integer Linear Programming (MILP) sub-problems, solved sequentially using the Duran and Grossmann (1986) outer approximation algorithm. The NLP sub-problem was solved by Successive Linearizations (SL), due to the fact that most of the modelling equations were linear. The same code LINDO was used to solve the NLP by SL and the MILP master subproblem. Petracci et al. (1993) selected the operating conditions of the chemical and utility plant simultaneously solving a nonlinear programming problem, but discrete variables of the utility sector were not included. In this work continuos and discrete operating conditions of the utility sector are selected simultaneously solving a Mixed-Integer Nonlinear Programming problem with the code GAMS. When the utility plant is in operation different configurations are due to the presence of alternative drivers for certain power demands and pumps and boilers that can be on or off. In this work, most of the discrete options available in the operation of the real plant have been included in the superstructure. The continuous and discrete operating conditions of the utility sector are evaluated for fixed conditions of the ethylene plant.
PROBLEM FORMULATION A Mixed-Integer Nonlinear Programming (MINLP) problem arises, which can reformulated, so it is linear in the binary variables y and nonlinear in the continuous variables x as follows:
min ~ x,y
s.t.
cixi
Objective function
h(x) = 0 g(x) <_o
Model equations Design constraint s
Ay+Bx<_b
Logical constraint s
x~X~{x~R~
~_x~_x~}
P1
Variable bounds
y ~ {0,1} The modelling equations h(x) of the plant include mass and enthalpy balances, equipment models, enthalpy and entropy functions of temperature and pressure for each steam header and de-aerator tank. The design inequality constraints g(x), represent real plant data, such as maximum and minimum equipment capacities and operating restrictions. Upper and lower bounds on the continuous variables x, such us pressures, temperatures, flow rates are also operating plant data. The binary variable y vector allows the selection of the boiler, pumps and drivers. The logical constraints relate the existence of an equipment with nonzero flows, alternatively if the equipment has not been included in the configuration, the corresponding flow is zero. In this work, most of the model equations h(x) are nonlinear due to the fact that pressure and temperature of the steam headers are optimisation variables. Linear expressions to correlate variable equipment power versus flow rate have also been used in pumps and air fans. The MINLP problem was formulated using the program GAMS (General Algebraic Modelling Systems) by Brooke et al.(1988). The original MINLP problem is decomposed in
477 successive NLP sub-problems and MILP sub-problems. GAMS interfaces the code DICOPT++ of Viswanathan and Grossmann (1990), an Outer-Approximation algorithm that solves an alternating sequence of NLP and MILP master sub-problems.
UTILITY PLANT The continuous and discrete operating conditions of the utility sector are evaluated for f'Lxed conditions of the ethylene plant. The power demands for the main compressors: cracked gas (TC1), ethylene (TC2) and propylene (TC3) are provided by steam turbines as shown in Figure 1. There are twenty nine inequalities constraints g(x) and a hundred and twenty six equalities constraints h(x). The numerical example analysed has hundred and forty eight continuous variables. The continuos optimisation variables are: temperature and pressure of high, medium and low-pressure vapour headers, the deaerator pressure, flow rates such us the letdowns from high and medium pressure steam headers and vent. The following twenty four binary variables are selected optimally: 9 eight binary variables corresponding to pmnps that have two drivers, electrical motors and steam turbines 9 eight binary variables to select the pumps which are in operation with a fixed driver 9 four binary variables to select the boilers in operation 9 four additional binary variables to select the air fan driver of each boiler. Alternative objective functions can be used. The operating cost is a linear expression that adds the partial costs of natural gas, electricity, water make up and cooling water. It is a weighted sum that includes natural gas and water consumption, where the cost coefficients are the weights. It also includes the electricity that is provided by a national interconnected system and could be generated by thermo-electrical, hydro-electrical o nuclear power plants. An environmental impact is associated to the electricity generation, and should also be included in the objective function. Some other weighted function representing the environmental impact could also be used.
NUMERICAL RESULTS Problem P1 has been solved with the code GAMS. The NLP sub problem was solved with the code CONOPT2 and the MILP sub problem with the code OSL. The main continuos operating variables at the initial point, NLP and MINLP solution points are shown in table I. The NLP and MINLP solutions shown in table I, quantify the contribution of the continuous and integer operating variables to the operating cost reduction.Although this results vary with the initial point selected. Different initial points were tried and the same solution has always been found for the MINLP problem. This is an indication that the global solution has been found. At the solution point the HPH (High Pressure Header) letdown, the MPH (Medium Pressure Header) letdown and the LPH (Low pressure Header) vent are zero. A reduction of 30 % in the water make up is observed in table I. The natural gas consumption has been reduced by 16 %. Both improvements have been obtained simultaneously by the selection of the continuos operating variables. A reduction in combustion emissions such as: Carbon
478 monoxide, Carbon Dioxide, Nitrogen Oxides, Total Organic Compounds and Particulate Matter proportional to the natural gas consumption is observed. The combustion emissions have been calculated, although they are not reported. The simulation is very sensitive to the prediction of water enthalpy and entropy. Therefore a good evaluation of enthalpy and entropy at different conditions of pressure and temperature is required. Enthalpy and entropy are polynomials of third order of temperature and pressure of the steam headers. The deaerator enthalpy is a second order polynomial of the deaerator pressure. Table I - Optimal NLP and MINLP operating conditions and improvement achieved.
HPH Temperature ~ HPH Pressure Bar MPH Temperature ~ MPH Pressure Bar LPH Temperature ~ LPH Pressure Bar Deaerator Press Bar HPS boilers ton/hr HPS furnaces ton/hr Operating Cost $/hr Fuel Gas ton/hr Make UP water ton/hr Electricity HP LPH Vent (ton/hr)
Initial Point 410 49 330 18 230 4.5 1.5 87.29 102.46 1452.95 9.36 31.42 2925.66 9.42
NLP Reduction % Solution 450 52 356.98 22.52 150 4.94 1.2 65.23 25.27 (25) 98.42 -3.95 1244.31 14.36 (14) 7.87 15.90 (16) 22.00 29.98 (30) 2698.43 7.77 (8) (lOO) 0.0
MINLP Solution 450 52 359.31 23.05 150
Reduction %
,,
3
69.93 103.63 1136.25 7.84 22.00 755.06 0.0
19.89 0.16 21.80 16.22 29.98 74.19
(20) (22) (16) (30) (74)
(lO0)
OPERATING CONDITIONS High pressure steam header The temperature of the high-pressure steam header is the variable that has the biggest influence in the operating cost, natural gas and water consumption. The operating cost, natural gas and water consumption have a monotonic decreasing functionality with respect to temperature and pressure of the high-pressure steam header. The enthalpy drop through the propylene compressor turbine (TC3) increases as the temperature and pressure of the inlet turbine flow increase. Therefore the vapour consumption can be reduced increasing temperature and pressure of the HPH and their optimum values of lie at the upper bounds. As the HPH temperature rises, the HPH enthalpy rises and the required vapour decreases. The reduction in vapour flow is more important than the enthalpy increment, resulting in a smaller heat requirement in boilers, a reduction in natural gas consumption and operating cost. The natural gas consumption and operating cost in boilers also decrease when the HPH pressure increases, due to the fact that the vapour flow through the propylene turbine and its enthalpy decreases.
479 Medium pressure steam header The Medium Pressure Header (MPH) steam is generated, as shown in Figure 1, by the propylene compressor turbine TC3. The optimum values of temperature and pressure of the MPH are observed inside the feasible intervals. The enthalpy at the optimum values of 359.31 ~ and 23.05 bar is equal to the enthalpy of the stream leaving the propylene turbine. When the MPH enthalpy is smaller than the TC3 exit enthalpy, de-aerated water is added to the MP header, to reduce its temperature and enthalpy. If the MPH enthalpy is bigger than the TC3 exit enthalpy, an increment in the flow going through the high-pressure letdown valve is observed, to raise the MPH temperature. If the HPH letdown flow is bigger, more HP steam is generated and the operating cost, natural gas and water consumption increase. Therefore for the structure of this plant, the optimum operating conditions of the medium pressure header correspond to the pressure and temperature that leads to the same enthalpy as the exit enthalpy of the TC3 turbine. Low pressure steam header The operating cost is monotonic increasing with temperature and pressure of the Low Pressure Steam Header (LPH). The enthalpy drop through the turbines increases as the exit pressure decreases and the required vapour generation decreases. The same tendency is observed when the exit temperature of the LPH decreases. Thus the optimum values of the temperature and pressure of the LPH header lie at their lower bounds. An extra operating constraint imposes that deaerator pressure is lower than the pressure of the LPH. Discrete variables The continuos operating variables were first selected, and reported as the NLP solution in table I. Then, the MINLP solution shows the contribution that can be achieved when both the continuos and discrete variables are calculated simultaneously. Nine electrical drivers and seven steam turbines are in operation at the initial point, while at the solution point of the MINLP problem, three electrical drivers and twelve turbines are in operation. Comparing the NLP and MINLP solution points the following results can be mentioned. The switching from electrical motors (nine to three) to steam turbines (seven to twelve) justify the reduction in the electrical demand of 74 % observed at the solution point. This leads to a further reduction in the operating cost of 52 % with respect to the NLP solution. An increment of 7% in the high-pressure steam generated in boilers is observed, with a relatively small increment of 0.4% in the corresponding natural gas consumption. One of the three boilers that were initially on is off at the solution point, due to the 20 % reduction in High Pressure Steam generated.
CONCLUSIONS By a proper selection of the operating conditions of the utility sector, reductions in the order of 22 % in the operating cost, 16 % in natural gas consumption, 30 % in water makeup and 74 % in electricity demands can be achieved in this plant as shown in table I. These results show the potential for improvement that can be expected in this sector, when MINLP codes coupled with rigorous modelling of the main processes are used. This approach reduces the environmental impact minimising the use of a non-renewable resource
480 like the natural gas and the combustion emissions generated in boilers. It also reduces the water consumption, a scarce resource in many places, including the city and site where this petrochemical complex operates. These objectives have been obtained simultaneously with an important reduction of the operating cost.
.
Boilers 1-4.
.
.
.
RGas esdi_uaL~ l ~ ~wast e-& h .~t Boiler
.
.
.
I
h
TCI
TC3
~
h
Motors
H.P. STEAM
"
l HPs DFM,,,Nr~}
TB1
M.P. STEAM
Fuel ~a~
I
il
~11~
~TC2
nTBn=2,3,..9
] [''~DEMAND }M'P'S"
L.P. STEAM VENT Vacuum condenser
[]
C.W.2
L. P- S- DEMAND} _ ...........
PI +P2
DEAERATOR
~up .....
Feedwaterpumps ~
( ~
Figure 1
REFERENCES Brooke A., Kendrick D. and Meeraus A., 1988, GAMS-A Users Guide. Redwood City, Scientific Press. Duran, M.A. and Grossmann, I.E. ,1986, An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Mathematical Programming, 36, 307. Petracci N., Eliceche A.M. and E. Brignole, Optimal Operation of an Ethylene Plant Utility System, Computers Chem. Engng., 17, ps147, 1993. Petracci N., Eliceche A.M. and E. Brignole, Utility System Optimal Operation, Comp. Oriented Process Engineering edited by L. Puigjaner and A. Espuha, Elsevier, p387, 1991. Viswanathan J. and Grossmann I.E., A combined penalty function and outer approximation method for MINLP optimisation, Computers Chem. Engng., 14, 769, 1990.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
481
Dynamic optimization of chemical and biochemical processes using restricted second order information. Eva Balsa-Canto", Julio R. Banga a, Antonio A.
Alonso b
and Vassilios S. Vassiliadis c
Chem. Eng. Lab., IIM (CSIC), Eduardo Cabello 6, 36208 - Vigo, Spain. bDept. of Chem. Eng., Universidad de Vigo, Aptdo. 874, 36200 - Vigo, Spain. c Dept. of Chem. Eng., Cambridge University, Pembroke Street, Cambridge CB2 3RA, UK. a
ABSTRACT. The extension of a recently developed method for the dynamic optimization (DO) of chemical and biochemical processes is presented. This method is based on the control vector parameterization approach and makes use of the calculation of first and second order sensitivities to obtain exact gradient and projected Hessian information. In order to achieve high discretization levels of the control variables with a moderate computational cost, a mesh refining technique is also presented here. The robustness and efficiency of this strategy is illustrated with the solution of several challenging case studies. 1. INTRODUCTION. Many biochemical and chemical processes are operated in batch or semi-continuous mode. In order to increase the productivity and/or profitability of these processes, many efforts have been devoted to their model-based optimization and control. Typical examples are the optimal control (OC) of batch polymerization reactors (Wang and Chiou, 1997) or fed-batch bioreactors (Banga et al, 1997). Most of these processes have highly nonlinear dynamics, and constraints are also frequently present on both the state and the control variables. Thus, efficient and robust dynamic optimization methods are needed in order to obtain their optimal operating policies. Here, we present the extension of a recently developed method based on the control vector parameterization (CVP) approach, which makes use of first and second order sensitivities to obtain exact gradients and projected Hessians of the objective. A mesh refining technique is used to achieve high levels of control discretization, which is of special interest here, as most optimal control policies of (bio)chemical units present a quite wild shape. 2. GENERAL OPTIMAL CONTROL PROBLEM (OCP) AND SOLUTION APPROACHES. The general open loop OCP, considering lumped parameter processes, can be stated as: Find u(t) and v to minimize (or maximize) a performance index J(z,u) : tf
J(z,u) = V(z(tf ))+ Iq~(z(t),u(t),t)dt
(1)
to
subject to: f(z, z,u, v ) = 0 ,
Z(to) = Zo(V)
(2-3)
482 h(z(t), u(t), v,t) = 0 u L(t) < u(t) < uV (t)
and
and
g ( z ( t ) , u ( t ) , v , t ) _< 0
(4-5)
vL < v < vv
(6-7)
where z ~ R n is the vector of state variables, u~ R a the vector of control variables and w R ~ the vector of time invariant parameters; Eqns. (2-3) are the system of DAEs with their initial conditions, Eqns. (4-5) are the equality and inequality path and/or point constraints, and Eqns. (6-7) are the upper and lower bounds on the controls and time invariant parameters. Basically, existing methods for the solution of OCPs can be classified in two classes: indirect and direct approaches (see review in Vassiliadis, 1993). Indirect approaches use the necessary conditions of Pontryagin while direct approaches transform the original problem into an outer non-linear programming problem (NLP), either using control vector parameterization (CVP; Vassiliadis, 1993) or complete (control and states) parameterization (Cuthrell and Biegler, 1989). In this work the CVP method was used due to its ability to handle large dynamic optimization problems without solving very large NLPs. It proceeds dividing the duration of the process into a number (9) of elements and approximating the control function using a low order polynomial (u=u(v)). In this work piecewise constant approximations were used. 4. SOLUTION OF OPTIMAL CONTROL PROBLEMS USING A SECOND ORDER METHOD. 4.1. S e c o n d o r d e r i n f o r m a t i o n analysis.
The use of sensitivity equations within CVP methods allows the efficient computation of the exact gradient and Hessian of the objective w.r.t, v, as presented in Vassiliadis (1993) and in Vassiliadis et al. (1999). These sensitivities can be obtained by a chain rule differentiation applied to the original DAE system with respect to the time-invariant parameters: ~ f b~
Of 0 z
a ,. a v
~z ~v
----+
+
af 0u
~u ~v
+
af
~
-0,•
~z
~(to)
'
av
=
~z o
(v)
(8,9)
where f,z and z ~ R n, U E R a and v~ R ~ with p =ap +77 and the initial conditions given by Eqn.(9). And a second differentiation (as presented in Vassiliadis et al., 1999) yields:
a2f ~-~| 3z I. |
02f 0 z +
JLazazav
~v
azauav
I
o ,oz
~
- - + ~ - - + ~
~
a~ auaz av
o ,ou --
azauav ~ a v
c)2z (to) = 02go 0v 2 0 _ ~ (v)
au
a 2f
l
J
auaz a~ + a~az +
~v2+ ~zz|
+
avau
avaz
+ ~ - -
azavav
+
~u
Nip
o ,oz
+ ~ - -
azavav
~
+ I,|
~v
o ,ou o ,l
+ ~ - -
auavav
+
av2j
x
(10)
= 0n.pxp
(11)
where | is the Kronecker matrix-product operator, and the second order derivatives result in tensors of third order.
483 To clarify the similarity between the first and second order sensitivities, we present the product of a time-in variant vector p c R p with the Hessians for each state. The result of Eqn. (10) is post-multiplied by p and by comparing terms the equivalent form is derived:
__ 0fST Oz
+--0f
S T +A(z , z,u,v)=0..~•
Oz
S(to)=(pT[02Z~
2 (v) 1 ...
pTI~2Zon ( v ) l / [ Ov ~
(12,13)
where S, S ~ R px, and the set of correspondent initial conditions is given in Eqn. (13). From the above results, it is clear that Eqn. (12) is a set of first order sensitivity equations, similar to Eqn. (8). The system Jacobian in Eqn. (2) is the partitioned matrix: I Of Of 1 O-z-' Oz
(14)
which is the same as the one in the resulting equations used in the integration of Eqn. (8), as noted in Leis and Kramer (1985) and the integration of system Eqn.(12), as noted in Vassiliadis et al (1999). This allows efficient integration of the extended initial value problem (EIVP) (Eqns.(2,8,12)) using a simultaneous evaluation technique. In order to fully automate the application of this procedure, a Mathematica TM 3.0 notebook has been developed to derive symbolically the EIVP and to transform the resulting set of equations into a suitable format (e.g. Fortran or C) ready to be used by standard solvers.
4.2. Solution of the outer NLP problem. Mesh refining technique. In order to efficiently solve the outer NLP problem, the TN (Truncated Newton) large-scale optimization code of Nash (1984) was selected because it makes use of gradient and Hessian dot vector product. This TN code was modified so that the first order and projected second order information are calculated exactly via the solution of the EIVP mentioned above. To achieve high discretization levels (9) with moderate computation times, we have also developed a mesh refining approach consisting of successive re-optimizations of increasing P values. The basic steps of this approach are outlined in the following pseudo-code: Step 1.- Set initial guess (usually a flat control value) and choose initial and final values for the discretization and the integration tolerance. Choose also a parameter (rp) used to increase the discretization level from one run to the next. Step 2.- Compute the number of refining optimizations (NRO) and the tolerance reduction step 8tol using the expressions: P_L _(NRO-1)and 6,o~= N R1o (int_toli - int_tolf ) Pi = 'p Step 3.- Initial optimization run (k=l): compute, using TN, the optimal control profile Ua(t) corresponding to the initial control discretization Pi and the tolerances int_toli and opt_toli. Step 4.- Optimization loop refining the control discretization From k=2 to k=NRO Adjust discretization level and tolerances:pk=pk_l rp Adjust the tolerances: int_tolk=int_tolk_l-Stol, opt_tolk=int_tOlk tol_ratio Optimization run k: using uk-1 as initial guess, compute uk using the modified TN Next k
484 5. CASE STUDIES.
In order to illustrate the performance of the method presented here, several case studies taken from the literature are considered. Due to space restrictions, only brief descriptions of the problems are given. Their mathematical statements can be found in the given references. Results are presented in Tables 1 to 3. For the sake of comparison, results from other authors are also reported. The computation times (CPU time) were converted from the original values reported by the authors to values of the PC Pentium/III (450 MHz), computer used in this work, based on their relative performance values as measured by the Linpack-100 benchmark. Examples of the optimal control profiles obtained are presented in Figures 1 to 4. 5.1. Case I: Park-Ramirez (PR) bioreactor.
This case study deals with the optimal production of secreted protein in a fed-batch reactor (Park and Ramirez, 1988). The dynamic model accounts for host-cell growth, gene expression and the secretion of expressed polypeptides and the objective is to maximize the secreted protein through the control of the nutrient feed rate. Two different sub-cases are presented (PRa, PR-b) which differ in the time horizon and control bounds used. Table 1. Summary of results for case I. Sub-case Authors' Yang and Wu (1994) PR-a Luus(1995) Banga et al (1998) This Work Tholudur and Ramirez (1997) PR-b This Work
Performance Index 32.681 32.686 32.562 32.632 (p=40) /32.691 (p=120) 32.31-32.47 32.388(p=40) / 32.480 (p=320)
CPU time (s) Not reported (NR) 3651 13 - 4 0 9/200 NR 23/316
5.2- Case II: Lee-Ramirez Bioreactor.
This problem considers the OC of a fed-batch reactor for induced foreign protein production by recombinant bacteria, as presented by Lee and Ramirez (1994) and modified afterwards by Tholudur and Ramirez (1997). The objective is to maximize the profitability of the process using the nutrient and the inducer feeding rates as the control variables. Three different values for the ratio of the cost of inducer to the value of the protein production (Q) were considered. Table 2. Summary of results Authors' Tholudur and Ramirez (1997) This work
for case II. Performance Index (Sub-case) 6.16 (Q=0.0) / 5.77 (Q=2.5) / 5.58 (Q=5.0) 6.15 (Q=0.0)/5.76 (Q=2.5) / 5.57(Q=5.0)
CPU time (s) NR 5.0/3.5/4.0
5.3. Case HI: Non-linear CSTR.
The problem consists of determining the optimal controls of a chemical reactor in order to obtain maximum economic benefit. The system dynamics describe five simultaneous chemical reactions taking place in an isothermal continuous stirred tank reactor. Luus (1990) and Bojkov et al. (1993), who considered two sub-cases using three and four control variables respectively, also solved this case study.
485
Table 3" Summary of results for case III.
Sub-case CSTR 3 controls CSTR 4 controls
........
Authors Luus(1990) .
.
.
.
.
.
.
CPU time (s)
20.0895 (p=10), 20.0953 (p=40)
This Work .
......................
Performance Index
1278
20.0895 (p-lO)/20.0953 (/)=40) .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3.4/22.0 .
.
.
.
.
.
.
.
.
.
.
Luus(1990) Bojkov et al. (1993)
21.757 (p=l 1) 21.744
NR 7.8
This Work ...
21.757 (/).=!!)/21.8 07 (t9=80)
17.0/119.0
2.5
.
u2
1.0 o
2.0
o
0.8
b.,
1.5
o
0.6 o
1.0
o
z
_,= 0.4 0
0.5
0.2
0.0 0.0
1
i
1
5.0
10.0
15.0
Time, h
Fig. 1. Optimal control profile for case I (PR-b). 20.0
0.0
,,
0.0
[i,
i
2.0
4.0
6.0 Time, h
8.0
i
10.0
Fig. 2. Optimal control profile for case II(Q=0.0) 20.0 .. . . . . . . . . . . . . . . . . . . . . . . . .
..........
', : ,: '
r162
= 15.0
15.0
b~ 10.0 o
10.0 ,,,
=~ 5.0 r,.) 0.0 0.00
................ i
i
'
~ul "'-u2 ----u3 ---u4
O
5.0 O 0.05
0.10 Time, h
~ 0.15
~'~ 0.20
Fig. 3. Optimal control profile (case III, 3 u)
0.0 0.00
0.05
0.10 Time, h
0.15
0.2
Fig. 4. Optimal control profile (case III, 4 u)
7. C O N C L U S I O N S . The use of exact first and second order information resulted in two major differences with existing CVP algorithms. Firstly, a significant reduction in function and gradient evaluations was observed due to the use of second order information, which although requires the more expensive second order information evaluation, results in overall computational savings. Secondly, the ability to consider very fine discretization levels for the underlying controls has been enhanced both by the high precision and speed of convergence but also by the use of the mesh-refining technique.
486 In terms of performance and quality of solutions for the examples presented here, results achieved are comparable or better than the ones found in literature. The results also indicate that the algorithm is capable of identifying clearly singular arcs in the optimal control profiles (e.g. in the case of fermentation processes) and is also efficient enough to be used in low-cost computing platforms with very competitive CPU times. ACKNOWLEDGEMENTS This work was supported in part by EU (project FAIR CT96-1192) and the Spanish Government (CICyT project ALI97-1939-CE). Author Balsa-Canto thanks the Diputaci6n Provincial de Pontevedra, Spain, for a pre-doctoral fellowship. REFERENCES.
Banga, J. R., Alonso, A. A. and Singh, R. P. (1997) Stochastic dynamic optimization of batch and semicontinuos bioprocesses. Biotechnol. Prog. 13:326. Banga, J. R., R. Irizarry and W. D. Seider (1998) Stochastic optimization for optimal and model-predictive control. Comput. Chem. Eng., 22(4-5):603-612. Bojkov, B., Hansel, R. and Luus, R. (1993) Application of direct search optimization to optimal control problems. Hung. J. Ind. Chem. 21:177-185. Cuthrell, J.E. and Biegler, L.T. (1989) Simultaneous optimization and solution methods for batch reactor control profiles. Comput. Chem. Eng. 13:49. Lee, J. and Ramirez, W.F. (1994) Optimal fed-batch control of induced foreign protein production by recombinant bacteria. AIChE J. 40(5): 899 Leis, J. R. and Kramer, M. A. (1985) Sensitivity analysis of systems of differential and algebraic equations. AIChE J. 9(3): 93-96. Luus, R. (1990) Application of dynamic programming to high-dimensional non-linear optimal control problems. Int. J. Control. 52(1): 239-250 Luus, R. (1995) Sensitivity of control policy on yield of a fed-batch reactor. Presented at the IASTED Int. Conf. on Modelling and Simulation, Pittsburg, PA (USA) Park, S. and W. F. Ramirez (1988). Optimal production of secreted protein in fed-batch reactors. AIChE J. 34(8): 1550-1558. Nash, S.G. (1984) Newton-Type Minimization via the Lanczos Method. SIAM J. Num. Anal. 21:770-778. Tholudur, A. and Ramirez, W. F. (1997) Obtaining smoother singular arc policies using a modified iterative dynamic programming algorithm. Int. J. Control. 68(5): 1115-1128. Vassiliadis, V.S. (1993) Computational Solution of Dynamic Optimization Problems with General Differential-Algebraic Constraints. PhD Thesis: Imperial College, University of London, U.K. Vassiliadis, V.S., E. Balsa-Canto and J. R. Banga (1999) Second order sensitivities of general dynamic systems with application to optimal control problems. Chem. Eng. Sci. 54:38513860. Wang, F.S. and J.P. Chiou (1997) Optimal control and optimal time location problems of differential-algebraic systems by differential evolution. Ind. Eng. Chem. Res. 36(11):53485357. Yang, R.L. and Wu, C. P. (1994) Global optimal control by accelerated simulated annealing. Presented at the First Asian Control Conference, Tokyo (Japan)
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
487
Interaction Between Process Plant Operation and Cracking Furnaces Maintenance Policy in an Ethylene Plant E. Schulz, S. Diaz and A. Bandoni Planta Piloto de Ingenieria Quimica- PLAPIQUI (UNS-CONICET) Camino La Carrindanga Km 7 - 8000 Bahia Blanca- Argentina e-mail: abandoni@plapiqui, edu.ar, [email protected] ABSTRACT This work addresses the problem of the determination of a maintenance policy for cracking furnaces in an ethylene plant, taking into account the interactions between the entire process plant operation and furnace performance. An important recycle stream (ethane from units downstream the furnaces) constitutes part of the feed, so nonlinear models of units from the entire ethylene process have been included in the optimization problem. The resulting model is a mixed integer nonlinear programming problem where integer variables are associated to the number of time cycles that each feed is processed in a furnace. 1. INTRODUCTION The increasing challenge of international competitiveness in the oil and petrochemical industry at a global level makes no longer possible to define optimal production policies only on operating conditions. Pinto et al. (1999) have derived and applied planning and scheduling models to optimal operation in oil refineries. Furthermore, in plants processing different feedstocks and requiring cyclic shut downs for cleaning, as it is the case of ethylene plants, a correct assigning of feeds to units and time cycles is essential to reach global optimal operation. Jain and Grossmann (1998) have studied the scheduling of multiple feeds on parallel units with decaying performance. In this paper, we formulate and solve the problem of the determination of a maintenance policy for cracking furnaces in an ethylene plant, taking into account the interactions between the entire process plant operation and furnace performance. Thermal cracking of ethane produces depositions of coke on the tubes internal surface, so furnaces performance decay with time and they have to be periodically shut down and cleaned with a certain stopping policy. Downstream process information is obtained through the addition of nonlinear models for the entire process plant (Bandoni et aL, 1989; Diaz and Bandoni, 1996) because an important recycle stream (ethane from units downstream the furnaces) constitutes part of the feed. In this way, it is possible to have a more realistic insight of the influence of the main scheduling decisions on the entire plant performance. Ethylene production process has been modeled with ten parallel furnaces and multiple feeds together with the entire plant model that includes: quenching, hydrogenation reactors, separation train, ethane recycle, ethylene recycle, cracked gas compressor, ethylene and propylene compressors and utility plant. These plant sectors are represented by nonlinear correlations for unit parameters and mass balances.
488 The resulting model is a mixed integer nonlinear programming problem where integer variables are associated to the number of time cycles that each feed is processed in a furnace. The objective is to maximize profits, calculated as the difference between income (production of ethylene, propylene, propane, butane, gasoline and residual gas) and cost (ethane, natural gas consumption for boilers and furnaces, electricity and cleanup costs for furnaces). The problem has been solved using GAMS (Brooke et al., 1992). 2. ETHYLENE PLANT DESCRIPTION The ethylene plant under study (Fig. 1) produces 260,000 ton/y of 99.9% pure ethylene. It consists of ten pyrolysis furnaces, a cracked gas compressor, heat recovery network, separation system, refrigeration system and steam generation system. The feed, which has high ethane content, is cracked in the pyrolysis furnaces to produce ethylene and subproducts. Previously, the feed gas at high temperature is diluted with a steam stream to minimize coke deposition in the tubes. Cracked gas is compressed from 0.2 to 30 kg/cm2, and afterwards it is cooled to -100~ Before to the cooling stage, water is fully removed from the process stream and acetylene, one of the reaction side products, is totally converted to ethylene. Finally, these gases at high pressure and cryogenic temperature enter the separation train; where the first unit is a demethanizer column that produces residual gas (mainly hydrogen and methane) as top product. This residual gas is used as fuel in the plant boilers. Ethane and ethylene are obtained as top products from the deethanizer column and are then separated in a splitter; ethane is recycled to the pyrolysis furnaces and ethylene is obtained as the plant main product. Propane and propylene constitute the top stream in the depropanizer column and are sent to a splitter. Butane and gasoline are separated in a debutanizer column. ......................
High Pr essu~e S t e ~
L
Furnaces
l
~J'"
9
CrackedOas
erie
I
l
._.3_.
"~
s
~ T ~ Hyok o genalac~n .Reactors
Re~imle, ]
Water
Gas
U
t
~
c~,,
i i
..............
i.... ~'//
c~.
i
"
'.... ~1 t~'-. E~e~" i Recycle
I
1
................................................................................................................................... / ~ . .................................. ~
|
, J
i
I I Oopro~.~er k_~
L_~ k . J
I P'~176176
le"e Prop~,
I I l
Oeb~-.
I
Figure 1' Process plant and alternative gas feeds.
I ~
l* ......................... o
489 There are two main process recycles in the plant: a) ethane recycle, which is the bottom stream from the ethane-ethylene splitter and is sent to the cracking furnaces after mixing with the ethane fresh feed; b) ethylene recycle, that is recycled from the gasoline stabilization column. 3. CYCLIC ASSIGNMENT OF FEEDS TO CRACKING FURNACES AND MAINTENANCE POLICY In this plant, ethylene is obtained by thermal cracking of ethane. Ethylene yield and, consequently, net profit are closely related to feed composition. Diaz and Bandoni (1996) have shown optimal profit variations as function of propane content in the feed; mixtures with higher ethane concentration render higher profit. The coke produced as a byproduct in the thermal cracking of ethane deposits on internal tube surface with two undesirable effects that make ethylene yield decrease. One of them is the insulation of the reacting mixture (a strong negative effect as the reaction is highly endothermic) and the other is the decrease of tubes cross sectional area. Consequently, cracking furnaces must be periodically shut down and cleaned with the associated cost of clean up and loss of production. Therefore, there is a compromise between operating the furnaces for very long periods of time with decreasing plant performance against higher clean up costs with better performance. In this work, we have addressed the problem of cyclic assignment of ethane mixtures with different ethane content to cracking furnaces and the determination of optimal operating conditions and clean up times. The model takes into account the important ethane recycle stream that is sent back to cracking. The entire plant model comprises correlations to represent furnaces operation and the separation train, together with the utility system and the three main compressors (cracked gas, ethylene and propylene compressors). These correlations (Bandoni et al., 1989) have been obtained from plant data and results of runs with rigorous unit models. 4. MATHEMATICAL MODEL The cyclic assignment of different feeds to cracking furnaces and determination of operating times, as well as clean up periods, has been formulated as a MINLP problem, as proposed by Jain and Grossmann (1998). These authors formulated the scheduling of multiple feeds to different cracking furnaces as a MINLP problem with linear constraints and a pseudoconcave objective function (maximization problem), which guarantees global optimality. To obtain linear constraints, they have approximated mass balances around mixers as linear equations. The scheduling was performed without a downstream plant model, and consequently, the ethane recycle has not been included in the optimization model. In this work, the objective function is the maximization of profit, defined as the difference between the sales revenue and the total operating cost. The revenue from products is defined over all products generated in the plant (ethylene, propylene, propane, butane, gasoline, residual gas); the feed costs are mainly associated to ethane feed cost. The model for the optimal assignment of feeds to furnaces, operating times and number of cycles for each feed in the furnaces includes the following equations: Component Mass Balances at Furnaces Entrance F1sT~y~te + RV. T, = D~s-T,
Vi,j
(1)
F,- = ZFv.
Vi
(2)
490 Recycle Stream V i,j
(3)
Vi
(4)
Vi
(5)
V i,j
(6)
n; : ZTcy;k
Vi
Zy,~= 1
Vi
(7) (8)
Vi
(9)
R,j = f(Rugi, Pdem~, Reli, FsepO
Tube Roughness Rugi = Cli + C2iTi/ni
Furnaces Inlet Pressure Pin~ = f(D~, Rd,, Pouti, Conv~,RouO
Furnaces Production Ffj. = f(D,, Rd,, Pin~, ConvO
Integrality Constraints for Number of Cycles
Total Processing and Clean up Time Dt, = n,v, + T, Z D 6 =Tcycle Plant Process Streams
(10)
V i,j, u
Ftu,j = f(Pdem, Reli, fSu~
(11)
Fbu,j. = f(Pdeml, Reli, fsu,.j) V i,j,u (12) Equations (1) and (2) stand for mass balances at the furnaces entrance; the term R,j T, that represents the recycle stream is not taken into account when the entire plant model is not included. Equations (3) to (6) are correlations that determine internal tube roughness and furnaces production as function of operating time, furnace inlet pressure and component molar flowrates. Equations (7) to (10) represent timing and integrality constraints (Jain and Grossmann, 1998). Equations (11) and (12) are product stream flowrate of the different units that constitute the separation train. The model also comprises correlations that evaluate main plant compressors consumption and utility system. 5. NUMERICAL RESULTS
We have analyzed the cyclic assignment of alternative gas feeds to a cracking furnace that represents ten existing furnaces in the plant. The model has been implemented in GAMS (Brooke et al., 1992). A comparison of results has been performed between the optimal maintenance policy with and without the entire plant model. We have studied problems with two and three gas feeds. Feed molar compositions are shown in Table 1, together with clean up times for each one. Table 1 Feed molar compositions and clean up times Feed A B C
Methane 0.40 3.73 0.40
Ethane 99.40 90.90 84.80
Propane 0.20 5.37 14.80
z'/(days) 2 3 2 '/2
Conversion is function of internal tube roughness and it is consequently a linear function of processing time. Table 2 shows main timing variables for the studied cases. The same trend has
491 been observed in the assignment of two and three alternative feeds to a furnace when not taking into account the ethane recycle stream (Cases 1 and 3). In these cases, each feed is processed only one time cycle in the furnace, but feed A is the one with longer processing time due to its high ethane content. Additionally, mean conversion is almost the same for all of them. When nonlinear models representing the entire plant are included in the problem (Cases 2 and 4), a different time schedule is obtained and the trend is the same for the processing of two or three feeds. In both cases, numerical results confirm the convenience of processing feed A for a longer time, as it is the one with higher ethane content. Moreover, as it has been associated to lower roughness coefficients, it only requires one shut down for cleaning (hA = 1). Feeds B and C are processed for shorter times and both of them require two time cycles; i.e., two shut down periods each (n8 - n c = 2). As regards profit, the model that includes the entire plant model predicts higher values because molar ethane flowrates at the furnaces entrance are parameters; i.e., there are fixed feed loads to furnaces. In cases 1 and 3, a higher flowrate of fresh ethane is needed, as there is no recycle. Table 2 Optimal assignment of feeds and processing times Case 1
2 3
4
Feed A B
A B A B C A B C
Recycle
T (d)
Tcycle
n
No
63 60
128
Yes
86 71 62 55 53 94 73 54
No
Yes
165 189
235
1 1
Mean Conversion 0.700 0.700
Fresh Feed (Kmol/h) 970 875
1 2 1 1 1 1 2 2
0.700 0.698 0.700 0.700 0.686 0.700 0.697 0.653
707 577 675 587 522 544 415 278
Profit ......... (U$S/h) 9463 10678
10359
11044
The optimal assignment of the three feeds (cases 3 and 4) is shown in Fig. 2, along with processing and clean up times. A
B
A
C
B 94
.....
Case 3
B
C 173
C Case 4 235
........
Time (days)
Figure 2. Processing times distribution and optimal assignment of feeds (Cases 3 & 4). Table 3 Main plant operating variable values for case 4 (three feeds) Variable A B C Inlet Furnace Pressure (bar) 3.548 3.393 .... 2.778 Ethane Recycle (Kmol/h) 578.76 529.33 538.44 Roughness 0.007 0.006 0.002 Ethylene Stream (Kmol/h) 1124.36 1077.72 1024.93 Dilution Ratio 0.27 0.27 0.27 ..................
.
.......
.
.
.
.
.
.
.
.
..................
.
.
.
.
.
.
.
.
.
.
.
.
492
Table 3 presents a comparison of main plant operating variables associated to the processing of each alternative feed for case 4. This problem contains 260 equations, 415 continuous variables and 24 binary variables and has been solved with GAMS (using DICOPT++ with CONOPT and OSL) in four major iterations.
6. CONCLUSIONS This paper has presented a model for the determination of optimal maintenance policy in ethylene plants with multiple feeds with a downstream nonlinear plant model to take into account operating decisions in the ethane recycle to furnaces feed. Different cleaning up and stopping policies are obtained when the entire plant model is added to the scheduling model. These results show that the ethane recycle should not be neglected when determining optimal operating policies. Acknowledgement. The authors gratefully acknowledge financial support from CONICET and Universidad Nacional del Sur, Argentina. Notation Fv Fresh feed flowrate of component j in feed i (Kmol/h) T~de Totalcycle time (h) R oFlowrateof component j in ethane recycle stream for feed i (Kmol/h) D o. Inlet flowrate of component j and feed i to furnaces (Kmol/h) T, Total processing time for feed i (not including clean up) Rugi Internal coil roughness for feed i Pdemi Demethanizer column pressure (bar) Rel~ Ethylene/Ethane ratio at the entrance of separation train Fsepuo. Separationfactor in unit u for feed i C1;,C2i Internal coil roughness coefficients n~ Number of subcycles of feed i in the furnaces Pini Furnace inlet pressure for feed i (bar) Rd~ Dilution ratio for feed i Pout~ Furnace outlet pressure for feed i (bar) Conv i Ethane conversion for feed i Ffj Molar flowrate of component j and feed i in furnaces product stream (Kmol/h) y,-k Binary variable (y~k=1 if feed i is processed k subcycles in the furnace) Dt, Total processing and clean up time for feed i v~ Clean up time for feed i Ftuo Molar flowrate of component j and feed i as top product in unit u (Kmol/h) Fb~o Molar flowrate of component j and feed i as bottom product in unit u (Kmol/h) REFERENCES Bandoni, A., A. Eliceche, G. Mabe, E.A. Brignole, "Optimal Operation of Ethylene Plants", Com. Chem. Eng., 13, 587-594, 1989. Brooke, A., D. Kendrick, A. Meeraus, "GAMS: A users guide", Scientific Press, Palo Alto, 1992. Diaz, S., A. Bandoni; "A Mixed Integer Optimization Strategy for a Real Large Scale Chemical Plant", Com. Chem. Eng., 20(5), 531-545, 1996 Jain, V., I. E. Grossmann, "Cyclic Scheduling and Maintenance of Parallel Units of Decaying Performance",AIChEJ., 44, 1623-1636, 1998. Pinto, J., M. Joly, L. Moro, "Planning and Scheduling Models for Refinery Operations", II Pan. Workshop on Catalysis and Process Systems Engineering, Santa Fe, Argentina, September 2-3, 1999.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
493
Convergence Refinement of Stochastic Optimization by Coupling a Genetic Algorithm and a Simulated Annealing Procedure A. Davin, C. Azzaro-Pantel, P. Floquet, L. Pibouleau and S. Domenech Laboratoire de GOnie Chimique- UMR - CNRS 5503
ENSIGC INPT 18, Chemin de la Loge - 31078 Toulouse Cedex 04 - France Numerous trials have to be carried out to refine the convergence of Simulated Annealing (SA) and Genetic Algorithms (GAs). A procedure that couples both GAs and SA algorithms using the ability of GAs to reach quite optimal solutions, and SA procedure for scanning the neighboring solutions to efficiently improve the objective function value. The probability of reaching the true optimal solution, can be significantly increased by this coupling. This approach is illustrated by the problem of synthesis of distillation sequences of mixtures involving 10 to 16 hydrocarbons. 1. INTRODUCTION The probability of a fast convergence towards the global optimum of stochastic algorithms, namely Simulated Annealing (SA) and Genetic Algorithms (GAs) from randomly selected initial solutions, strongly depends on the values of control parameters of procedures (initial temperature, length of stages, probabilities of crossover and mutation . . . . ). In practice, numerous trials have to be carried out to refine the convergence of such procedures. In the framework of GAs, the quite optimal solutions are rapidly approximated by the systematic selection of strong individuals, but after which the convergence towards the optimal solution may be arduous, due to the mutation process that only generates solutions in the neighborhood of the current one. Conversely, for SA procedures the selection of neighboring solutions which improve the objective function value is very efficient, whereas the scanning of other solutions of the search space that could both improve the criterion and reach the optimum, highly depends both on the initial solution and on the policy implemented for degrading the objective function. We propose, in this paper, a procedure that couples both GAs and SA algorithms. A statistical study is then carried out on the well known problem of synthesis of separation sequences of mixtures involving 10 to 16 hydrocarbons, using distillation columns with or without sidestreams. 2. STOCHASTIC PROCEDURES The procedures implemented in this work are the well-known SA and GA algorithms. The simulated annealing procedure mimics the physical annealing of solids: the slow cooling of a molten substance, that redistributes the arrangement of the crystals
494 following some probabilistic rules. In the annealing of solids the goal is to reach given atomic configurations that minimize internal energy, while in a rapid cooling or quenching, the final result would be a metastable structure with higher energy. Based on this physical property, Kirkpatrick et al. (1983) have first implemented the simulated annealing (SA) procedure. The goal is to generate feasible solutions of an optimization problem that minimize a given objective function. Like careful annealing leads to the lowest internal energy state, the SA procedure can lead to a global minimum, and like rapid cooling generates a higher energy metastable state, the SA procedure may also block on local minima. Indeed, the cooling scheme is one of the predominant parameter. The simulated annealing algorithm implemented in this study is presented on Figure 1. Select an initial structure M = M ~ Select an initial value for the SA- temperature Tsa ~ Select the length o f cooling stage Nsa Do while the search can evolve Do Nsa times Generate a new structure M" neighbor o f M
Evaluate
Aj = j(M') - j(M)
Accept or reject the new configuration according to the Metropolis algorithm : p=l if A j < O
or p=exp
Tsai
if A j>__0
I f M ' accepted : M = M ' I f M ' rejected cycle keeping M unchanged Decrease Tsa according to the SA-cooling schedule : Txai+ 1 = a . T x a i
(o<~1)
7inal structure 9M
Figure 1 9General Algorithm of SA Procedure GAs are search procedures based on the mimesis of the mechanics of natural selection and genetics. Theoretically developed by Holland (1975), GAs emulate the biological evolutionary theory to solve optimization problems. Unlike the SA algorithm, a GA computes a set of individuals, the population, evolving through a set of biologically inspired operators constituting the reproduction scheme. In this way new individuals are generated 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 is enclosed in the chromosomes of the individuals represented in an optimization problem by a specific encoding strategy. The suitability of each element according to the optimization problem considered is evaluated via a fitness value, directly derived from the objective function. The evolution mechanisms are constituted by three specific procedures : selection, cross-over and mutation. The evolution is generally repeated until a predefined number of generations is reached. The GA implemented here follows the classical steps of GAs (Figure 2).
495
Generation of the initial population Estimation of the fitness of the initial population While the total number of generations is not reached do ." Generate the offspring population Selection of surviving individuals Synthesis of offspring obtained through cross-over Mutation of individuals Enddo
Figure 2 9General A l g o r i t ~ of GA Procedure The two main features of GAs is that they provide several good altematives at the end of the search and have appeared more robust than SA on various test problems according to the implicit parallelism notion involved in the population generations (Koza, 1993). Conceming the SA algorithm, the search initialization is the basic point to ensure a good convergence, and in many cases several preliminary trials are required to determine this initialization. In opposite, in GAs implementation, this initialization step does not play a so predominant role. 3. COUPLING GA/SA PROCEDURES In the first step on the approach the GA is used to rapidly approximate the optimal solution by flying over a large search space. Then the SA procedure is implemented for a detailed scanning of the solutions provided by the GA. Figure 3 shows the coupling procedure.
Genetic Algorithm
1
Coupling parameters : uracy of GA, ice of intermediate individual, initial temperature of SA
l
Simulated Annealing Figure 3 : Schema of Coupling Procedure For GA,the population size, the survival and mutation rates are independent parameters of the coupling procedure. Likewise, for the SA algorithm, the cooling scheme and the stopping criteria are also independent of the coupling procedure. On the other hand, the two predominant parameters of the coupling procedure are the number of generations (for GA) and the choice of the intermediate individual to select in the final population. For the SA, the algorithm evolution is directly connected to the initialization and to the initial temperature. The choice of these parameters is studied in the following sections.
496 4. SEPARATION SEQUENCE SYNTHESIS PROBLEM To illustrate and select relevant parameters of the coupling procedure, a statistical study was carried out on the well-known separation sequence synthesis (SSS) problem. The SSS problem can be stated as follows: <. The number of possible separation sequences highly increases with the number of components to be separated (Floquet et al., 1994, Laquerbe et al.., 1997). 5. RESULTS AND DISCUSSION In the case of a 10-component mixture (continuous distillation of an equimolar mixture of 10 hydrocarbons n-C7 to n-C16 - feed flowrate of 1000 kmol.h~), where the optimal solution is well known, because the set of all the solutions can be exhaustively enumerated, the results of GA and SA can be obtained separately. The results presented below, were obtained on a 50 trial sample. 5.1. GA Results
The results of GA (population size: 60, survival rate: 0.5; mutation rate: 0.1) are shown in Table 1, and the evolution of the average gap from the optimal solution, is presented on figure 4. It shows the capability of the genetic algorithm to quickly come close to the optimal solution. Indeed, after 50 generations, the average of costs found is about 6% of the optimal sequence cost. Generation Number 400 300 200 100 50 30 15
Optimal Cost Mean Cost of best sequences obtained (MS/year) (50 trials) (MS/year) 15.848 16.499 15.848 16.400 15.848 ........ 16.533 ........ 161006 16.789 .......... 15.848 ..16.897 15.848 17.396 16.416 17.834 ........
Confidence Interval (%) 0.8 0.6 0.8 1.0 ....1.0 1.1 1.1
Table 1 9Statistical Results (AG) for the 10-component example 30
Average Distance
(%)
25 20 15 10 5
~,
, I 0
50
100
150
200
A A v
250
300
v
350
Generation Number Figure 4 9Accuracy of GA procedure
400
497 An increase of the generation number improves progressively the results while reducing the distance from the optimal solution. After 400 generations, the probability to reach the global minimum is always of 4%.
5.2. SA Results The relevant parameters of SA algorithm are given in Table 3, and the results obtained from 50 trials randomly generated are reported in Table 2. Initial OptimalCost obtained MeanCost of best sequences Confidence Temperature (MS/year) (50 trials) (MS/year) Interval (%) 10,000 15.848 16.861 1.9 100,000 15.848 16.701 2.0 Table 2 9Statistical Results (SA) for the 10-component example .
.
.
.
.
.
.
In any case, the convergence of SA towards the optimal solution is better than the GA one, despite a less number of criterion computations. Nevertheless, in a weak number of cases, SA blocks itself on local minima far from the optimal solution (about 35%). The initial temperature of 100,000 allows of better space scanning than a temperature equals to 10,000. The confidence interval confirms this tendency: in the case of the SA solution, the confidence interval is doubled compared with GA, and exhibits the initialization influence.
5.3. GA/SA Coupling Strategy Parameters of GA/SA coupling algorithm are given in Table 3. GA Generation Number Population size Mutation rate Survival rate
SA Initial temperature (Tsa) Decreasing temperature coefficient (~) Maximum number of tested configurations Maximumnumber of accepted configurations
15~30 - 50 60 0.1 .... 0.5 .
.
. . . . . .
.
.
104 - 105 01'95 100 10
.
Table 3 9GA/SA Parameters The probability of obtaining the optimal solution (38% with a randomly selected initialization) is increased up to 60% by using the best solution provided by the 50th generation of the GA as the initial one for the SA, and the computation time was reduced by 30% with the same control parameters used (initial temperature, cooling procedure, length of stages, stopping criteria). On no account, the gap versus the optimal solution does not exceed 10%, even when this gap reaches 35% with a random initialization. When the GA is used alone, the optimal solution is found only in 4% of trials after a limited number (50) of generations, but this result was not improved by 350 supplementary generations. Furthermore, if 50 generations allow to obtain all the solutions lying in a range of 15% with respect to the optimal one, at least 300 generations are required to reduce this gap at 10%, while the CPU time is multiplied by a factor 6. All these statistical results are summarized in Figure 5.
498
Figure 5 : Statistical analysis of results A similar study was carried out for mixtures involving 14 and 16 components (in that cases, the optimal solutions are unknown, because of the combinatorics of the problems), and the same tendencies were observed. Let us note that in all cases, when equimolar mixtures are considered, the optimal solution only involves columns with sidestreams (except the last separators), but this topology can change with the feed composition. Even if the convergence towards the global optimal solution is not guaranteed by the coupling GA/SA, the probability of reaching it is significantly increased by this approach. REFERENCES Floquet P., Domenech S., Pibouleau L., I. E. C. Res. Vol. 33, n ~ 2, 444, (1994) Holland J., <~,MIT Press, (1975) Kirkpatrick S., Gellat C.D. and Vecci M.P., Science, n ~ 220, 671, (1983) Koza J.R., <>,MIT Press, (1993) Laquerbe C., Floquet P., Domenech S., Pibouleau L., RAIRO RO Vol. 31,n ~ 4, 397, (1997)
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
499
Fuzzy Modeling of Catalytic Multi-phase Reactor Freitas Jr, B. B.; Maciel Filho R. Laboratory of Optimization, Design and Advanced Control (lopca), College of Chemical Engineering- State University of Campinas PO Box" 6066, Zip Code: 13081-970, Campinas- SP, Brasil e-maih basilino@lopca, feq.unicamp.br
ABSTRACT. The mathematical modeling and computational simulation have been used as indispensable resource in the analysis of most processes, including catalytic reactors. Among the available tools for process representation. The fuzzy modeling has na increasing interest. The fuzzy logical is the logic that supports the reasoning manners that are approximate instead of exact. Fuzzy modeling of systems is a technique for the treatment of qualitative information in a rigorous way. Derived from the concept of fuzzy set, the fuzzy logical constitutes the base for the development of methods and modeling algorithms, allowing the reduction of the project complexity and implementation. The mathematical modeling of three-phase reactors is not still as developed as some other processes of the chemical industry. Therefore this work has as objective the development of a model mathematical of a three-phase system, using the foundations of group operations and fuzzy logical. The system in subject is a three-phase slurry catalytic reactor operating in steady-state regime, and with a tubular geometric configuration and refrigeration. The fuzzy model was able to represent very well. 1. INTRODUCTION The elaboration of sophisticated mathematical models is sometimes very difficult to be carried out when only physical-chemical laws and mass/energy balances are used. In fact this is more apparent becomes when the phenomena involved in the process are not elementary. Difficulties appear not only in the mathematical complexity of the models and consequent resolution but also in selection to the parameters which are normally restrict to some operation range and ever, sometimes not available. Bearing this mind, it helpful to have alternative ways to represent the system in a more flexible and representative fashion. An attractive manner to do that is through the fuzzy logical concepts. The concept of set fuzzy was introduced, in 1965, by Lotfi A. The systems fuzzy was applied with success in several types of processes, as in cement plant, nuclear plants, refineries, biological and chemical processes, machines diesel, treatment of water and automatic operation of trains. The development of techniques of artificial intelligence in the last years, has been occupying important position in the area of control of industrial processes, and slowly, they begin to be implanted in industrial plants with enormous success. The process of production of the methylcyclohexanol belongs to the first generation petrochemical industries, it is a intermediate product for the obtaining of industrial products of great commercial value as the nylon. In this case, the cost of production of final products, as nylon, depends on the appropriate operation not only of the plant of the product itself, but also of the that process the intermediary products, among these it is the methylcyclohexanol.
500 Several works are mentioned, applying fuzzy logical in industrial processes. Hanai et al (1997), proposed fuzzy-neural modeling (FNN) for the Ginjo process of sake fermentation. The this process of sake fermentation possesses a slow dynamics and complex modeling, making unfeasible the control feedback and the deterministic modeling. The model fuzzyneural proposed, build up with a large amount of information, was able to foresee the temperature of the system. The adaptability of the model was confirmed by the simulation. Yamata et al (1991), proposed fuzzy model for the process of fermentation of the coenzyme Q10. The deterministic modeling of the fermentation processes is difficult due the many biochemical reactions. The control objective was to obtain larger productivity as well as to control the rate of aeration of the process. After modeling the process with 70 fuzzy rules, it was possible to verify that fuzzy controls the a system with stability resulting in a high productivity operation. 2. LOGICAL FUZZY IN PROCESS REPRESENTATION The fuzzy logical is related with the formal beginnings of approximate reasoning. The importance of this in relation to the fuzzy logical comes from the fact that most of the manners of human reasoning is pared on the inference of answers approached for subjects based on inexact, incomplete knowledge or sometimes completely not reliable information. An universe of the speech X is a collection of objects denoted for {x}. a membership function is ~ta:X---~[0,1]. A = { ( x , ~ta(X))lx E X}
1
A n
A
0 ,.-'~X Fig. 1. Classic set
i~Y Fig. 2. fuzzy set
In agreement with the theory of fuzzy set an element doesn't simply belong or it doesn't belong to a set as in the classic theory, but it can belong to a set with membership degree that varies in the interval [0,1 ]. The fuzzy logical uses linguistic variables instead of numeric variables, linguistic expressions (or we have primary), as " very big ", " not very cold ", younger ", that are represented by set fuzzy. The main operations of fuzzy set A,B C are: Union: P AuB (X) -- ~,~A (X) V ~'[B(X) Intersection: PA~B(X) = ~tA(X) ^ PB (X) Complement: p_ (x) = 1--pA (X) A
A relationship fuzzy R is a mapping among universes, expressed for the membership function, pR(x,y) in the Cartesian space XxY inside of the interval [0,1 ], The relationship among two sets is determined by the Cartesian product. R=AxB where XxY={(x,y)/x~X, y ~Y} pR(X,y)=~tAxB(X,y)----min(~tA(X),~tB(y)) Supposing R is a relationship fuzzy in the space XxY, S is the relationship fuzzy in the space YxZ, T is the relationship in the space XxZ. The max-min composition fuzzy is defined as"
501 T=RoS ~T(X,Z)-- V(gR(X,y) ^gs(y,z))
The implication material among two crisp set p-> q is interpreted as (p A q). p->q is true if t (p)c_ t(q) where the t(p) and t(q) they can be 0 or 1. Then p - > q =
{~
if t(p) c t(q) otherwise
The fuzzy implication is an extension of the implication material. Several operators exist for fuzzy implication among them we have: Larsen
A->B= ~tax ~tb
Mamdani
A->B = min (~ta, ~tb)
Kleene- Dienes
A->B = max (1-~ta, ~tb)
The triangular norms were introduced by Schweizer and Sklar to model the distance probabilistic metric space. In the theory of fuzzy set, the triangular norm (t) are extensivamente used to model the logical connective and, already the conorm (s) are used to model the logical connective or. Norma t: Minimum---> min (a,b) = min {pa,~tb} Product->Tp(a,b) = ~ta x ~tb Lukasiewiez->T1 (a,b) = max {~ta+Pb-1,0} Conorm s: Maximum -> max (a,b)= m a x {~ta,~b } Lukasiewicz -> S1 (a,b)= min {~ta+~tb, 1 } Probabilistic ---> Sp (a,b) = ~ta + ~tb - ~ta X ~tb The last goal of the fuzzy logic is to form a theoretical foundation about imprecise proposition of the reasoning (approximate Reasoning). In the fuzzy logical and in the approximate reasoning, the most important inference rule is the one of Modus Ponens generalized (GMP). The rule of inference classic modus ponens says: if p is true and p-> q is then true q it is also true. The inference implication fuzzy is based in it rules it inference compositional for approximate reasoning suggested by Zadeh. Where consequent B' is determined as a composition of the fact A' and the operator fuzzy implication. B ' = A' o ( A - + B )
The mechanism of reasoning fuzzy can be seen as, given a set of rules:
RI" if x 6 A l e y 6 B 1
ent~oz6C1
R2: if x 6 A 2 e y 6 B2 e n t i o z 6 C2
Rn: if x 6 An e y 6 Bn ent~o z 6 Cn x6xoey6yo z6c ........
502
n
C= E
Ci = Al(xo)/~Bl(yo)---~Cl(w) v...v An(xo)ABn(yo)-->Cn(w)
The desfuzzification of value C can be found by the method of mass center, maximum value, minimum value, etc. 3. INDUSTRIAL PROCESS OF M E T H Y L C Y C L O H E X A N O L PRODUCTION A typical unit of methylcyclohexanol production is formed and for a reactor, which is formed by tubular modules immerged in a boiler. Basically two reactants are involved, namely, phenol and hydrogen. The reaction of hydrogenation of phenol producing methylcyclohexanol is exothermic, and depending on the temperature of operation of the reactor and of the used catalyst, several products can be formed as ketones and acyclic alcohol, aromatic and acyclic hydrocarbon. 'The phenomenological equations of the system was written in a possible more general form so that the models can represent the most several situations, by means of pertinent simplifications for each specific case. The hypothesis for the formulation of the model: -
-
-
Stead state Plug-flow for the mixture reagent and for the thermal fluid Suspension (liquid- solid) homogeneous, considered as a pseudo one - fluid. Worthless variations of the pressure Reaction of the type A(g) + vB(/) ~ v C(/), happening in the catalyst and with a kinetics dependent of the concentrations of A and B It doesn't happen phase change in the system Gradient intraparticle is worthless hc=0
4. RESULTS AND DISCUSSION The mass and energy balance, values of the coefficients and methods of resolution of the system of differential equations are found in [Santana p.1, 1999]. The results of the simulation of the model which was validated with industrial data are exhibited in the tables 1 and 2 were used for the generation of fuzzy rules and for the test of the predictions of the set of rules. The membership function of the temperature as well as of the concentration for each point of the reactor was considered triangular. The applied inference was that based on Mamdani and the method of desfuzification was of the mass center. Rule 1: if T is 460 then B1,1 is 0,99429 and B1,2is 0,98863...and ... Bl,lO is 0,94302 Rule 2 : if T is 490 then B2,1 is 0,98573 and B1,2is 0,97183... and ...BI,lO is 0,86005 Rule 14: if T is 660 then B14,1 is 0,8973
and B14,2is 0,81136... and ...BI,10 is 0,16985
503 Table 1 Set of data for the generation of the fuzzy rules .......... Dimensionless concentration of B Ti (K) Z1 Z2 ..... Z3 Z4 Z5 460 0,9942 0,9886 0,9829 0,9715 490 0,9857 0,9718 0,9579 0,9439 0,9300 505 0,9784 0,9587 0,9385 0,9182 0,8980 510 0,9763 0,9536 0,9310 0,9084 0,8857 520 0,9704 0,9426 0,9147 0,8869 0,8591 540 0,9571 0,9176 0,8783 0,8392 0,8003 550 0,9499 0,9046 0,8594 0,8145 0,7700 570 0,9359 0,8793 0,8230 0,7673 0,7122 585 0,9264 0,8623 0,7988 0,7359 0,6738 590 0,9235 0,8572 0,7914 0,7263 0,6622 610 0,9133 0,8393 0,7659 0,6933 0,6217 630 0,9054 0,8255 0,7463 0,6678 0,5904 650 0,8996 0,8153 0,7317 0,6488 0,5668 660 0,8973 0,8113 0,7259 0,6412 0,5574
6,9772
alon~ the len~h (Z) of thereactor Z6
0,9658 0,9160 0,8778 0,8631 0,8314 0,7617 0,7259 0,6579 0,6128 0,5991 0,5514 0,5143 0,4862 0,4749
Z7
0,960i
Z8
..... 0,9544 0,9020 0,8880 0,8576 0,8374 0,8406 0,8181 0,8038 0,7764 0,7234 0,6856 0,6824 0,6395 0,6046 0,5523 0,5529 0,4946 0,5373 0,4771 0,4826 0,4158 0,4399 0,3677 0,4072 0,3305 0,3940,,0,3154
Z lo
0'9487 0,8740 0,8172 0,7956 0,7491 0,6482 0,5973 0,5013 0,438 0,4187 0,3515 0,2984 0,2571 0,2401
0,9430 0,8600 0,7971 0,7733 0,7220 0,6114 0,5559 0,4518 0,3834 0,3627 0,2903 0,2330 0,1883 0,1698,
Table 2 Set of data for training of the fuzz5, rules .... Dimensionless concentration of B'"along the lengih'""(Z) of ihe reactor T!, (K)
470 480 500 515 530 545 555 560 580 600 605 620 625 640
Z1
0.9920 0.9892 0.9814 0.9733 0.9640 0.9535 0.9464 0.9428 0.9294 0.9181 0.9156 0.9091 0.9072 0.9023
Z2 0.9842 0.9787 0.9634 0.9482 0.9304 0.9111 0.8981 0.8917 0.8677 0.8477 0.8433 0.8319 0.8286 0.8201 iiiiiiiiiiiil
........ Z 3
0.9764 0.9682 0.9455 0.9231 0.8969 0.8688 0.8500 0.8408 0.8065 0.7779 0.7717 0.7554 0.7507 0.7384
.....Z4
0.9686 0.9576 0.9275 0.8979 0.8636 0.8268 0.8023 0.7904 0.7459 0.7088 0.7008 0.6797 0.6735 0.6576
Z5
Z6
0.9608 0.9470 0.9095 0.8727 0.8303 0.7851 0.7551 0.7404 0.6861 0.6407 0.6309 0.6050 0.5974 0.5778
0.9529 0.9364 0.8915 0.8477 0.7972 0.7438 0.7083 0.6011 0.6272 0.5738 0.5622 0.5316 0.5226 0.4992
Z7
Z8
0.9450 0.9372 0.9258 0.9152 0.8735 0.8555 0.8226 0.7977 0.7644 0.7317 0.7029 0.6624 0.6622 0.6168 0.6425 0.5947 0.5694 0.5130 0.5084 0.4447 0.4951 0.4298 0.4599 0.3902 0.4495 0.3786 0.4224iiiiii0.3478
.........
Z9
Z9
0.9293 0.9045 0.8376 0.7729 0.6994 0.6226 0.5723 0.5479 0.4581 0.3832 0.3669 0.3233 0.3104 0.2764
........ Z lo...
0.9214 0.8939 0.8196 0.7481 0.6674 0.5835 0.5287 0.5023 0.4052 0.3245 0.3069 0.2599 0.2460 0.2092
The results of the behavior of the process (represented by the deterministic equations) and of the behavior foreseen by the fuzzy modeling are depicted in the figures 3 arid 4. As can be seen the fuzzy modeling approach was able to predict very well the reactor behaviour. 5. CONCLUSION In this work a multiphase reactor model based on fuzzy logical was developed. In this approach it is not necessary to represent the system through deterministc balance equations,
504
1.0
t.01 0.91
~0.8 0
~o.8
~..,
~0.7
~0.6
~
~o.g r~ ~0.4
0.5 N 0.4
~ Process ..... 9..... Fuzzymodel
~.~
.~ 0.3
0.3
T-
0.2 t 0.1 , ' , ,,I 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 ..............
'
........ . . . . .
' ,
' ,
' ,
'
,
Dimensionlesslengthof the reactor Fig. 3. Behavior of the concentration of B along the reactor.
0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Dimensionless length of the reactor
Fig. 4. Behavior of the concentration of B along the reactor.
-which sometimes bring difficulties due to the model parameters identification as well as solution procedure, fuzzy modeling, on the other hand, is a more flexible and universal approach to represent the system, but requires a good knowledge about the process. The results shown in this work allow to conclude that the fuzzy logical approach is a reliable way to represent the system. In fact, the proposed fuzzy model with 14 rules was shown quite efficient in the representation of the process multi-phase of methylcyclohexanol production. REFERENCES 1. Chang w. c.; ouyang c. f.; chiang w. 1. n a d hou c. w. slugde pre-recicle control of dynamic enhaced biological phosphorus removal system: an application of on-line fuzzy controller. water reseach, v. 32, n. 3, p.727-736, 1998. 2. Draeger a.; engell s. and ranke h. model predictive control using nerural networks, ieee control systems magazine, dortmund, v.15, n.5, p.61-66, 1995. 3. Hanai t.; katayama a.; honda h.; kobayashi t. automatic fuzzi modeling for ginjo sake brewing process using fuzzy neural networks, journal of chemical engineering of japan, v. 30, n. 1, p.94-100, 1997. 4. Jamshidi rn.; vadiee n. and ross timothy j. fuzzy logic and control, new jersey: ptr prentice hall, 1993.0 5. Kartalopoulos stamatios v. understanding neural networks and fuzzy logic: basic concepts and applications, new york: ieee press, 1996. 6. Kuo r, j. and cohen p. h. intelligent tool wear estimation system through artificial neural networks and fiazzy modeling, artificial inteell eng, v. 12, n, 3, p.229-242,1998. 7. Santana p.1. modelagem matem~itica para reator trif~tsico: modelos deterministicos, neuronais e hibridos, campinas, 1999. tese de doutourada- departamento de processos quimicos, unicamp. 8. Yamada y.; haneda k,; murayama s.and shiomi s. application of fuzzy control system to coenzime ql0 fermentation, journal of chemical engeneering of japan, v. 24, n. 1, p94-99, 1991.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
505
Strategy and Mathematical Development for Scale-Up of Molecular Distillators for Recovering Carotenoids from Palm Oil Batistella, C.B.; Moraes, E. B.; Wolf-Maciel, M. R. and Maciel Filho, R. Separation Process Development Laboratory (LDPS). Faculty of Chemical Engineering. State University of Campinas, P.O. BOX 6066, ZIP CODE 13081-970, Campinas-SP, Brazil. Molecular distillation is a powerful method of separation, which happen at extremely low pressures and, therefore, at reduced temperatures. Consequently, molecular distillation finds usefulness in the separation and purification of materials with molecules of high molecular weight as well as of those that are thermally sensitive as vitamins A, E, K, many pharmaceutical intermediates, oils of vegetable origin, etc. Studies for recovering carotenoids from palm oil were developed in the LDPS (FEQ-UNICAMP) through molecular distillation (modeling, simulation and experiments) (Batistella and Maciel, 1998; Batistella, 1999). All the studies, however, have involved distillators with dimensions in laboratory scale, without consideration of distillators with industrial dimensions. Aiming designing molecular distillators with industrial dimensions starting from the simulation of a reduced one or even establishing an operating condition starting from a smaller equipment, a methodology was developed looking for an easy and fast form for scaling-up. 1. INTRODUCTION A development for scaling-up necessarily needs an analysis of dimensionless process parameters and of the distillator dimensions. However, in the molecular distillation, a fundamental restriction exists: the risk of thermal decomposition of the material. It is exactly due to this risk that the molecular distillation is used. When a scale-up study is developed based on analysis of dimensionless parameters this fact is usually not taken into account. Therefore, for a safe operation, without danger of thermal decomposition, the time of thermal exhibition is an important variable to be considered in the elaboration of a scale-up procedure. Bearing this in mind, the following approach was developed for scaling-up: the mean speed and the thickness of the liquid film in the periphery of the evaporator are the same for both distillators (Bhandarkar and Ferron, 1988). With the equations of speed and of thickness of the liquid films draining off on the evaporator plus the conditions established previously, it was possible to correlate the variables for both equipments: the one of reference and the equipment to be scaled-up. 2. THEORETICAL BASIS It was considered for the reference distillator (1) and for the larger distillator (2) the following conditions in the extremity of the evaporator (Bhandarkar and Ferron, 1988): W1-W2 (1) S 1 -- S 2 (2)
506 where W is the mean speed of the liquid film in the periphery of the evaporator and they are the same for both distillators and S is the thickness of the liquid film in the periphery of the evaporator and they are also the same for both distillators. The first condition requires that the time of exhibition be approximately the same for both distillators, and the second condition assures that the thickness of the liquid film is different from zero in any point of the evaporator, avoiding risk of thermal decomposition (thickness zero = infinite time). The studies of scale-up for the centrifugal and falling film distillators are presented following.
2.1. Centrifugal Molecular Distillator The mean velocity of the liquid on the evaporator is given by (Batistella, 1996): W=
$2~22x sen2 ~
(3)
3/.t where: W represents the angular speed, x the distance from the center of the evaporador, kt the viscosity and ~bthe angle of half cone. Considering equation (1) and using the equation of velocity (3), it can be obtained:
S12~212x_._ Sen2 # = 822~"222x___ Sen2 #
(4)
3/t 3// Considering: the same angle for both (the smallest and the largest) distillators, equation (2) and the same liquid, equation (4) becomes:
n2gl =n2L2
(5)
where L is the total length of the evaporator (periphery of the evaporator). The equation of the thickness of the liquid film is given by (Batistella, 1996):
[(mo- ~ C~Ei~i"rAxsen#(2x+Ax)l~ / /I ! k"
(6)
x=x~
Txs2 Considering, again, the same angle for both (the smallest and the largest) evaporators and considering a same liquid, the thickness of the liquid film is proportional to the:
1 S oc
i0 2Lm2 1
(7)
where x = L in the periphery of the evaporator, p is the density, E is the evaporation flow rate and m the rate of liquid on the evaporator. In the procedure, it is important to observe that: 2x sen ~b>> Scos ~b Therefore, the thickness of the liquid film in the periphery of the evaporator is given as a function of the mass flow rate of the liquid on the evaporator, of the dimension of the evaporator as well as of the speed of the rotor. Here, the proportionality constant is function of the properties of the distilled materials as well as of the temperature. For same materials and supposing the same mean temperature, equations (2) and (7) become: ml _ m2 (8)
2 2 niL,
Using equations (5) and (8) and dividing by the thickness of the liquid film
(S):
507 ml
m2
_
L1S
(9)
L2S
where: S = $1 = $2 It can be seen through the equation (9) that the flows in the liquid film on the evaporators are the same. Considering that both distillators present the same temperature profiles, it can also be said that the distillate flow rate are also the same, that means: D1 _ D2 (10) AI A2 where: D = distillate flow rate and A - evaporation area. As, A - 7eL2 then: D1 _ D2 L~ L~ The distillate flow rate can also be expressed by:
(11)
D = mo - m
(12)
Considering equations (8), (11) and (12), it is possible to write: mo2
_........_...:..~'
~
~
mo, '
D,(1
..~ ~
1)
(13)
Finally, for the scale-up calculation, it is necessary to know the rotation of the rotor, the feed flow rates and the distillate flow rate of the reference distillator as well as the dimensions of the evaporator of the reference distillator and of the distillator to be scaled-up. Therefore, to apply the proposed procedure, for the larger distillator, it is necessary to know the dimension of its evaporator (considering the mentioned assumptions). In this way, being known the dimension of the larger distillator, the speed of the rotor can be calculated by equation (5), and then, the feed flow rate can be determined by equation (13). The distillate flow rate can be determined by equation (11). After having developed the equations for the scale-up procedure, simulations were carried out and applied to the process of recovery of [3-carotenoids from palm oil for the two main equipments of molecular distillation, and the results are presented below. Starting from the dimension (column 1 of table 1) of the rotor to be scaled-up, the equations for scale-up calculation allow to determine: the speed of the rotor (column 2); the feed flow rate (column 3) and the distillate flow rate (column 4). Finally, the simulation is made being considered the predicted feed flow rates and the speed of the rotor. The simulated distillate flow rate is presented in the last column of table 1. Table 1. Results of the S c a l e - U p procedure for the Centrifugal Molecular Distillator. Diameter of the Rotor, cm
Speed of the Rotor, rpm
7.5
1300
Feed Flow Rate kg/h
Distillate Flow Rate, kg/h: Predicted Simulated
Reference
0.63
0.62
Scaled-Up
10.0 20.0 30.0 40.0 50.0
1126 796 650 563 503
1.12 4.43 9.94 17.64 27.55
1.10 4.39 9.89 17.58 27.47
1.09 4.35 9.76 17.31 27.03
508 Table 1 shows an appreciable agreement between the predicted and simulated values of the distillate flow rate. These results show that the procedure developed for scaling-up is quite satisfactory in predicting operating conditions for larger distillators. It should be noted that for a distillator processing 27.55 kg/h, 44 times larger than the reference distillator, the prediction deviations of the distillate flow rate compared to the result of the simulation were ofjust 2%. For intermediate scale-up dimensions, the deviation was still smaller.
2.2. Falling Film Molecular Distillator The velocity equation of the liquid flowing on the evaporator is given by (Kawala and Stephan, 1989): = r/
~ S
-
S
(14)
where r/is the viscosity, g the gravity acceleration and R the evaporator radius. The velocity is maximum in r = R + S, or:
Wmax -
gP S 2 2q
(15)
The mean velocity is given by the following equation (Welty et al., 1976)"
~_2Wmax
(16)
3 Using equations (15) and (16), it is obtained:
gp S 2
w=5-~-
m
(17)
Considering equation (1) and using the equation of velocity (17):
gP
gP
377 $12 = - ~ $22
(18)
And assuming the same liquid, equation (18) becomes:
S1 - 8 2
(19) The equation of the thickness of the liquid film is given by:
m-
2rcRgp2 S3 ~
(20)
3r/ where m is the liquid flow rate on the evaporator. Considering, again, a same liquid, the thickness of the liquid film is proportional to the:
1
S oc
(21)
Therefore, the thickness of the liquid film in the exit of the evaporator is given as a function of the liquid mass flow and of the evaporator radium (R). The expression (21) shows that the proportionality constant is a function of the properties of the distilled materials as well as of the temperature. For the same materials and the same mean temperature, equations (19) and (21) give: m___x~= m___x~ R, R~ The global mass balance for the equipment 2 provides:
(22)
509 m~ = mo2 - D 2
(23)
Combining equations (22) and (23):
m~ =
R~ m + D 2 R, '
(24)
In the same way as was considered in the centrifugal distillator, where it was considered that both distillators present the same temperature profiles, it can be said that the distillate flow rate is also the same, or: D, D~ A~
= ~ A~
where: D = distillate flow rate and A = evaporation area. As A = 2~rRL, then: D1 D2 =~ 2rcR1L 1 2nR2 L 2 or"
(25)
(26)
RzL2
D 2 - R1L1 D 1
(27)
Combining equations (24) and (27), it can be obtained: R2 mo2 = -~1 (ml +-~1 L2 D1)
(28)
The global mass balance for the equipment 1 provides: m, = too, - D~
(29)
Combining equations (28) and (29), it is obtained: m~ = R1 \L1 o l + m~ - D1
(30)
Therefore, for the scale-up calculation, it is necessary to know the feed and the distillate flow rates, the dimensions of the reference and of the larger distillators. In this way, knowing the dimensions of both distillators, reference and larger, plus the feed and distillate flow rates of the reference distillator, equation (30) provides the flow rate which the larger distillator should operate. Equation (27) will provide the distillate flow rate of the larger distillator and equation (23) will provide the concentrated flow rate. In the case of mixtures, equation (27) can be used for each component. The relationship between the diameter and the length of the evaporator was maintained constant, as it can be observed in table 2, with the objective of do not alter the proportionality between the studied distillators. Starting from the dimensions (column 1 and 2, table 2) of the evaporator to be scaledup, the equations for the scale-up calculations allow to determine: the feed flow rate (column 3) and the distillate flow rate (predicted) of the larger distillator. Finally, the simulation is made considering the predicted feed flow rate and the dimensions of the evaporator. The simulated distillate flow rate is presented in the last column of table 2. Again, it is possible to observe an appreciable agreement between the predicted and simulated values of the distillate flow rate. These results show that the scale-up procedure adopted for the falling film distillator is also quite satisfactory. It should be noted that for a distillator processing 82.5 kg/h, 24 times larger than the reference distillator, the prediction deviations of the distillate flow rate were ofjust 1.4%.
510 Table 2. Results of the Scale-Up procedure for the Falling Film Molecular Distillator for carotenoids recovering. Diameter of the Evaporator, cm 5 10 15 20 25 .
.
.
.
.
Length of the Feed Flow Rate Evaporator, m kg/h Reference 0.30 3.5 Scaled-Up 0.59 13.3 0.88 29.8 1.18 53.1 1.47 82.5 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Distillate Flow Rate, kg~: Predicted Simulated 3.4
.
.
.
13.1 29.6 52.8 82.2
13.0 29.3 52.1 81.0
3. CONCLUDING REMARKS It can be said that the methodology developed for scaling-up in this work presents good results and, thus, it can be used to proceed with scale-up studies, projects and simulations, and even to study the behavior of molecular distillators of larger dimensions for recovering carotenoids from palm oil. It is also worthwhile mentioning that is available the software to carry out these studies, which can be considered as another contribution of this work, since it can be used for other systems. NOTATION D Distillateflow rate, [kg/s] E Evaporation rate, [kg/mZ.s] g Gravitational acceleration, [rn/s2] L Evaporator length, [m] m Mass flow rate, [kg/s] r Radial coordinate, [m] R Outer radius of condenser, [m] S Film thickness, [m] W Velocity in film, [m/s] x Distance along rotor surface, [m]
Greek Symbols 7/ Viscosity, [Pa.s] r Kinematic viscosity, [m2/s] p Density, [kg/m3] ~b Cone half-angle, [rad] .(2 Rotor speed, [rad/s] Subscripts i Compound. o Initial.
REFERENCES
Batistella, C.B., 1999, PhD Thesis, LDPS/FEQ/UNICAMP, Campinas, Brazil. Batistella, C.B. and Maciel, M.R.W, 1998, Recovery of Carotenoids from Palm Oil by Molecular Distillation. Computers & Chemical Engineering, 22, $53-$60. Batistella, C.B., 1996, Master's Thesis, LDPS~EQ/UNICAMP, Campinas-SP, Brazil. Bhandarkar, M. and Ferron, J. R., 1988, Transport Process in Thin Liquid Films during HighVacuum Distillation. Ind. Eng. Chem. Res.,27, 1016 - 1024. Kawala, Z. and Stephan, K., 1989, Evaporation Rate and Separation Factor of Molecular Distillation in a Falling Film Apparatus. Chem. Eng. Tech., 12, 4 0 6 - 413. Welty, J.R., Wicks, C.E., Wilson, R.E., 1976, Fundamental of Momentum, Heat and Mass Transfer, John Wiley and Sons, New York.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
511
Characterization and quantification of liquid distribution in a packed column on a pilot scale M.S. Kobayasi a, M.R.W. Maciel, F.A.N. F e r n a n d e s b, D. Moraes Jr. c and S.M. Pizzo d Faculdade de Engenharia e Ci~ncias Quimicas, UNIMEP, Rodovia Santa Bfirbara-Iracemfipolis, km 01,CEP 13450-000, Santa Bfirbara d'Oeste (SP), Brazil b Faculdade de Engenharia Quimica, UNICAMP, Cidade Universit/tria Zeferino Vaz, Bargo Geraldo, DPQ/FEQ/UNICAMP, Caixa Postal 6066, CEP 13081-970, Campinas (SP), Brazil c Departamento de Engenharia Quimica, UNISANTA, Rua Oswaldo Cruz, 266, Boqueirgo, CEP 11045-907, Santos (SP), Brazil d Departamento de Engenharia Quimica, UFSCar, Rodovia Washington Luiz, km 235, Caixa Postal 676, CEP 13565-905, Sgo Carlos (SP), Brazil a
A simple method for quantification of liquid distribution efficiency was developed to characterise a packed column on a pilot scale, operated with 1 in. plastic Pall ring. The study variables were the water flow rate, the packing section height and the liquid distributing device: a tube and two distributor models, representing commercial applications (100 distribution points/m2). The liquid distribution was correlated through a model with 5 parameters. Through the model presented for the quantification of the distribution efficiency, it was verified that at the experimental conditions employed, the packing without a distributor was not capable of to distribute the liquid efficiently. The two distributor models do not present a significant difference in terms of global efficiency and dispersion. 1. INTRODUCTION The efficient use of a packed column is directly related to its liquid distribution. Despite of the fact that some manufacturers recommend the use of liquid distributors at the top of packing columns, it is quite common to find columns that do not have them in research centers (Haure et al., 1992 and Metzinger et al., 1992) and in many industrial units, since it is believed that the packing itself promotes the distribution of the liquid. It is fundamental that the distribution of liquid in contact with the gas be homogeneous. The formation of inoperative or stagnation zones in the columns must be avoided, because the mass-transfer processes take place in the effectively wetted regions of the bed (Leva, 1953; Treybal, 1980; Kister, 1992). Thus, support plates, besides distributors and redistributors, are designed to allow the passage of the gaseous phase with minimal head loss as well the liquid spreading among the packing modules (Chen, 1984). Bemer and Zwiderweg (1978) utilized a packed column with diameter of 0.2 -m, filled with Teflon- or gas-coated Rasching rings, besides water-butanol solutions of different compositions in the feeding. They concluded that the bed length, the irrigation rate, and even the wettability did not influence significantly the behavior of the column. The size of the packing elements was the main variable in the variation of the liquid radial distribution.
512 Computer simulations of liquid distribution were utilized to test several types of distributors, such as drip pans, single sprays, and seven-spot sprays, in a packed column After using different values for the ratio between the diameter of the column and that of the packing elements, it was concluded that every type of packing has a natural distribution of the liquid flow. Studies of initial distribution and heterogeneity of the bed for optimum design distributors were presented by Albright (1984). Kouri and Sohlo (1996) observed the flow patterns in a 0.5-m-diameter column filled with plastic Pall rings of 25 or 50 mm diameter or with Intalox ceramic saddles. They especially studied the development of the flow on the walls. They concluded that the flow profiles were a function of the initial flow rates and distributions of the liquid and gas as well as of the packing section height. Inside this context, the objective of this work is to quantify the liquid distribution in a pilot column filled with 1-in. plastic Pall rings, with and without a distributing device, through the application of a mathematical model to determine efficiency. 2. MATERIALS AND METHODS 2.1. Equipment The equipment built for the experiment (Figure l a) is composed by a pumping unit and the collecting modules. Which has an assembly of 21 acrylic tubes (4 mm thickness, 800mm height, and 52-mm intemal diameter) disposed in a square pitch (Figure 1b). It has points at the bottom for the collection of the liquid samples. A Cartesian x-y orientation was established, with the origin of the system being tube 11 (at the center). After statistical considerations, the pitch step is established to be 1.5. At the top of the assembly there is a middle head, and above it there is a column of the same material than the tubes, with 400-mm internal diameter and 1800-mm height. In the study module, the packing investigated (1-in. plastic Pall ring) is introduced. On the top of this column was an upper head for the feeding of water, with orifices are distributed in a square pitch, identical to that of the collecting tubes module. As shown in Figure la, the liquid phase (water) is aspired from a 250 L reservoir, by a centrifugal pump. Then, it goes through a rotameter and returns to the reservoir; a three-way valve with total passage deviates the water flow from the tank to the column in study.
..1..
'lll/lO,/,
l: ]lIII'"
2
~7
"
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
!
I. 0
E F 63-~---ee-~
(a) (b) (c) Fig.1. (a) Schematic representation of packed column - 1. Reservoir, 2. Suction duct, 3. Centrifugal pump, 4. Gate valve, 5. Gate valve (recycle), 6. Recycle duct, 7. Rotameter, 8. Discharge duct, 9. Three-way valve, 10. Flexible hose, 11. Top head, 12. Feeding tube, 13. Packing section, 14.Middle head, 15. Acrylic tubes, 16.Bottom head.; (b) Representation of the tube numbering of
513 the sample collecting module; (c) Representation of the utilized device - dimension are in milimeters
2.2. Experimental Methodology The experiments are based on a complete three-level statistical planning. The study variables are the water flow rate, the feeding point on the upper part of the column, the height of the random packing section, and the distributing devices ( tube and two models of distributors). Four series of tests were carried on.: in the first one just one pipe of 0.5 in. of PVC was employed to feed the column, to demonstrate that the packed column with 1 in. plastic Pall rings used without distributing device liquid is not enough to supply a homogeneous distribution of the liquid. The chosen values were 1.0; 2.0 and 2.5 m3/hfor water flow rate, and 30, 40, 60, 75 and 90 cm for the packing bed height. The liquid feeding was made at the center of the upper head and at two different positions, equivalent to the radial distances 7.5 and 15.0 cm. In the second series, again, the water flows mentioned previously were used. The tests were run with a pipe lateral distributor of 0.5 in. of PVC- type I. These experiments were performed in two stages. The first one was to determine the influence of the distributor on the liquid spread. Thus, the distance between the distributor and the sampling module was varied. These tests were run without packing. Four positions were chosen for analysis: the top and bottom of the column and the positions at 55 and 85 cm from middle head. In the second stage, the experiments were run with packed column with 1 in. plastic Pall ring, and 30, 40, 60 and 90 cm for the packing height. The third series was performed with a pipe lateral distributor of 1 in. of PVC - type II (Figure l c) with 3.0 mm of the orifices diameter representing commercial applications (100 distribution points/m2). The chosen values of liquid flow rate were 0.1, 0.3 and 0.6 m3/hfor all tests (new range of flow rates, given the increase in number of process that operate below of that range utilized in first and second series, which is present in most distillation process at atmospheric pressure). Firstly, the experiments were run with packing column (1 in. of plastic Pall ring) and after without packing. Finally, the fourth series consisted on the comparison of the two distributors (type I and II), under the same experimental conditions. Throughout the results of the experiments, a statistical treatment of the data was made aiming the development of a mathematical model, allowing the correlation of the liquid heights in each tube with its positions x and y. The output of each one of the tests was the mass of water collected at the sampling points located at the base of the sampling module points (21 tubes numbered as shown in Figure l b). These values were normalized and they were used in Sigmaplot software in order to obtain the parameter values (representing the studied variables influence: packing bed height, rate and operation point) of the normal tridimensional distribution model, given by equation 1 (Pizzo et al., 1998).
Z+aexpE '2]exp[ 2
(1)
In equation 1, a is the coefficient of the normalized value, b is the displacement of the curve in relation to the base z =0, c is the multiplier of the variance of the distribution and d and e are the distribution variance displacements in relation to x and y, respectively, z is the
514 normalized height for a given pair (x, y). The signal preceding the parameter a is positive when the distribution displays a pick (maximum point), and it is negative denoting the presence of a depression in the final liquid distribution (minimum point). From the physical significance of the parameters defined in Eq. 2, it can be established two measures of efficiency of the liquid distribution in the packed column. The first measure is the global efficiency (Eft), given by equation 2, which provides an indication of the distribution quality in terms of the height differences of the water obtained. Hence, a high global efficiency corresponds to a liquid distribution without significant differences in the water level of the collecting tubes. The second efficiency defined is the dispersion efficiency (EfD), given by equation 3. In that case, the degree of the liquid spreading is analyzed in the collecting section. Low dispersion efficiency means a situation in which the collected water has concentrated in a certain group of tubes, depending on the feed position on the upper part of the column. EfG = (1- a).lO0%
(2)
EfD : (1- c)100%
(3)
To locate the regions eventually favored by the irregular distribution of liquid, two other factors were also defined, given by equations 4 and 5. Equation 4 is the representation of the displacement radial position. It refers to the radial position (tube 11 as the origin) of the pick or depression (maximum or minimum point) of the distribution in each experiment. Equation 5 provides the angle measured from the positive axis of the abscissas counterclockwise, in relation to the pick or depression. R=
+
O= arccos/3-~)
(4) (5)
3. RESULTS AND DISCUSSION
The values of global and dispersion distribution efficiencies, as well as those values for the displacement of center from the first series of tests, with a pipe are shown in Table 1. Table 1 Values of efficiency distribution for the experiments of the first series Experiment Bed Height FlowRate Feeding Efficiency(%) Point (cm) (m3/h) (cm) Global Dispersion 1.1 30.0 1.0 Center 81.53 45.42 1.3 30.0 2.5 Center 84.27 50.37 1.7 30.0 1.0 15.0 64.09 0.0 1.19 60.0 1.0 Center 84.52 47.00 1.21 60.0 2.5 Center 88.39 59.77 1.24 60.0 2.5 7.5 84.18 52.55 1.28 75.0 1.0 Center 86.25 53.27 1.34 75.0 1.0 15.0 67.76 0,00
of tests with pipe Displacement from the Center Radius(cm) Angle(o) 0.05 180 0.09 102 0.85 17 0.17 111 0.17 167 0.43 6 0.19 47 0.75 0
515 The correlation coefficient (r2)for the adjustment of Eq. 1 varied between 0.90 and 0.96. It can be noted that the efficiencies increase as the flow rate increases, probably due to the decrease in the liquid phase channeling. The efficiencies also increases as the packed-bed height is increased, especially in the 30-75 cm range. The increase of the efficiencies is smaller in the 75-90 cm range, corroborating perhaps the existence of natural distribution of the liquid flow from the 90 cm bed. (see Albright, 1984). The efficiencies diminish as the feeding goes from the center to the periphery of the upper head. Table 2 Values of the efficiency distribution for the the second series of tests with distributor type I Experiment Bed Flow O r i f i c e Distributor Efficiency (%) Displacementfrom the Height R a t e Diameter Position Center (cm) (m3/h) (mm) (cm) Global Dispersion Radius(cm) Angle(o) 2.4 0.0 1.0 2.0 top 94.68 87.01 0.21 80 2.13 0.0 1.0 2.0 55 94.68 90.61 0.26 43 2.22 0.0 1.0 2.0 85 93.02 87.33 0.11 97 2.23 0.0 2.0 2.0 85 93.92 91.63 0.15 83 2.49 40.0 1.0 2.0 5 95.79 94.15 0.21 99 2.51 40,0 2.5 2.0 5 96.55 93.32 0.08 168 2.69 75.0 1.0 2.0 5 96.22 92.81 0.29 119 2.70 75.0 2.0 2.0 5 98.14 99.81 0.77 101 Table 3 Values of the efficiency distribution for the third series of tests with distributor type II Experiment Bed Flow O r i f i c e Distributor Efficiency (%) Displacementfrom the Height R a t e Diameter Position Center (cm) (m3/h) (ram) (cm) Global Dispersion Radius(cm) Angle(o) 3.1 30 0.1 3.0 5.0 86.62 91.34 0.28 44 3.2 30 0.3 3.0 5.0 88.81 88.94 0.21 45 3.4 60 0.1 3.0 5.0 90.00 91.44 0.37 12 3.5 60 0.6 3.0 5.0 91.81 88.32 0.22 70 3.7 90 0.1 3.0 5.0 91.43 91.71 0.53 172 3.8 90 0.3 3.0 5.0 95.48 86.43 0.20 155 3.15 0.0 0.6 3.0 55 86.72 82.73 0.17 107 3.19 0.0 0.1 3.0 top 78.57 90.02 0.31 92 3.20 0.0 0.3 3.0 top 88.70 77.56 0.14 57 In the second and third series of experiments, a perforated-pipe distributor was built using materials that are easy to find and to handle, such as PVC tubing and accessories of 0.5 and 1.0 in. nominal diameter. The results of efficiencies are shown in Table 2 and 3 (distributor type I of 0.5 in. and type II of 1 in. of PVC, respectily). By comparing the values presented in Tables 1, 2 and 3, it is clear that a better water distribution may be achieved using distributing device, because the efficiencies are higher than those resulting from the tests in which only one distribution point was used (a pipe). It is probable that the greater number of distribution points compensated the reduction in the range of flow rates applied. In geral, the efficiencies increased with an increase in the bed height. The global efficiency increase slightly when the flow rate was raise from 1 to 2 and 0.1 to 0.3m3/h, and remained stable or diminished a little from 2 to 2.5 and 0.3 to 0.6 m3/h. The dispersion efficiency tended to
516 decrease as the flow rate increased. As shown in Table 4, the distributor models did not present a significant difference in terms of global efficiency and dispersion. Table 4 Values of the efficiency distribution for the fourth series of tests with distributor type I and II Experiment Bed Flow Orifice Distributor Efficiency (%) Displacementfrom the Height R a t e Diameter Type Center (cm) (m3/h) (mm) Global Dispersion Radius(cm) Angle(~ 41 30 0.3 3.0 I 91.52 80.12 0.26 25 4.3 30 0.7 3.0 I 94.86 85.57 0.058 84 4.14 75 0.3 3.0 I 91.25 76.07 0.19 118 4.16 75 0.7 3.0 I 95.97 94.32 0.30 146 4.4 30 0.3 3.0 II 89.83 76.34 0.02 111 4.6 30 0.7 3.0 II 92.41 87.13 0.084 146 4.16 75 0.3 3.0 II 91.17 80.22 0.071 119 4.18 75 0.7 3.0 II 94.36 75.85 0.042 180 4. C O N C L U D I N G R E M A R K S Despite the simplicity of the model presented, Eq 1 and the equations derived from the characterization proposal of the liquid distribution, i.e., Eqs 2 to 5, they proved to be very useful in determining the distribution efficiencies of the experiments carried on. It was verified, from the results obtained, that on the experimental conditions employed, the packing without a distributor was not capable of distributing the liquid efficiently, which may be verified through the values of efficiency and of the center displacement factors. The homogeneity of the distribution may be obtained by necessarily employing the proper liquid distributors and redistributors for this use at the top and between the packing units, besides employing distributing plates. This hypothesis is confirmed by observing the results obtained in the experiments in which the distributing device was utilized. This distributor was responsible for the improvement of distribution efficiencies on the analyzed conditions. Therefore, special attention should be paid to the project of a distributor, since it affects significantly the operation and efficiency of a packed column. REFERENCES Albright, M.A. Hydrocarbon Processing, (1984) 173. Bemer, G.G.,Zwiderweg,F.J. Chem. Eng. Sci., 33 (1978) 1637. Chen, G.K. Chem. Eng., 91 (1984) 40. Haure, P.M.; Hudgins, R.R.; Silveston, P.L. Can. J. Chem. Eng., 70 (1992) 600. Kister, H.Z. Distillation design; McGraw-Hill Inc.: New York, 1992. Kouri, R.J.; Sohlo, J. Chem. Eng. J., 61 (1996) 95. Leva,M. Towerpacking andpacked tower design; The US Stoneware Co.: Akron, OH, 1953. Metzinger,J.; Hasokowati, W.; Hudgins, R.R.; Silveston,P.L.; Gangwal,S. Chem.Eng.Sci., 47 (1992) 3723. Pizzo,S.M.; Moraes Jr.,D.; Fernandes, F. A N.; Kobayasi, M.S. Ind. Eng. Chem. Res., 37 (1998) 2844. Treybal, R.E. Mass-transfer operations; McGraw-Hill Inc.: New York, 1980.
European Symposiumon Computer Aided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
517
Sensitivity in Optimization of a Reactor System with Deactivating Catalyst .Ingvild Lgtvika, Magne Hillestad b and 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. An optimal operating strategy for a fixed bed reactor system with a slowly deactivating catalyst is found. The process studied is Lurgi's methanol synthesis. A rigorous model of the reactor and the loop has been used and the actual control variables have been optimized. In this paper, we look specifically at sensitivity in the optimization. The parameters in the catalyst deactivation model are believed to be uncertain. The effect of variations in the deactivation parameters on the optimal operating variables and the objective function has been studied by a first order error propagation-approach.
1. Introduction Catalyst deactivation occurs in practically all fixed bed reactors. The two main questions in operation of fixed bed reactors with deactivation are when to change catalyst, and how to compensate for deactivation between the catalyst changes. This work looks at the last problem only, because in the methanol synthesis the time for catalyst change is decided by factors outside the process section. Much work has been done on optimal operation of fixed bed reactors undergoing catalyst deactivation. Some central references are [ 1-5]. Most of the earlier work [3-5] has focused on optimal distributed control, e.g. optimization of reactor temperature distributed in time and space. A more realistic approach is taken in this work. The actual time varying control variables in the reactor system, the recycle rate and coolant temperature, are optimized. This study also uses a detailed, realistic model of the total reactor system. Parts of this work are published earlier [6,7]. This paper looks specifically at sensitivity in the optimization with regards to the deactivation model. In the methanol synthesis, synthesis gas (CO, CO2 and H2) is converted to methanol over a Cu/ZnO/A1203 catalyst. The following exothermic reactions occurs [8]: CO 2 +3H 2 r CO + H 2 0 r
CH3OH+ H20 CO 2 + H 2
In the Lurgi reactor [9] the catalyst is packed in vertical tubes surrounded by boiling water. The reaction heat is transferred to the boiling water and steam is produced. Efficient heat transfer gives small temperature gradients along the reactor. Typical operating conditions are 523 K and 80 bar. The pressure of the boiling water is controlling the reactor temperature. Because o f the quasi-isothermal reaction conditions and high catalyst selectivity, only small amounts of byproducts are formed. Methanol conversion is limited by equilibrium. Unreacted synthesis gas is separated from crude methanol, compressed and recycled. The Lurgi reactor system consists of two parallel reactors with a common steam drum, a feed/effluent interchanger, a cooler, a methanol separator and a recycle
* Author to whom correspondence shoud be adressed, email: [email protected]
518
compressor. The control variables are the coolant temperature and the recycle rate. A flowsheet of the process is shown in Fig. 1. ....................................... S.'/n|hesk~ ~t"~
MP Steam
Cruc!e melhonol
Fig. 1: The meth~ol synthesisl..... The Cu/ZnO/A1203 catalyst can deactivate irreversibly because of chemical poisoning or thermal sintering [10-12]. Sintering is the cause of deactivation under normal operation. The catalyst poison sulfur is removed earlier in the process. Chlorine and heavy metals act as catalyst poisons but is not likely to occur in the process gas. Sintering is caused by high temperatures and increases when the catalyst is exposed to high partial pressure of water [ 10,11 ]. Copper is the active phase in the catalyst. During sintering, copper atoms migrate to larger agglomerates. This leads to increasing crystal size and decreasing active area. The sintering mechanism changes at higher temperatures; copper crystals migrate together, causing severe deactivation and loss of selectivity. Reported temperatures for when severe deactivation starts range from 543 K [9] to 670 K [11 ]. The lowest temperature is chosen as a constraint in the optimization. The catalyst deactivates slowly under normal operating conditions, and after 3 to 4 years, the activity is so low that the catalyst has to be replaced. A shut down of a part of the plant is necessary to change catalyst. The maintenance plan of the plant and the natural gas supply determine when to replace the catalyst. This is why the catalyst lifetime is not optimized in this study. A common opera7ion strategy is to increase the temperature at the end of the catalyst lifetime to compensate for decreased activity [9]. The decisions regarding temperature increase are based on the experience of the operators. Increased temperature gives higher reaction rates, but also higher deactivation rates. This makes coolant temperature an interesting optimization variable. Increased recycle rate leads to lower conversion per pass in the reactor, but higher overall conversion in the loop.
2. Modeling and Optimization The catalyst deactivation was the only dynamic effect included in the model while steady state was assumed for the other states. This pseudo steady state assumption is reasonable because the dynamics in composition, temperature and pressure are much faster than the deactivation dynamics. A twodimensional, heterogeneous reactor model with the following assumptions was used: * Dispersion of mass in axial and radial directions is negligible. 9 Dispersion of heat in axial direction is negligible. 9 Isotherm catalyst pellets. 9 Inter-facial temperature gradients are negligible. 9 Viscous flow in catalyst pellets is negligible. The LHHW-type reaction kinetic by Bussche and Froment [8] was selected. The fifth order deactivation kinetic is based on Skrypek et al. [ 11] with deactivation energy Ed from Cybulski [ 13]. The activity is scaled to fit a temperature profile for a Lurgi reactor with deactivated catalyst [9].
519
1)/
a,t, dt
~Rg
- "~"0 "a(t)5
a(0) =a0 a ' ( t ) = 1 - ~'-atto)a ao
This model was selected because it predicts a reasonable long catalyst lifetime. Few deactivation models for this catalyst are published, and they predict quite different deactivation rates. Most of the models were developed in laboratory scale, and therefore predicts too fast deactivation [13-16]. One model [ 17] considers the reaction gas composition, but the mechanism that is assumed is in conflict with other literature. This model also predicts too fast deactivation. Lumped steady state models were used to describe the remaining unit operations in the reactor loop. Soave-Redlich-Kwong equation of states was used to find the phase equilibrium in the separator [ 18]. The task of finding an optimal operation strategy was formulated as a nonlinear dynamic optimization problem: tl
Max "Profit = ] (FMeOH " PMeOH + Fsteam " esteam )it
Tc(t),R(t)
to
s.t.:
Zrmeaa~ctor <_ 5 4 3
K
513K
2
I~
:
:
:
300
:
:
:
600
Time
:
:
:
900 [days]
:
:
:
1200
:
53~ t 525~
,
,.........
L .........
..
300
600
Time
Fig. 2: Optimal coolant temperature and recycle rate (- opt ' ref)
900 [days]
1200
520
3. Sensitivity Analysis We wanted to investigate the sensitivity in the optimization results with regards to the deactivation model. Two factors were studied, the rate constant Kd and the activation energy Ed. The effects and cross effects of the factors were found by varying the factors according to the 22 experimental design in Table l [23]. The effects and cross effects of varying the deactivation parameters on the optimal coolant temperatures and the scaled objective function are shown in Table 2. No effects were observed on the optimal recycle rate. The Kd effect on coolant temperature is positive. The temperatures are increased to compensate for the increased deactivation caused by increased Kd. The Ed effect on coolant temperatures is not significant in all intervals. It is negative and shrinking in the tree first intervals, resulting in a steeper profile that starts at a lower temperature. The temperatures are lowered in the start were most of the deactivation takes place, because the increase in Ed makes the deactivation rate more sensitive to temperature. Both effects on the objective function are negative as expected. A positive cross effect on the objective function exists, meaning that the Kd effect is lager when Ed is at the low level. Both the objective function and the coolant temperature were checked for nonlinear effects, with a negative result. The center point was close to the response surface for all responses.
Table 1" Experimental design. 0.8*Ed Ed 1.2*Ed 0.8*Kd
Kd 1.2*Kd
--
-'+
00 +--
Table 2: Effects of the deactivation parameters on the optimization results.
++
Effects
Response Obj TJ Tc2 TJ Tc4 Tc5 Tc6 Tc7 TJ
Mean 21.00 520.3 523.9 526.1 527.6 529.0 530.9 530.9 531.7
Kd -1.39 0.7_-/-0.4 0.9_+0.4 0.9_+0.4 1.1_+0.4 0.8_+0.4 0.8_+0.4 1.0-+0.4 0.9_+0.4
Ed KaxEd -0.73 0.19 -1.4_--t-0.4 -1.0-+0.4 -0.7_+0.4
The propagation of statistical uncertainty from the deactivation parameters to the optimization results were calculated from the equation [23]"
Var(f)=l~i2Var(Kd)+ ~~f-~l)~21Z. Var(Ed)+2I_~f d Y_)f) Var(Kd ) =
Var(Ed ) =CYEd
Ea)
CoV(Kd ,Ed ) = P Kd,Edt~Kd~ Ed
The partial derivatives were estimated from the effects in Table 2. Different cases with combinations of uncertainties and correlation factors in the deactivation parameters were used, see Table 3. The propagated uncertainties in objective function and coolant temperatures are shown in Table 4. The uncertainties in the scaled objective function is relative large. Converted to percent increased profit from optimization, the uncertainties are +0.2 percent in case 4 and +0.1 percent in case 2. The uncertainties in the optimal coolant temperatures are relative small. Varying the correlation coefficient had small effects on the uncertainties in both the objective function and the coolant temperatures. It is interesting that the uncertainties in coolant temperatures in case 3 and 4 are of the same size as the optimization accuracy of +0.4 K. This point is illustrated in Fig. 3. From these results, it can be concluded that 20 percent standard deviation in the deactivation parameters is
521
sufficient for optimization purposes. More accurate deactivation parameters will not lead to a more accurate optimal operation strategy. This can be used as a target for uncertainty in a future model development. In adition to uncertanty in calculation of the optimal coolant temperatures, there is also uncertainty in implementation of the optimal coolant temperatures. The implementation uncetanty is estimated to +0.5 K. This is another reason not to use a more accurate deactivation model for optimization. Table 3: Standard deviation in the deactivation parameters Case (YKd (YEd PKd,Ed 10 10 20 20 40 40
1
2 3 4 5 6
10 10 20 20 40 40
Table 4: Standard deviation in optimization results. Center O'respo.se Response Point Case Case Case Case 20.86 520.03 523.47 525.86 525.86 527.34 528.67 529.94 530.89
Obj
0.5 0.95 0.5 0.95 0.5 0.95
Tc1 Tc2
Tc3 Tc4 Tcs Tc6 Tc 7
Tc8
532
'
3
4
Case 6 +2.10 +0.77 +0.32 +0.32 _+0.73 +0.61 +0.80 +1.00 +1.29
........ i|
528
r.
524
-
.
.
.
.
-
.
- : _- I: "
.....
.
~-" - = "
522 520
2
'
53O ~'
Case 5 +0.47 +.52 +0.93 • +1.86 +0.30 +0.19 +0.61 +0.38 + 1 . 2 1 +0.24 +0.08 +0.48 +0.16 +0.95 _+0.21 _+0.08 +0.41 +0.16 +0.82 _+0.24 _+0.18 _+0.48 +0.37 _+0.96 +0.18 +0.15 +0.36 +0.31 +0.72 _+0.20 _+0.20 +0.40 +0.40 +0.80 +0.25 +0.25 +0.50 +0.50 +1.00 +0.29 _+0.32 _+0.58 _+0.64 +1.15 1
......
4
0
I
500
!
.........
T i m e [ d a y s ] 1000
1500
Figi3" Standard deviation in optimal coolant temperature in case 4 ( m prop. err. ' opt. acc.).
4. Conclusions The operation of the methanol synthesis with catalyst decay has been optimized, and the sensitivity in the optimization results with regards to the deactivation model has been investigated by a first order error propagation - approach. The optimization results have been published earlier [7]. The effects of variations in the two parameters in the deactivation model on optimal coolant temperature profile and the objective function was found. Both the rate constant and the activation energy had a negative effect on the objective function, and a cross effect between the to factors was observed. The rate constant had a positive effect on the optimal coolant temperature profile. The activation energy had a negative effect on the first intervals of the coolant temperature profile, resulting in a steeper profile. The propagated uncertainties was relative large in the scaled objective function and relative small in the optimal coolant temperatures. It was found that the uncertainties in coolant temperatures with 20 percent standard deviation in the deactivation parameters are of the same size as the optimization accuracy. From these results, it can be concluded that 20 percent deviation in the deactivation
522
parameters is sufficient for optimization purposes. This can be used as a target for uncertainty in a future model development.
5. Notation a a
Ed [J/mol] Kd [da y-i] Tc(t) [K] Tre actor max
[K]
R [mol/mol] Profit [USD]
Catalyst activity Scaled catalyst activity Activation energy for deactivation Rate constant for deactivation Coolant temperature Maximum reactor temperature Recycle rate Profit over catalyst lifetime
FMeOH[tons/day] [;'Steam[tons/day] Pi [USD/ton] Qcomp[ k W ] tl [days]
Var09 Cov~ g) (yf pf, g
Production rate of methanol Net production rate of steam Price of product i Compressor power Catalyst lifetime Variance Co variance Standard deviation Correlation coefficien
References 1. A.F. Ognue and W.H. Ray, AIChE J., 17(1), (1971) 43. 2. G. Buzzi-Ferraris, E. Facchi, P. Forzatti and E.Tronconi, Ind. Eng. Chem. Proc. Des. Devel. 23 (1984) 126. 3. J.R. Gonzalez-Velasco, M.A. Gutierrez-Ortiz, J.I. Guiterrez-Ortiz and J.A Gonzales-Marcos, Canad. J. Chem. Eng., 7 (1987) 65. 4. M.T. Asrar and A.S. Moharir, Comp.Chem.Eng., 15 (1991) 7. 5. R.S Dixit and N. Grant., Canad. J. Chem. Eng., 74 (5) (1996) 652. 6. I. Lcvik, M. Hillestad and T. Hertzberg, Comp. Chem. Eng. Suppl., 22 (1998) 707. 7. I. Lcvik, M. Hillestad and T. Hertzberg, Comp. Chem. Eng. Suppl., 23 (1999) 839. 8. K.M. Vanden Bussche and G.F. Froment, J. Catal., 161 (1996)1. 9. E. Supp, Hydrocarbon Proc., 3 (1981) 71. 10. H.H. Kung, Catal. Today, 11 (1992) 443. 11. J. Skrzypek, J. Sloczynski and S. Ledakowicz, Methanol synthesis, Polish Scientific Publishing, Warszawa, 1994, Chap.II.2.2.4. 12. J. Ladebeck, Natural Gas Conversion W-Stud. Surf. Sci.Catal., 107 (1997) 73. 13. A Cybulski, Catal. Rev.-Sci.Eng., 36 (4) (1994) 557. 14. C. Keuchen and U. Hoffmann, Chem. Eng. Sci., 48 (22), (1993). 3767. 15. M. Sahibzada, D. Cadwick and I.S. Metcalfe, Natural Gas Conversion 1V - Stud. Surf. Sci. Catal.., 107 (1997) 29. 16. G.W. Roberts, D.M. Brown, T.H. Hsiung and J.J. Lewnard, Ind. Eng. Chem. Res. 32 (1993) 1610. 17. M.R. Rahimpour, J. Fathikalajahi and A. Jahanmiri, Canad. J. Chem. Eng., 76, August, (1998) 153. 18. R.C.Reid, J.M Prausnitz and B.E Poling, Properties of gases and liquids, McGraw-Hill, 1988 19. Methanex, Current methanol price, http://www.methanex.com. 1998. 20. T.F. Edgard and and D. H. Himmelblau, Optimization of Chemical Processes, McGraw-Hill, New York, 1989. 21. PSE, gPROMS User's Guide 0.2 ed., London, 1997. 22. PSE, gPROMS Advanced User's Guide, 1.1 ed., London, 1998. 23. G.E.P Box, W. G. Hunter and J. S. Hunter, Statistics for experimenters, John Wiley, New York, 1975.
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
523
Detailed M a t h e m a t i c a l Modelling of M e m b r a n e M o d u l e s J. I. Marriott, E. SCrensen* and I.D.L. Bogle Department of Chemical Engineering, University College London Torrington Place, London, WC 1E 7JE, UK Membrane technology is used for a wide range of separations from particle-liquid separations to gaseous and liquid-liquid separations. In this paper, we introduce a detailed model that describes a general membrane separation. The model disregards many common assumptions such as plug flow; constant temperature and pressure; steady-state conditions; and constant physical properties. Our approach is applicable to any membrane separation and in this paper we demonstrate its application to both liquid mixture separation (pervaporation) and gas separation in hollow-fibre modules. Both cases are seen to exemplify the need for a detailed model. 1. I N T R O D U C T I O N During the last thirty years, the search for alternatives to traditional energy intensive separation methods such as distillation has led to the introduction of processes based on membranes. Membrane technology is used for a wide range of separations from particle-liquid separation, such as reverse osmosis, to gaseous and liquid-liquid separation. To achieve the desired separation, a large membrane area is often required. In an industrial plant, this area is supplied in a modular configuration for which there are a number of variations. Historically, plate and frame modules have been used, but more recently, spiral-wound and hollow-fibre modules have attracted a lot of interest as these offer much higher packing densities [ 1]. In other areas, the use of mathematical models for process design, optimisation and for control studies has shown significant benefit. The application of these methods to the design of membrane processes is now starting to be considered [2,3,4]. However, to minimise the technical risk that is inherent in the design of any new process, it is essential that detailed unit models are used. Unfortunately, to the best of our knowledge, a detailed model of a general membrane separation process is not currently available from published literature. Although a number of models do exist (e.g. [1,5,6,7]), due to the complex nature of flow through membrane modules, these usually rely on a number of assumptions. These assumptions include: plug flow; constant temperature and pressure; steady-state conditions; and constant physical properties. As a result, existing models are typically process specific and are only valid for a limited operating range. However, with better solution algorithms and the improved computer power that is now available, such assumptions are no longer necessary. The objectives of this paper are to describe a detailed approach to modelling a general membrane separation process and illustrate its use for different systems, in particular, pervaporation and gas separation. Author to whom all correspondenceshould be addressed; email: [email protected].
524 2. M A T H E M A T I C A L M O D E L We have developed a consistent modelling approach that can consider spiral-wound, plate and frame, tubular, and hollow-fibre membrane modules. However, in this paper we will limit the discussion to the latter. Hollow-fibre modules contain a large number of membrane fibres housed in a module shell. Feed can be introduced on either the fibre or the shell side and permeate is usually withdrawn in a co-current or counter-current manner. The flow pattern in a hollow-fibre module is illustrated in Figure 1. To simulate a membrane module three sub-models are required; two to describe the flow either side of the membrane and a third model which characterises the separative properties of the membrane and any porous support material. Two alternate flow sub-models have been developed: a one dimensional flow model and a two dimensional flow model. Both models are developed from rigorous dynamic mass, momentum and energy balances, and thus are applicable to any membrane separation. The model characteristics are summarised in Table 1. Although the 2-D model provides the most rigorous approach, it is limited to nonturbulent flow on the fibre side of the membrane. In contrast, the 1-D model is more versatile but requires an additional empirical parameter to describe the radial variation in concentration (concentration polarisation). Common assumptions such as isothermal flow, negligible axial diffusion, steady-state conditions, and constant physical properties have all been disregarded (Table 1). Therefore, it is apparent that model accuracy is now mainly constrained by uncertainty in parameter values, such as molecular diffusivity, and uncertainty in the module design, e.g. the non-uniformity of the fibres. 2.1 Solution Method
To simulate the membrane module, the 1-D flow sub-model which describes shell side conditions is coupled (using an appropriate membrane sub-model) with either the 2-D submodel or a second 1-D sub-model describing fibre side conditions. The complete model is solved on an IBM RS6000 using the gPROMS simulation software [8], and fluid properties are provided externally by Multiflash [9]. To approximate the spatial variation of the distributed variables both finite difference and orthogonal collocation methods have been investigated. The latter method is preferred as it uses the least number of equations and thus minimises solution times. Typically, using orthogonal collocation, these are 0.4 CPUs for the 1-D model and 2-3 CPUs for the 2-D model. Table 1. Characteristics of the 1-D and 2-D flow sub-models Model characteristics Sub-model 1-D 2-D Fibre-side flow V/ 9 Shell-side flow v/ X X Turbulent flow v/ Non-isothermal flow v/ Viscous and diffusive axial flow v/ 9 Viscous and diffusive radial flow X 9 Radial variation of concentration and velocity X 9 Multicomponent separation V' 9 Physical properties provided externally v/ 9 Steady-state and dynamic simulation v/
525 3. MODEL APPLICATION In the following section, the generality of the model is demonstrated by considering two different systems: a pervaporation separation and a gas separation.
3.1 Pervaporation Pervaporation can be used to separate liquid mixtures. In this process, the membrane forms a semi-permeable barrier between the liquid feed and a low pressure gaseous product. Consequently, the heat of evaporation must be supplied to the permeating material, typically resulting in a feed stream temperature drop. To accurately simulate pervaporation, both the temperature drop and radial concentration variations must be modelled. In this section, we show how the detailed model proposed earlier formally accounts for both of these factors. Concentration Polarisation
In many membrane processes, such as reverse osmosis and pervaporation, a concentration gradient, or boundary layer, is formed due to the slow rate of molecular diffusion to the membrane surface. This effect, commonly termed concentration polarisation, is particularly apparent in the flux of dilute organic material though pervaporation membranes. In such cases, the resistance of the boundary layer to the flux of organic material can be significantly larger than that of the membrane, and thus often controls the overall mass transfer rate [10]. Clearly, therefore, the effect of concentration polarisation cannot be neglected. Concentration polarisation is usually modelled using a 1-D flow model coupled with an empirical mass transfer coefficient [11]. The development of the 2-D model enables this effect to be described more rigorously, as it does not assume a boundary layer and enables full radial variations in both concentration and velocity (This is shown in Figure 2). In separate work, Cote and Lipski [10] and Psaume et al. [12] have described the removal of trichloroethylene from waste-water using lab-scale silicone rubber hollow-fibre modules. The authors calculate the experimental mass transfer coefficient for a range of feed flows from the overall trichloroethylene flux rate. In this paper, we use the 1-D and 2-D flow models to simulate the liquid feed flow through the fibre bore, and compare the results with the experimental data. Diffusion coefficients are estimated for both models using the Wilke correlation [13] (Di-- 9.9 xl0 -1~
Fig. 2. Radial concentration profile for flow through a hollow fibre. Fig. 1. Flow pattem in a hollow-fibre module
526 7E-05 - - This work , 1D model - - T h i s work, 2D model [] Cote and Lipski 9 Psaume et al.
6E-05 5E-05 E
4E-05
~ '5
3E-05
~ o
2E-05
'~
u ~
F
9
]
[]
~]
1E-05 0E+00 1
10 Reynolds number
100
. t 1000
Fig. 3. Mass transfer coefficient as a function of Reynolds number mZ/s), and for the 1-D model the empirical mass transfer coefficient is estimated as a function of Reynolds number using the Leveque correlation [ 10]. The accuracy of the simulation results is assessed in Figure 3, which shows the mass transfer coefficient as a function of Reynolds number. The average mass transfer coefficient is calculated from the ratio of permeate flux to the mean feed-side concentration. Unfortunately, uncertainty in the value of the diffusion coefficient and in the experimental results, which show a wide degree of scatter (Fig. 3), limits the scope of this analysis. However, both models are seen to predict the experimental results reasonably well and the predictions of the two models show a close correlation with each other (Fig. 3). It can be concluded that, though the 2-D flow model is computationally more expensive, it provides a rigorous and more general approach to describing concentration polarisation than the 1-D model. Furthermore, an empirical mass transfer coefficient does not need to be estimated for the 2-D flow model. Temperature Changes
The model has also been used to describe ethanol dehydration in hollow-fibre modules. Excellent agreement is found between simulation results from the detailed (2-D) model and experimental results reported by Tsuyumoto et al. [ 14]. For a feed of 94.0 wt% ethanol and a feed rate of 44.8kg/hr, the model predicts a product concentration of 97.3% compared to the experimental value of 97.2%. This small error is most likely due to uncertainty in the membrane characterisation parameters [14]. This case also highlights the importance of a detailed approach. Our model predicts that the associated temperature drop for this case study is 26~ If an isothermal model were used, product purity would be significantly overestimated to be 98.4wt% ethanol (an error approaching 40%). Thus, isothermal flow should not be assumed at the design stage as it could lead to a significantly under-sized plant.
3.2 Gas Separation Membrane technology is commonly used to separate gas mixtures. Here the membrane forms a barrier between the high pressure feed gas and the low pressure permeate gas. Hence, to accurately describe gas separation, both concentration variations and pressure changes along the module must be described. To aid in the design of membrane systems, Smith et al. [15] and Krovvidi et al. [1] have recently proposed two approximate models which describe the separation of a binary gas
527 Table 2. Gas separation system (30000 fibres: 0.76m length, 50 lam i.d.,100~tm o.d.) Case A B co-current counter-current Flow pattern 20% 85% Feed composition, fraction H2 Feed temperature (K) 308 308 137.9 137.9 Feed pressure (bar) Permeate pressure (bar) 1.379 27.58 H2 Permeability (10 -l~ mol/m 2sPa) 109.3 109.3 N2 Permeability (10 -l~ mol/m 2sPa) 10.93 0.607 mixture in a hollow-fibre module. Both published models neglect pressure drop and simplify the mass balances. Smith et al. [15] assume that the permeate concentration is constant along the entire length of the module, whereas Krovvidi et al. [1] assume a linear relationship between the feed-side and permeate-side concentrations - the operating line m e t h o d [ 1]*. Smith et al. [15] use their model to describe the separation of hydrogen and nitrogen in a shell fed hollow-fibre module. This system will also be used in this work. (It should be noted that the detailed model presented in this paper has also been verified against experimental data, presented by Pan [7], for a similar, but multicomponent system.) Two cases with different operating conditions, membrane selectivities and feed compositions are considered. For each study the retentate purity and the stage-cut (the ratio of permeate to feed flow) are calculated for several feed flows. System details are given in Table 2. Simulation results from the two approximate models for Case A are compared with the results from the detailed model in Figure 4. This shows the retentate product purity as a function of the stage-cut. The approximate model results are generally poor, showing a significant deviation from the detailed model results over the entire operating range. This error is primarily due to the effect of neglecting the pressure profile, when in fact, there is a significant pressure build up on the fibre side. The simulation results for Case B are shown in Figure 5. In this case, retentate product purity is plotted as a function of mass feed rate. The permeate pressure is higher than in Case A, and as a result the pressure drop is much lower. Consequently, the approximate model results show a good agreement (_+ 1-2%) with those from the detailed model, particularly at high feed rates (i.e. low stage-cuts). This is because the concentration profiles along the length of the module are relatively constant. However, at low flowrates, the Krovvidi [1] model predicts an infeasible permeate concentration, and the results from the Smith [15] model diverge significantly from the detailed model results (Fig. 5). The reason for this is that the Smith [15] model assumes constant permeate concentration along the entire length of the module. This assumption is particularly invalid at low flowrates, when there are significant concentration changes along the length of the module. Even for this simple system (ideal gas and constant membrane permeability), the approximate model results are unreliable, showing relatively poor agreement with the detailed model. It is seen that only at high permeate pressures and low stage-cuts do the approximate models provide a reasonable level of accuracy. In contrast, hollow-fibre modules are usually connected in parallel [16], with high stage-cuts, and low pressure operation will often be optimal. However, the biggest disadvantage of the approximate models presented in the
* Krovvidi et al [I] have also described a second model. The authors report that though it does not perform well for co-current operation it is a better model for counter-current operation. Unfortunately, there appear to be a number of mistakes in the equations given in the paper, and it was impossible to reproduce the author's results.
528 9O% 8O%
12% 10~176 .=_
:s
-~ 70%
-,,- This work + Smith
~-~
.,.., c-
8%
:ff ..-
6%
,t-
o to ..~ o
4%
.-
LI.
2% 0% 15%
60%
-,,- This work l -*- Smith -.- Kroviidi
- 9 50%
/
40% 30% 20% 10%
t
I
20%
25%
1
I ....
30% 35% Stage Cut
t
I
j
40%
45%
50%
Fig. 4. Predicted product purity (Case A)
0%
.......
0
t
200
J
J
I
400 600 800 Mass feed rate, kg/hr
~........................... I
1000
1200
Fig. 5. Predicted product purity (Case B)
literature is that, unlike the detailed model, they are limited to binary separations. As design methods based on rigorous models now exist (see [2]), the use of approximate models is in any case mostly unnecessary. 4. CONCLUSIONS In this paper, a detailed model that describes a general membrane separation has been introduced. The use of the model to investigate complex modelling issues in a liquid separation system has been demonstrated. The model has also been used to assess the performance of approximate models in a gas separation system. In both these cases it is seen that a detailed model is essential. The use of this model for the optimal design of membrane systems will be considered in future work. REFERENCES [1] K.R.Krovvidi, A.S.Kovalli, S.Venury and A.A. Khan, J. Membrane Sci., 66 (1992) 103. [2] J.I.Marriott, E.SCrensen and I.D.L.Bogle, FOCAPD Proceedings, Colorado, 1999. [3] M.M.E1-Halwagi, AIChE J., 38 (1998) 1185. [4] R.Qi, and M.A. Henson, J. Membrane Sci., 148 (1998) 71. [5] D.T.Coker, B.D.Freeman, and G.K. Fleming, AIChE J., 44 (1998) 1289. [6] R.Qi and M.A.Henson, Ind. Eng. Chem. Res., 36 (1997) 2320. [7] C.Y.Pan, AIChE J., 29, (1983). [8] Process Systems Enterprise Ltd., London. gPROMS Advanced User's Guide, 1999. [9] Infochem Computer Services Ltd., London. Mulitflash, 1996. [10] P.Cote and C.Lipski. In R.Bakish (ed), Proc.3 ra Int. Conf. Perva. Proc. Chem. Ind., 1988. [11] C.J. Brouchaert, and C.A.Buckley, Water SA, 18.3 (1992) 215. [12] R.Psaume, P.Aptel, Y.Aurelle, J.C.Mora and J.I.Bersillon, J. Membrane Sci., 36 (1988) 373. [13] R.B.Bird, W.E.Stewart, and E.N.Lightfoot, Transport Phenomena, John Wiley, New York, 1960. [14] M.Tsuyumoto, A.Teramoto and P.Meares, J. Membrane Sci., 133(1997) 83. [15] S.W.Smith, C.K.Hall, B.D.Freemann and R.Rautenbach, J.Membrane Sci.,118(1996) 289. [16] W.S.W.Ho and K.K.Sirkar (eds), Membrane Handbook, Van Nostrand Reinhold, New York, 1992.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
529
A novel approach to the analysis of distillation columns for multicomponent mixtures Alessandro R. Giona, Massimiliano Giona and Lidia L. M. Lombardi Dipartimento di Ingegneria Chimica Universit~t di Roma La Sapienza Via Eudossiana 18, 00184 Roma, Email: [email protected] .it Starting from the definition of the Total Mixture Exiting (TME) each section, a simple and novel approach is proposed for the rigorous analysis of distillation columns involving multicomponent mixtures based on the simultaneous solution of material and energy balances. A new algorithm for the evaluation of the minimum extemal reflux ratio is also developed in the case of ternary mixtures and compared with classical methods. 1. I N T R O D U C T I O N
In historical terms, computational methods for the analysis of distillation columns for multicomponent mixtures started in the early 1960s, when massive use was first made of computers to solve engineering problems. These methods are still applied nowadays. In addition to stagewise methods [1] (see also Hanson et al. [2] and Holland [3]), other approaches have been proposed (e.g. by Arnundson and Pontinen [4]) that are based on the application of extensive matrix algebra to the simultaneous solution of the balance equations describing the theoretical stages forming a distillation column. These methods were very popular in the past and still form the central core of process analysis codes (e.g. ASPEN) as regards distillation equipment. The analysis of distillation columns received fresh impetus in the wake of articles by Doherty and coworkers [5,6,7] and of the textbook by Stilchmair and Fair [8] .All these approaches assume the total number of stages to be fixed and then evaluate distillate and residue compositions on this basis. These methods can therefore be correctly described as checking methods since the number of stages is given a priori. This article introduces a simple rigorous stagewise method of analysis in which the simplified assumption of constant flowrates is removed, and proposes a new criterion for evaluation of the minimum external reflux ratio. The application of the method is exemplified for a ternary mixture (which is the standard benchmark in the field [6,7]), and is compared with classical methods. The computational issues related to its application in codes for process analysis are also briefly addressed.
530
2. T O T A L M I X T U R E EXITING: BASIC D E F I N I T I O N S
Throughout this article we consider a distillation column with a rectifying and a stripping section. All the stages are assumed to be theoretical ones, i.e. equilibrium conditions between the liquid and vapor phases are achieved.The TME of the rectifying section is a fictitious stream (P', xp., Hv.) expressed by the distillate including the heat exchanged at the condenser: P'= D,
Hp, = hD + qc.
xv , = XD,
(1)
For the stripping section, the TME is expressed analogously in terms of residue flow rate, composition and enthalpy by considering the heat exchanged at the reboiler: P" = R ,
HI," = hR-qs.
Xp" = XR,
(2)
A relation between the TME of the two sections is given by the energy balance over the entire column: FhF = P'Hp, + P"Hp,,.
(3)
The TME can be viewed as the difference between the overall mass and energy flowrates, and between the flowrates associated with each component referred to the vapor and liquid phase. By introducing the internal reflux ratio, given by Ek=Lk/D, the energy balance for the rectifying section becomes: Hp, = 0Ek + 1)Hk+l - Ekhk
(4)
Once the external reflux ratio has been specified, Hp, can be obtained by enforcing Eq. (4). 3. T H E T M E A P P R O A C H TO THE N U M B E R O F STAGES R E Q U I R E D
TME provides a simple stagewise framework in order to approach the analysis of distillation columns and to determine the number of stages required for a given separation. The computations can be carried out by following a stagewise procedure. Let us consider a generic stage k of the rectifying section for which the composition, enthalpy and flowrate of the liquid phase exiting the stage are known. The corresponding quantities referred to the vapor phase entering the stage can be thus obtained by solving the following system of equations: Ek
Yk+l = E~ k x+1
k+
E k = Hr"-HK+ I . Hk+1 - h k '
xD ~ Ek+
1
TK+I = Tbp(YK+1).
(5)
531 An iterative method can be used to solve this system as the composition Yk+l can only be estimated if the internal reflux ratio Ek is known, and the latter quantity is ultimately a function of the composition vector yk+l itself due to the dependence of the enthalpies Hk+a on composition. The iterative procedure can be set up by fixing an initial conjectural value for the external reflux ratio Ek and thus computing the corresponding vapor composition by means of the first equation of system (5) together with the corresponding enthalpies. Finally, a new value of Ek is obtained through the second equation of system (5), and the procedure is repeated until convergence is reached. Once the vapor conditions (composition, enthalpies and flowrate) have been determined, it is possible to evaluate the corresponding quantities of the liquid phase in equilibrium with the vapor phase.The procedure continues with the next stage.The initial condition is given by the liquid stream L1, which is in equilibrium with the vapor stream V1, a saturated vapor of composition XD. When the liquid phase Lk is in equilibrium with the vapor stream of the next stage Vk+l, i.e. when the temperatures of the two stages Tk and Tk§ coincide, the concentration profiles collapse, thus corresponding to a pinch-point condition. For the stripping section, the internal reflux ratio is given by Sk=Vk/R. Since Hp, is known from Eq. (4), Hz. can be obtained from the energy balance over the entire column and stage analysis can be performed by solving the system of equations analogous to Eq. (5). The initial condition is given by the vapor Vn (where n is the last stage of the column), which is in equilibrium with the liquid exiting the last stage (x~=XR). This system of equations can be solved stagewise until the pinch point is reached. Figure 1 displays the numerical results obtained in the case of a ternary benzene(1)-toluene(2)-xylene(3) system represented by a triangular diagram. Points A and B are the liquid and vapor phases in equilibrium with each other at the point at which the feed is introduced. The stage counting for the rectifying section starts from the representative point of the distillate and terminates at point B. Analogously, for the stripping section, it starts from the point associated with the residue and terminates at point B. By making use of this stagewise approach, the resulting point corresponding to the feed-stream introduction is properly evaluated, since the number of stages is minimum for this separation. 4. MINIMAL EXTERNAL REFLUX RATIO As shown in Figure 2, reduction of the external reflux ratio causes the composition profiles of the sections to start deviating from one other. As a result, the number of stages required for separation increases and becomes infinite when the pinch-point of the stripping section coincides with the point at which the feed is introduced. For lower values of the external reflux ratio, the composition profiles of the stripping and rectifying sections do not intersect and separation is not feasible. This observation motivates the following definition: the minimum external reflux ratio is the lowest
532 1
1
0.8
01
0.6
O.
2:2
;i"2
0.4
02
0.2
0
O0
I
0.2
0.4
0.6
0.8
:I; 1
Fig. 1, Liquid and vapor composition profiles.
1
00
0.2
014
0'.6
0.8
1
:1; 1
Fig.2, Liquid phase composition profiles: [] Eo=2.2, o Eo,m=l.23, * Eo=l
value (infimum) of the reflux ratio for which the composition profiles intersect each other. This condition implies that separation is feasible, albeit with an infinite number of stages. Following this definition, the minimum reflux ratio can be obtained as the pinch-point of the stripping section corresponding to the justtouching conditions with the concentration profile of the enriching section. The pinch-point of the stripping section can be determined by considering the balances for stage k of this section and enforcing not only equilibrium between the liquid Lk and the vapor Vk+l of the next stage k+ 1 but also the condition that the sum of the liquid composition is equal to unity through application of the parameter OC,k--RfL k instead of Sk, which attains values in the interval [0,1 ]:
3
cXt'xi'R = 1. ~--,1- K, (T) + o~,K i (T)
(6)
For a fixed temperature T, the parameter Ctk can be evaluated through Eq. (6) thus enabling us to estimate the composition of the liquid and vapor in equilibrium at temperature T, i.e. the pinch-point. By introducing the parameter Otk into the energy balance of the stripping section, the values for Hp,, can be estimated while Hp. is obtained from the energy balance over the entire column Eq. (3). Once Hp, is known, the external reflux ratio can be obtained by enforcing Eq.(4). The reflux ratio, as evaluated by means of this procedure, determines the pinch-point of the stripping section at temperature T. When the reflux ratio thus obtained and the temperature T also satisfy the rectifying balances, the reflux ratio is the minimum external reflux ratio for this separation. In order to establish the minimum condition, we allow temperature to assume all the values between the bubble point of the distillate and the dew point of the residue, and enforce the condition that the pinch
533 point of the stripping section will belong to the composition profile of the rectifying section. Figure 3 compares the results obtained by means of this procedure in the case of the ternary system benzene-toluene-xylene with those furnished by Underwood's [3] and Gilliland's [9] methods, which are commonly employed in commercial packages for process analysis. It can be observed that these traditional criteria give lower values for the minimum external reflux ratio than the present approach, which may conceptually imply the use of an erroneous design criterion. If separation is defined with respect to residue composition (as regards the heaviest component), pinch-point conditions should of course be sought in the rectifying section, and the minimum external reflux ratio corresponds to the condition that this pinch point belongs to the stripping composition profile. I
i
I
I
i
!
I
I
i
1
la,
i
I
I
I
I
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0
I
90
1()()
110
120 130 T [~
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Fig.3. Eo,m as a function of the TF: a) proposed method, b) Underwood's method, c) Gilliland's method. 5. CONCLUDING REMARKS This article has developed a new stagewise approach to design (evaluation of the number of stages required) in a multicomponent distillation column. Unlike most of the existing methods available in the literature, the method is based on the stage-bystage solution of the material and energy balances through reference to the TME of the sections of the column. The application of TME makes it possible to account quite easily for the changes in the liquid and vapor flowrates throughout the column. With respect to the methods usually employed in process analysis codes,starting from the number of stages and from a fixed location of the feed stream, the method starts from the prescribed separation specifics for the residue and the distillate, and
534 obtains the number of stages required and the optimal position of the feed stream. The method has been exemplified in the case of ternary mixtures since meaningful graphical representation on a triangular diagram is feasible in this case. The computational method can, however, be extended to any number of components in the mixture. Moreover, it also is possible to extend the method to non-ideal systems by introducing the expressions for the activity coefficients into the liquidvapor equilibria. This method has been described for a simple column, but can also be applied in the general case.It is also possible to consider a column with more than one feed and sidestreams by introducing for each new section its own TME. As a byproduct of this approach, an alternative definition of the minimum external reflux ratio has been proposed, the evaluation of which reduces to a simple algebraic algorithm. It can therefore be used as an alternative to these methods in packages of process analysis. NOTATION D 9distillate molar flow rate V : molar flow rate vapor phase E ' reflux ratio rectifying section x : composition liquid phase F : feed molar flow rate y : composition vapor phase H : specific enthalpy vapor phase h: specific enthalpy liquid phase subscript K : equilibrium ratio bp: bubble-point L" molar flow rate liquid phase c :condenser P': TME molar flow rate rectifying section D: distillate P" TME molar flow rate stripping section k: generic stage q : specific heat flow referred to a TME F: feed R : residue molar flow rate m: minimum S : reflux ratio stripping section R: residue T: temperature s : reboiler
REFERENCES 1. J. S. Bonner: Chem. Eng. Prog. Symp. Series, 55 No 21(1959) 87. 2. D. N. Hanson-J. H. Duffin- G. F. Somerville: Computation of multistage separation processes, Reinhold Publ. Co. N.Y., 1962. 3. C. D. Holland: Multicomponent distillation, Prentice Hall Inc., 1963. 4. N. R. Amundson- A.J. Pontinen: Ind. Eng. Chem., 50 (1958)730. 5. V. Julka, and M. F. Doherty, Chem. Eng. Sci., 45 (1990) 1801. 6. Z. T. Fidkowski, M. F. Malone, and M. F. Doherty, AIChE J., 37 (1991) 1761 7. S. G.Levy,D.B.Van Dongen, and M.F.Doherty, Ind.Eng.Chem.Fun..25(1986)279. 8. J. G. Stichmair and J.R.Fair, Distillation, Wiley-VCH, N Y , 1998. 9. J. B. Maxwell,Data Book on Hydocarbons, Van Nostrand Co. Inc. Toronto (1955).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
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ROME: A Repository to Support the Integration of Models over the Lifecycle of Model-based Engineering Processes L. von Wedel, W. Marquardt Lehrstuhl ffir Prozesstechnik, RWTH Aachen, Germany {vonWedel, Marquardt} @lfpt.rwth-aachen.de Process modeling is currently performed by using a number of incompatible tools. The heterogeneous storage formats and systems of these modeling tools limit the reuse of models because semantic relationships among models originating from different or even the same tool cannot be maintained. A repository is proposed as a central storage system for models from various sources represented in various formats. The repository holds models representing chemical engineering and physics concepts by a conceptual data model rather than as mathematical equations or some programming language code. The conceptual representation of a model is accompanied by tool specific representations and documentation, enhancing the use of models over the lifecycle of model-based engineering processes such as chemical process design. The conversion among different tool formats is simplified by reducing the number of required converters if tools are connected to the repository instead of integrating them pair-wise.
Process modeling, lifecycle, database, tool integration, data model 1. INTRODUCTION State-of-the-art tools for modeling and simulation used for the development of chemical processes are currently not integrated satisfactorily, thereby limiting the efficiency of modelbased approaches to solve process systems engineering problems. The reuse of information coded in models for different applications, a revision of the underlying modeling assumptions, or tracing modeling decisions and errors along the evolution of a model are almost impossible, even with a single tool. The situation in practice is often worse because process design usually involves different tools for steady-state simulation and optimization, for dynamic simulation, or for control studies. The main cause for these difficulties is that current tools mostly represent models on a level which suits their implementations, e.g. unit operation modules in flowsheeting tools such as Aspen+ or Hysys, mathematical equations in equationoriented modeling tools such as gPROMS or ROMEO, or input-output blocks in blockoriented tools such as Matlab/Simulink. From our experiences a model should be represented in a neutral format, independently from the tool specific requirements. We suggest a representation on a conceptual level by means of an object-oriented data model to directly capture the model content in terms of chemical engineering and physics concepts rather than as mathematical equations or even chunks of some programming language. Model representations specific for a certain simulator can then be derived easily from the conceptual model. The work process of modeling as conducted in chemical industries studied by Foss et al. (1997) shows that modeling on a conceptual level and documentation are of significant interest in practice. This field study also emphasizes the need for an integration of process modeling tools as industrial models are seldomly reused but rewritten from scratch instead, mainly due to the fact that information about the model is lost once it is represented in a tool specific format where e.g. assumptions are not explicitly represented.
536 Model development and maintenance must also be assessed from a lifecycle perspective to emphasize the integration of models on different scales of length, time, and chemical resolution and their evolution along the development of the process or plant to be built (Marquardt, et al., 1999). The complexity and range of scales considered in chemical engineering makes it unlikely that a single tool coveting a broad range of the lifecycle of model-based development processes can be built. Rather, an environment has to be developed where the advantages of individual tools for modeling and simulation are combined to result in synergies overcoming individual limitations. This contribution proposes a (conceptually) central storage service through which models can be shared among different, heterogeneous applications in such an environment. Such a shared information database system incorporating meta data is generally called a repository. The repository presented is called ROME (Repository of the MODKIT Environment) because of its close interrelationship with MODKIT (Bogusch et al., 1999). MODKIT aims at supporting modeling on a conceptual (semantic) level and is therefore well-suited to act as a tool for the development and administration of conceptual models in the repository. The simulation platform CHEOPS (von Wedel, Marquardt, 1999) is also tightly connected to the repository. It enables the integration of models implemented for different simulation paradigms (e.g. block-oriented and equationoriented) into a single simulator to thus exploit the individual strengths of existing simulation tools. The paper is structured as follows: Chapter 2 describes how the requirements formulated above can be satisfied by a model repository. The representation of models in the repository is presented in Chapter 3. Finally, the current status of the realization is presented in Chapter 4. 2. A REPOSITORY FOR THE MANAGEMENT OF PROCESS SYSTEMS MODELS
A repository can be described as a shared database of information about engineered artifacts produced or used by an enterprise (Bernstein, Dayal, 1994). An important feature of repositories is the metadata (data about other data) they employ in order to integrate tools to an environment, the most important task of a repository beyond providing persistent storage services. Thereby, different tools for modeling and simulation can be used cooperatively by i'etrieving or storing models in their native tool formats. A repository should further provide a mechanism to decouple concurrent work of different developers from each other and provide versioning capabilities to represent the evolution of artifacts during the workprocess. These features are required in order to support modeling as a collaborative design process. Repository structure: A repository can be implemented on top of a database which holds the
data itself as well as necessary metadata. Data comprises models themselves, their documentation and implementations, whereas metadata describe e.g. how models are interrelated, the capabilities of tools and the model representation formats accessible by them. The database schema managed by the model repository is a suitable representation of models in an object-oriented manner as mentioned above. This object model must cope with the fact that process models are not just equations but also have to include context information to understand and interpret these equations (Subrahmanian et al., 1993). Therefore process models must be described as semantically rich concepts, with semantic links between related artifacts. Using this data model, the repository can store conceptual model representations and associate them with the model documentation and tool-specific implementations. The repository contents can be partitioned into a model library providing reusable modeling concepts, a set of actual models, and metadata linking together the former two parts. The modeling concepts stored in the repository library span different levels of granularity from material property to unit operation models and are structured into a class hierarchy in order to
537 facilitate navigation and reuse. A modeling concept can be (re-)used by selecting an appropriate class from the hierarchy, instantiating, and finally configuring it. Hence, actual models are instances of the classes stored in the model library. Here, metadata is employed to represent instantiation and inheritance relationships. The model library can evolve over time by adding new models as they are developed because the metadata to represent the class hierarchy of the model library can be modified at runtime. The maintenance of inheritance and instantiation relationships is supported by means of description logics (Sattler, 1998) which enable reasoning about this kind of information. Also, learning new library elements from existing models is currently under investigation (Baader et al. 1999) in order to maintain and extend the model library. Using such a formally sound and algorithmic basis for structuring the model library is essential to maintain conceptual and structural integrity if multiple modelers contribute to the repository contents. Integrating Tools with the Repository: As mentioned above, models can be retrieved from the repository in tool specific formats, making it act like a model server (Britt, Pantelides, 1994). Data integration of tools is achieved by translating between the different scopes and model representations of tools. Usually, n 2 converters are required in order to integrate n different data representations. By using the repository as a central entity for the data integration of tools, only n converters are needed. Several code generators tailored for the export to a specific tool representation support the derivation of model implementations from the conceptual model representation (Bogusch et al., 1999). In order to support the full lifecycle of model-based development processes, the import of models from tool specific representations must also be supported, enabling modifications to a model being made in a certain tool to be reflected in the repository without additional effort.
Tool integration can be facilitated a lot by using upcoming standards for declarative or executable simulation models such as Modelica (Mattson et al., 1998) or CAPE-OPEN (Braunschweig et al., 1999), because they allow several tools to be connected to the repository without additional effort. Whereas a representation of a model in Modelica can be exported as an ASCII file for further processing, the model must be exported as a CAPEOPEN compliant software component if it shall be used in an appropriate simulation environment such as gPROMS or CHEOPS. The capabilities of a repository can contribute to the model development process as illustrated with the following scenario. A modeler builds a steady-state model in MODKIT on a conceptual level and adds documentation (probably a Microsoft Word document). He then derives a CAPE-OPEN compliant implementation from it for use in an appropriate simulation environment by means of a code generation step. When he finishes developing the model he publishes it in the repository, together with its implementation and documentation. Another modeler can now derive a gPROMS model implementation for the same model, again by means of a code generation step. This process fully supports the requirements mentioned above, integrating different tools by supporting their native formats and storing documentation and semantic information about models along with it. 3. THE REPRESENTATION OF MODELS IN THE REPOSITORY
A rich data model comprising the scope and concepts of all tools connected to the repository is important. This model must be developed from an ontological point of view, trying to formalize what is known about modeling concepts and their relationships rather than building a superset of constructs used by individual tools to represent a model. The latter approach would quickly yield a data model that is hard to extend because different tool representations may be contradictory in details.
538 From the description of the modeling process by Foss et al. we can identify four areas into which relevant concepts can be grouped by means of UML packages. We introduce a model documentation package defining concepts for model documentation and a model representation package describing semantic modeling concepts. The model implementations are defined in the model implementation package and a modeling project package contributes, among other concepts, the simulation experiment. The packages of the data model and the main concepts defined in them are shown in Fig. 1. Model Documentation: Previous work has shown that documentation can be structured based on the hypertext paradigm (Bogusch et al., 1999). This object-oriented representation of information is defined using hypertext nodes containing information and hypertext links specifying semantic relationships among two nodes (cf. Fig. 1). Subclasses of information nodes describe requirements nodes, general information units, and the IBIS-related concepts issues, position, and argument. IBIS (Issue-based information systems; Rittel, Kunz, 1970) is an approach to represent negotiation and decision-making processes. Hypertext nodes do not have to store actual text fragments, they can also refer to external Word documents or URLs in the internet to integrate heterogeneous sources of information about a model.
Figure 1: Overviewof the conceptual data model
Model Representation: For the conceptual representation of mathematical models we introduce the general system model which can aggregate submodels of any number and kind (Fig. 1). Such a generic model is associated to a set of mathematical relations and system quantities describing the model's behavior. Some of the model's quantities can be published, all others are considered to be private and cannot be accessed by other models. This encapsulation mechanism is important to enable different formulations of a model to be exchanged in a certain modeling context. The mathematical relations of a model can refer to any of its internal quantities (published and private) or to any published variables of its submodels. Structured models extend the general model class by adding a number of so-called ports that act as a plug to connect other structured models with via a coupling. Structured models have to publish quantities at their ports in order to make them available for information transfer across different models. A set of domain concepts based on the general notion of a model can be defined by subclassing the very general system model. The structured representation of chemical processes uses the concepts proposed by Marquardt (1996). On a finer level of granularity models for material properties or geometric relationships may be introduced (Marquardt et al., 1999).
539
Model Implementations: A set of possible model implementations is related to a system model of arbitrary granularity (cf. Fig. 1). As an example, a material properties model can be implemented by a physical properties system whereas a device model can be realized as a unit operation module in a classical flowsheeting tool. Since model implementations are toolspecific, a subclass is introduced for each tool considered. New tools can be integrated by adding appropriate subclasses. Through this mechanism, the model repository becomes an excellent counterpart to the heterogeneous simulation platform CHEOPS, allowing the definition of simulation experiments involving several simulation tools that can subsequently be executed by CHEOPS. Modeling Project: The case concept represents a modeling project, and is associated to a composite device that assumes the role of the process under consideration. Further, the case is associated to hypertext nodes which serve as general documentation. Finally, a set of simulation experiments (cf. Fig. 1) describe variable specifications (which of them are fixed in the sense of parameters, what computational units are being used, estimate values, bounds, initial conditions) and model implementation specifications (which implementation among the possible ones has been used in this experiment).
4. ARCHITECTURE, DESIGN, AND IMPLEMENTATION OF ROME The architecture of ROME follows the layers pattern, a common architecture for database applications (cf. Fig. 2). The repository manager is implemented as a layer on top of the database server layer. An object-oriented database management system is chosen because of its capabilities to store objects directly instead of implementing a mapping from objects to a relational database. Further, versioning and workspace capabilities are already provided to a certain degree.
Figure 2: Layersin the repositoryarchitecture The communication between the different parts of the environment is realized by CORBA, enabling applications to be connected to the model repository over an intranet, regardless of the programming language or platform they are realized in. This is an important feature when integrating existing applications. The repository manager consists of two layers in turn, one written in plain C++, the other mapping the C++ classes into the CORBA-based API. Clients are connected to the repository manager via the fine-granular API (e.g. ModKit) or via code generators on a more coarsegranular level of input files. Control integration among tools connected to the repository is provided through an event service. Such asynchronous integration of tools is far more flexible than a synchronous approach where tools would be directly notified. The current realization enables models to be stored according to the data model introduced above. Model implementations for Modelica and Cheops can be derived automatically by
540 using code generators. Modelica model implementations are described in a text file which is under control of the repository. Cheops unit operation modules can be generated by translating the model equations into C++ and compiling and linking a CORBA software component. Automatic differentiation techniques (Griewank et al., 1996) are used within this process to supply derivatives. Experiments can be defined and exported to Cheops to execute a simulation.
5. CONCLUSION AND FURTHER WORK An approach for storage and maintenance of process models in an environment of existing tools for modeling and simulation based on repository technology has been presented. It enables the representation of models on a conceptual level to be stored and shared among different tools, simplifying the maintenance and reuse of models over the lifecycle of process design as a model-based development process. Future versions of MoOKIT should use the model repository as a model storage and serve as an administration tool for the repository. The data model of the repository is currently being extended to cover hybrid discretecontinuous and spatially distributed models as well. The definitions of domain-specific concepts for physical and chemical phenomena, material properties, and geometric models must be extended. Further work has to concentrate on the integration of other tools for process development, preferably gPROMS and Aspen+. A Web-based interface shall enable users to execute simulation experiments via the internet and to retrieve the results in desired formats.
6. REFERENCES Baader, F., R.Kiisters and Molitor, R. (1999). Computing least common subsumer in description logics with existential restrictions. In: Proc. of the 16th International Joint Conference on Artificial Intelligence (IJCAr99). Ed. T. Dean. Morgan Kaufmann, 96-101. Bernstein, P.A., and U. Dayal (1994). An overview of repository technology. In: Proc. 20th VLDB Conference, Santiago, Chile, 705-713. Bogusch, R., B. Lohmann, and W. Marquardt (1999). Computer-aided process modeling with ModKit. Submitted to Comp. Chem. Engng. Braunschweig, B., H. Britt, C.C. Pantelides, and S. Sama (1999). Open software architectures for process modelling: current status and future perspectives. In: Proc. FOCAPD '99. Breckenridge, CO, USA. Britt, H.I. and C.C. Pantelides (1994), Multipurpose Process Modeling Environments, In: Proc. FOCAPD '94. Foss, B., B. Lohmann, and W. Marquardt (1998). A field study of chemical process modeling. J. Process Control 8, 325-337. Griewank, A, D. Juedes, H. Mitev, J. Utke, O. Vogel, and A. Walther (1996). ADOL-C: A Package for the Automatic Differentiation of Algorithms Written in C/C++. ACM TOMS 22(2), 131-167. Marquardt, W. (1996). Trends in computer-aidedprocess modeling. Comp. Chem. Engng. 20, 6-7, 591-609. Marquardt, W., L. von Wedel, and B. Bayer (1999). Perspectives on lifecycle process modeling, In: Proc. FOCAPD '99, Breckenfidge, CO, USA. Mattsson, S.E., H. Elmqvist, and M. Otter (1998). Physical system modeling with Modelica. Control Engng. Practice, 6, 4, 501-510. Rittel, H. and W. Kunz (1970). Issues as elements of information systems. Working Paper No. 131, Institute of Urban and Regional Development, Univ. of California, Berkeley, CA. Sattler, U. (1998). Terminological knowledge representation systems in a process engineering application. Dissertation. RWTH Aachen. Verlag Mainz, Aachen. Subrahmanian, E., S.L. Konda, S.N. Levy, Y. Reich, A.W. Westerberg, and I. Monarch (1993): Equations aren't enough: Informal modeling in design. Artificial Intelligence in Engineering Design, Analysis and Manufacturing 7, 257-274. von Wedel, L. and W. Marquardt (1999). Cheops: A case-study in component-based simulation. In: Proc. FOCAPD 99, Breckenridge, CO, USA.
European Symposium on Computer Aided Process Engineering - l 0 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
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Increase business benefits by using on-line models Industrial Application of Known Methods Dominik Dempf, Thomas List Wacker Chemie 84480 Burghausen (Germany) e-mail: [email protected] e-mail: [email protected] Abstract The obstacles of transferring theoretical methods to practical application in an industrial environment are easily underestimated. To tackle this problem technical, methodological as well as communicative aspects have to be considered. Cooperation and communication between plant management, engineering and software suppliers are of key importance to implement theoretical tools into a process environment. Models -reduced and rigorous- with plant adjusted complexity extract the maximum information from the raw process data. We have applied on-line models in four different continuously operated plants: A black box model in a 80 000 jato vinylacetate plant for mass-energy balance, plant scale-up and process follow-up. A dynamic model of a acetylacetone reactor for predicting optimal reactor performance.The Kalman-Filtering technique for change-point control in a diketene reaction system to meet safety requirements. A ketene cracker dynamic model for cracker design and plant revamping. The model approach reduces operating costs up to 15 % mainly by reduction of overall plant variance, which allows operation closer to limits. Keywords: on-line model, data reconciliation, Kalman-Filter, NDDR, GUI, noise reduction, change point control, plant safety, process variance, Computer Supported Cooperative Work (CSCW)
542
Introduction The current high level of activity in plant optimizing research disciplines as chemical engineering, electrical engineering, chemistry, mathematics can be compared with the excitement that would result if scientists from different but similar planets suddenly found themselves able to communicate. There would be a sudden flow of information as concepts on each side were recast within the other sides frameworks. Just as it would not be surprising that methods known for years on one side were being reformulated using a different language, it would be surprising that some of the ideas on each side were new to the other. These socio-psychological factors are as important as the technical ones in understanding what has happened in the last several years as interest in plant optimization has exploded. From the engineering view of Process Optimization, one is often looking for a model capable of producing numerical values that match values measured from the physical plant. This application of parameter estimation is known as system identification. The engineer assumes that the model has a specific mathematical structure expressed in terms of a set of unknown parameter values and then uses a parameter estimation method to find reasonable parameter values. The estimation method requires data in form of examples of how the system being modelled behaves for a collection of inputs. The large amount of different parameter estimation methods from different areas of application and their scientific background are like an avalanche of data, which hardly can be handled by plant managers and decision makers in the production units. Optimization Tools E~F
Prosim MINOS ~CMM~TLAB P i n c h Point N e l d e r M e a d RT-Opt OPX MATHCAD M I N L P KBF Scorecards VALI SIMU
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The decision making process for choosing an adequate approach must therefore
be improved. Elements of Plant Optimizati,on The essential elements of a plant optimization problem are
9
1. Problem representation--- This is an aspect of Plant Optimization that is guided the least by useful theory. What aspects of objects, inputs, states etc. should be taken into account for classification or modelling ? What is the exact definition of a successfull plant optimization ? 2. Class of decision rules or models - Select a choice that includes deciding how
many parameters to optimize and how they define the decision rule or model
543
3. Performance criteria - The best measure of performance might evaluate the overall, final performance of the system that makes use of the decision rule or model (e.g. in the commercial world ,,How well does the device sell ?"; in the biological world, ,,How well does the organism reproduce?". In practice, however operating a real plant, simpler criteria that are closely related with overall performance and yet can be measured easily and quickly have to be found. 4. Optimization methods- Two major types of optimization are distinguished: offline and on-line methods. Here, we restrict attention to on-line methods because plant operators are generally dealing with real time-data. 5. Use of a priori knowledge- The optimization process can be improved by the use of a priori knowledge, not only as it is incorporated into the selection of the problem specification and the class of decision rules or models, but also in the form of operator experience and sociological and environmental constraints.
Information model for on-line systems Dataflows from DCS systems are in itself of little value, unless structured and related to a plant model. In addition process models are underutilized, because plant engineers and plant managers do not have time modelling or refrain from spending time in building models,because of hands-on activities. Furthermore process models are not easily attainable and the question how accurate they should be in a given context is not clear. To extract information from process data, reduced models (black box) have been shown to be sufficient for mass-and energy balancing (data reconciliation). In this case spatial redundancy is used to improve data quality and estimate unmeasured variables. This method is limited to steady state operation and applied within time-windows of hours and days. For the dynamic case spatial and temporal redundancy is used simultaneously, which leads to 'Non Linear Dynamic Data Reconciliation'. In the case of linear systems the optimal estimate of noisy, or non measureable variables is obtained with the Kalman-Filter algorithm within time-windows of seconds (ref.:3).
Business benefits are the end of a sequence formed b y Data -) Information --) Knowledge ~ Implementation --) Business Benefits Business benefits reinforce, by feed back loop, the previous steps.
Application of known on-line techniques in chemical plants 1. Black-box model of vinylacetate plant (Ref.:3) Problem representation : The overall balance did not match the law of massconservation.. Class of decision rules or models :Black box model. Reliabilty of measurements is distorted by gross errors. These were removed by an off-line chi-square test. Optimization methods : On-line application of a steady state model.
544
Performance criteria : Minimum squared error sum. Use of a prion knowledge : A pseudo-measurement vector replaces missing measurements based on experience, via distribution functions from historical plant data.
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2. Dynamic model of an acetylacetone reactor
Problem representation : Plant scale up, best reactor design. Class of decision rules or models : Dynamic plug flow reactor, neglecting high temperature radical reactions (microkinetics). Optimization methods : Cost function minimization to determine the best ratio between energy input and feed flowrate (Nelder-Mead). Performance criteria : Specific energy consumption and selectivity Use of a priori knowledge : Thirty years of non-optimum operation. Macrokinetics unknown. I T,ull p l r l ~
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Fig.:2 Result: Run reaction adiabatically between reactor outlet and quenching unit (Ref.:6)
545
3.Dynamic model of a ketene crac.ker Problem representation : Decrease the specific acetic acid consumption, evaluating different cracker geometries or configurations in terms of maximum business benefits.Substitute old cracker by a new one. Create model based benchmarks for different cracker designs. Class of decision rules or models :Dynamic plug flow reactor, neglecting partially high temperature radical reactions. High temperature gas mass-spectroscopysampling device to determine kinetic constants (ref.:10). Optimization methods: Evolutionary Operation (EVOP) and determining key process variables (e.g. ---) catalyst injection --> inhibitor injection), On-line Data Reconciliation. Performance criteria : Ketene-Production in DM/hr visualized with an ,CASH Observer' (ref.:12). Functional dependence between temperature, energy input, catalyst rate and feed rate. Use of a priori knowledge : Internal experience + external on-line laboratory know - how and published information (ref.:4).
4. Kalman-Filtering for thermal control of a diketene reaction system. Problem representation : Investment for a blow down vessel for the diketene reactor in compliance with legal regulations. Class of decision rules or models : Diketene is characterized by a high potential decomposition risk. With the use of a reactor model and the Kalman-FilterAlgorithm a process-runaway can be detected within seconds, a blow-down vessel could be avoided, in compliance with legal regulations. Optimization methods: Implementation of a Kalman-filter to estimate the heat of reaction. The assumption of linearity in the process variables around a certain operating point allows a time independent solution of the Matrix-Ricatti Equation.(ref.:9) Performance criteria : Sensitivity to temperature changes in reaction vessel.. Use of a priori knowledge : History of time series in reactor variables.
CASE 1
% time on-line 80%
Model type Blackbox
CASE 2
10 %
CASE 3
CASE 4
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Project time 10 month
CSCW
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Used for
VALI Ill Belsim
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-
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PI Oil Systems
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mass/energy balance Dynamic model
80 %
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Dynamic model
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MATLAB
PI Oil Systems
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-
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Scale-up process follow up Optimal reactor control Reactor design benchmarking Detection of runaway
SYNOPSIS
9C A S E 1 to C A S E 4
The aim is to advance the state of the art in plant optimization by developping novel methods and demonstrating their application on non-steady state processes. Required is :The development of robust, efficient, large-scale methods for
546 It has been shown (CASE 1-CASE 4), that even without these sophisticated requirements, hidden productivity potentials in plants, matured along their lifecycle by trial and error, can be further exploited with already available methods and tools.
CONCLUSION At the shop floor level, according to our experience, the large amount of process data delivered by DCS, the laboratory and on-line analyzers, are converted to useful information by the plant operators in an intuitive way for ad hoc decisions. This leads by trial and error to an evolutionary plant optimization along the plants life-cycle. Ten to twenty percent of the productivity-potential is hidden however under process variance and cannot be exploited by the trial and error approach. Driving the process however closer to physical and monetary limits, under tighter time constraints, faster decisions are required, because of process dynamics and nonlinear system behaviour, to achieve a competitive advantage. One of the main challenges is to get reliable information out of the raw process data and to convince engineers and plant managers not only to look at budgets and costs, but also at the value of insight. This can be achieved by : - enhancing mutual communication between the different knowledge domains and plant supervision levels - use of mathematical process models (rigorous and reduced) - focussing all sources of process-data and map them to a graphical user interface. From a general point of view, information technology can help us transforming data into information and information into knowledge. Material and energy is thus converted in an economical,ecological and social acceptable way to valuable products. Literature
(1) Vaclavek, V.(1969) Studieson SystemEngineering.OptimalChoiceof the Balanceof Measurementsin ComplicatedChemical EngineeringSystems,Chem.Eng.Sci.24,947-955 (2) Cameron, D., (1991) DynamicDataReconciliationin the Contextof an Equation-OrientedFlowsheetSimulator,PhDThesis
Universityof CambridgeTrinityCollege
(3) D.Dempf, T.Ust, On-linedataReconciliationin ChemicalPlants.IndustrialApplicationof KnownMethodsESCAPE8 (4) M.G. Grottoli, O.Loiacono, E.Ranzi, SimulationModelfor KeteneProductionviaAceticAcid, Computer-OrientedProcess Engineering,Amsterdam1991,S.203-207 (5) E.Ranzi, M.Dente, S.Pierucci, S.Barendregt, P. Cronin, Oil andGasJ.,49.1985(Sept.2) (6) Process of Manufacturing 1,3-Diketones , WACKER CHEMIE, unpublished Patent Application, filedat GermanPatentOffice underDE 19938 341,expecteddateof publication15.2.2001 (7) Pierucci, S.,P.Brandani, Ranzi, E., Sogaro, A., An industrialapplicationof an on-linedatareconciliationandoptimizationproblem., Cornput.Chem.Eng.20,Suppl.,$1539-1544(1996). (8) Belsim s.a., VALI III users guide, Belsim,Alleedes Noistiers,1,B-4031AngleurBelgium,(1997) (9) Dynamics, H.Musch, Hegarstr.16,CH-8032Z(Jrich(Switzerland) (10) On-line Analyse der Prozessgase bei der Herstellung von Keten, Hoechst AG, patentfiled6.8.97underDE 19733 837,A1 (11) Heyen, G., Marechal, F., Kalitventzeff, B., Sensitivitycalculationsandvarianceanalysisin plantmeasurementreconciliation, Comp.Chem.Engg.,vol20S,pp539-544(1996b)
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
547
S y m b o l i c Discretization of P o p u l a t i o n M o d e l s for Process S i m u l a t i o n M.Brahmadatta, R. Ki3hler, A. Mitrovi6, E.D. Gilles, M. Zeitz Institut ftir Systemdynamik und Regelungstechnik, Universit~it Stuttgart, Pfaffenwaldring 9, D-70550 Stuttgart, Germany 1 INTRODUCTION A large number of chemical engineering processes are operated with one or more dispersed phases that can be modelled by the population balance approach comprising partial differentials of first order and integral terms over the population domain [1, 2]. The integral terms characterize intra particle phenomena e.g. breakage or agglomeration of particles. The population balance approach leads to partial differential equations (PDEs) or integro partial differential equations (IPDEs) which must be preprocessed for the application of standard numerical simulation or optimization tools. The PDEs, IPDEs and the related boundary conditions can be transformed into differential algebraic equations (DAEs) using the Method-of-Lines (MOL) approach [3]. Thereby, a special treatment of the integral terms has to be considered. For the resulting overall DAEs, efficient numerical routines are available in simulation tools like DIVA [4] or gPROMS [5]. For the simulation environment DIVA, the Symbolic PreProcessing Tool SYPPRoT has been designed for an application of different discretization methods on PDE and IPDE models [6]. SYPPRoT is implemented by means of the computer algebra system M A T H E M A T I C A . C o n figurable conventional discretization methods like finite-difference and finite-volume schemes are implemented in order to apply the MOL approach to PDEs and IPDEs. The architecture of SYPPRoT is designed as a toolbox that can be easily extended to more sophisticated discretization methods. In this contribution, the functionality and the application of SYPPRoT will be illustrated for a population model of a simple two-phase system with a continuous and a dispersed phase. Due to the hyperbolic character and the usually fully coupled jacobian, advanced discretization methods affecting the numerical treatment of IPDEs have to be reflected. Investigations of Essentially Non-Oscillatory (ENO) schemes [7] and the Robust Upwind method [8] with prototype simulation models of such discretization schemes have therefore be considered.
2
ARCHITECTURE OF THE SYMBOLIC PREPROCESSING FOR THE SIMULATION ENVIRONMENT DIVA
The DIVA architecture shown in Fig. 1 comprises three layers that are all accessible for editing and debugging by the user. The bottom layer contains the DIVA kernel with the numerical methods and the library of generic process unit models. The kernel as well as the model representation are implemented in FORTRAN. Process unit models as well as data files for the parameters and initial values are automatically generated by the code generator (CG) [9], which is the second layer of the DIVA architecture. The CG requires the complete model information written in a CG input file. This file and the numerical subroutines use the linear implicit DAE
548 ~ /
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Fig. 1: Architecture of the simulation environment DIVA with the Symbolic Preprocessing Tool SYPPROT [6], the Code Generator [9], the Model Library, and the DAE Numerical methods [4].
model formulation with a differential index of one:
dz
B(z, u, p, t) . ~
- f(z, u, p, t)
t>t0,
z(t0)-z0.
(1)
There, B denotes the usually singular left-hand-side matrix, and f the right-hand-side vector. The symbols z(t), u(t), y(t), p, and z0 are the vectors of the states, inputs, outputs, parameters, and initial values, respectively. The CG input file is either written by the user or is the result of the symbolic preprocessing of a PDE / IPDE model, which is located in the third layer of DIVA. The preprocessing tool includes MOL discretization of PDEs and IPDEs into DAEs, index analysis and reduction of DAEs as well as DAE transformation into the form (1). The preprocessing functions are implemented by means of the computer algebra system MATHEMATICA. For the definition of a population model, the user writes an input file using the developed MATHEMATICA data structure (MDS). A MDS input file contains the complete model information as well as the definition of the discretization methods to be used. The execution of the preprocessing steps is controlled by the data management of SYPPRoT (Fig. 1) and performs the transformation of the IPDEs into DAEs. Finally, the DAE writer of SYPPROT generates the CG input file. In this paper, the capabilities of SYPPRoT concerning the MOL discretization of population models described by IPDEs will be presented. 3 POPULATION BALANCE EQUATIONS The population balance approach characterizes particles of the dispersed phase by internal coordinates like the particle length in crystallization processes. This approach allows to integrate submodels for the microscopic phenomena on a macroscopic scale into a model for the overall process unit. A general population balance for a dispersed system using one internal length coordinate 0 < L < Lmax is given for the number density function F(L, t), t > 0:
OF O(vLF) fLmo= = - O------if-- + Fi, - Fo~t - ~ F + t~ ]L P(L, L')fl(L')F(L', t) dL'. Ot
(2)
This balance accounts for growth, in- and outflow and breakage of particles. The first term on the fight side of Eq.(2) describes the flux in direction of the internal coordinate L with the growth velocity vL(L). The inflow and outflow of particles are given by ~n(L, t) and Fo~,t(L, t). The sink due to breakage is given by the breakage rate/3(L) multiplied with F(L, t). The source
549 due to breakage of particles is described by the integral term using u as the number of fragment particles and P(L, L') as the probability for the formation of a particle with length L due to breakage at position L'. The considered process simulation is assumed to start with an initial profile Fo(L), and moreover, no particles are present at L = O:
F(L, O) = Fo(L),
F(0, t) = 0.
(3)
The continuous phase is modelled with a material balance for the total mass me(t) including an integration over the whole population to determine the mass transferred from continuous to dispersed phase due to growth:
dmc dt
. = rain -
. mout +
fL,,o O(VLF) kypc Jo ~L3 0L dL .
(4)
The functions thin(t) and (hour(t) describe the in- and outflow of mass, ky the volume shape factor and Pc the density of the continuous phase. The component material balances can be set up in the same manner as Eq.(4). Due to the hyperbolical differential growth term and the integral terms for breakage in Eq. (2) as well as for growth in Eq. (4), a discretization scheme has to be applied for use of numerical DAE simulation methods. 4 METHODS FOR INTEGRAL PDE DISCRETIZATION The focus in discretizing Eqs. (2) to (4) is on the partial differentials and the integral terms. The integrals are approximated by sums over the discretized population domain. Looking at Eq. (2), the approximation of the integral leads to a dense upper triangular jacobian matrix. An increase in the number of discretization points will lead to a large computational effort due to the dense jacobian matrix. For the discretization of the partial differentials, different discretization schemes can be used. Common discretization approaches are the finite-difference and the finitevolume schemes. Finite-Difference and Finite-Volume Schemes: The finite-difference and the finite-volume schemes subdivide the state space into uniform or non-uniform distributed grid points or control volumes, respectively. The finite-difference schemes approximate the partial derivates with respect to the state values on the grid points. Different approximation orders can be derived from Taylor series expansion or polynomial functions. The finite-volume method integrates the IPDE over each control volume leading to the integral IPDE form. The unknown state values at the control volume boundaries need to be approximated by so-called profile assumptions. These profiles can defined by constant, linear, or higher order approximations. Due to the hyperbolic nature of the model equations, only low order approximations for both methods can be used. Therefore, a large number of grid points is required for sufficiently accurate simulation results leading to a high computational effort. With respect to computational efficiency, high resolution methods are considered and have to be included into the preprocessing tool. High Resolution Schemes: High resolution methods must unify high numerical accuracy as well as numerical stability and efficiency. Two promising advanced finite-volume discretization schemes have been investigated concerning an increased performance: (i) Essentially NonOscillatory (ENO) schemes [7], (ii) Robust Upwind scheme [8]. In the following, the main ideas of both approaches are described without focussing on details. Both schemes are applied on the population model (2) to (4) and compared to the standard finite-difference and finitevolume schemes. ENO schemes are of high order accuracy and basically free of spatial oscillations [7]. The schemes use the so-called ENO interpolation which approximates the flux values using a fixed
550 to = 0.0 s It.
t 1 = 0.6 s
Table 1 Benchmark comparison calculating F(L, tl) in Fig. 2 using different MOL discretization schemes for 0 < L < 2.0.
1-
c
.9
"5o.8c
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9)
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,
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114
Fig. 2: Simulation result for simplified Eqs. (2)(4) for a pulse initial profile Fo(L).
Method Finite-Difference Finite-Volume ENO 3rd order ENO 4th order ENO 5th order ENO 6th order Robust Upwind
Points 1000 1000 130 100 70 70 80
CPU Time 4.83 s 4.73 s 4.05 s 4.69 s 5.70 s 11.81 s 1.78 s
number of interpolation points on solution dependent positions. The "optimal" position of the interpolation points is determined successively by the construction of a Newton interpolation formula of increasing order for approximation of the flux. This approximation procedure compares divided differences of the Newton interpolation formula built of adjacent interpolation points. The minimal magnitude of the compared divided differences determines the position of the next interpolation point. ENO schemes can be extended to any approximation order. By increasing the approximation order, the numerical accuracy is increased but also the computational effort. The Robust Upwind scheme provides a stable discretization scheme with an accuracy of 2nd to 3rd order [8], the application of which is also known in context of the finite-volume discretization. The robust upwind approximation of flux values at the control volume boundaries uses two upwind located control volume centre points. The Robust Upwind scheme is based on a limiter function that depends on the ratio of consecutive solution gradients. In contrast to the ENO schemes, the robust upwind method is described by only two simple equations. The computational effort is therefore be rather low compared to the ENO schemes. Benchmark Simulation Results: The finite-difference and finite-volume schemes and the two high resolution methods are studied for simulation of a population model (2) with (fl ~ 0) and without (fl = 0) breakage and a constant value for the growth velocity vL. The performance of the different discretization schemes regarding accuracy and computational effort is evident in the number of discretization points that are needed to generate dynamic profiles of the same discretization error. The pulse density profile Fo(L) in Fig. 2 is the initial profile for all four considered discretization schemes. The benchmark for the MOL series concern to calculate the density profile F(L, tl) in Fig. 2 with the same accuracy as the reference profile calculated by finite-differences with 1000 grid points. Therefore, different numbers of grid points in the different discretization schemes are required (see Tab. 1). There, the number of grid points and the CPU time on a SUN Ultra 60 are shown for each discretization method. Following Tab. 1, a rather high number of discretization points are only needed by the simple finite-difference and finite-volume schemes. The ENO scheme of 3rd order needs about seven times less discretization points than the two simple discretization schemes. Moreover, the CPU time is reduced from 4.7 seconds to 4.0 seconds. By increasing the order of the ENO schemes up to 6th order, the number of needed dicretization points is reduced while the consumed CPU time is increased. This is due to a decrease in integration time step size by using higher order
551 schemes and the required additional equations. As given in Tab. 1, the Robust Upwind scheme performs very well. This scheme only needs a very low number of discretization points and only 1.78 seconds of CPU time. The four discretization schemes have been also tested on the complete population model (2) to (4) including an equally distributed probability function P(L, L'). Almost all schemes perform well with the complete population model. But only high order ENO schemes like the 5th and 6th order have some numerical stability problems which cause negative particle numbers in the population. 5 SYMBOLIC PREPROCESSING OF POPULATION MODELS The symbolic preprocessing tool SYPPRoT of the DIVA simulation environment in Fig. 1 handles typical equations of distributed parameter models of chemical processes. These models include coupled one-dimensional PDE and IPDEs up to second order of the partial derivative with respect to the space or property coordinate and of first order in time. The full automatic symbolic discretization of SYPPRoT can be applied on Eq.(4) including its integral term with constant limits. In contrast to Eq.(4), the integral of the population balance in Eq.(2) has a variable lower limit L and a integral variable L'. This type of integral cannot be discretized automatically by the current version of SYPPRoT. The user has to modify the preprocessed Eq.(2) in order to alter the integral limit and the integral variable. Extensions of the symbolic preprocessing are planned for the definition and discretization of terms depending on L and L' for a fully automatic handling of Eq.(2). The available discretization methods are configurable finitedifference methods based on Lagrange polynomials and finite-volume schemes with different profile assumptions. The integral expressions of the model equations are discretized using sums over the population domain according to the chosen discretization method. The application of the symbolic preprocessing tool is illustrated on the population model (2) to (4). To simplify matters, only particle growth in the balance equation (2) and no intra particle phenomena are considered. The population model is written in an input file using the MATHEMATICA data structure (MDS). The MDS consists of several definition sections for parameters, variables, equations as well as methods and grids for state space discretization. The following extract of the MDS input file presents the definitions of (2) and (4): Scalar[
Scalar[
D[F[L,t],t] = = - D [vL [L] * F [L, t] , L] + F i n [ t ] - Fout[t], LowerBound -> 0 = = F [ L , t ] , Discretization -> " U p w i n d l " , Name - > "popul at i on", Comment -> " p o p u l a t i o n balance"] D [ m C [ t ] ,t] = = m i n [ t ] - m o u t [ t ] + kv*rhoC* Integral[ L^3*D[vL[L]*F[L,t] ,L] , { L , L m i n , L m a x } ] Discretization -> " U p w i n d l " , Name -> " g r o w t h " , Comment -> " m a s s b a l a n c e of the conti, phase"]
,
In the definition section, the initial conditions are specified for the state variables. But this section is not shown here. The simplified Eqs. (2) and (4) are defined in the MDS. Each equation consists of a S c a l a r [ . . ] expression, which contains the equation expression as first argument. Further arguments define the related boundary condition (LowerBound), the discretization method ( D i s c r e t i z a t i o n ) , as well as the name of the equation object (Name) and a description string (comment). In the equation formulation, derivatives are defined by the MATHEMATICA operator D [ . . ] and integrals are defined by the MDS function I n t e g r a l [ . . ]. The equations and the related boundary conditions are grouped together for the
552 application of a common discretization method. This method is defined by the Discretization statement, which refers to a user configured finite-difference or finite-volume scheme. The symbolic preprocessing for the population model (2)- (4) is performed by the developed tool SYPPRoT in three steps (Fig. 1): (i) MOL discretization of the IPDE into DAE, (ii) transformation of the resulting DAE model into form (1), (iii) generation of a CG input file of this DAE model. These steps are executed by the user within a MATHEMATICA session. The discretization step is performed by the command A u t o m a t i c D i s c r e t i z a t i o n [ . . ]. The command MDS2CG[.. ] performs the DAE transformation and the generation of a CG input file. Subsequently, the code generator compiles the CG input file to a DIVA simulation model (see Fig.l). 6 CONCLUSIONS The symbolic preprocessing tool SYPPRoT implemented in MATHEMATICA provides an automatic MOL discretization of IPDEs for population models. Due to the contained integrals and the hyperbolic nature of the population balance equations standardized discretization schemes can only be applied on very fine grids for satisfactory simulation results. Therefore, advanced discretization methods have been investigated with the objective to reduce the computational effort: Essentially Non-Oscillatory schemes and the Robust Upwind scheme. The main advantage of both methods is the maintenance of sharp profiles during the dynamic simulation. Significant gain in computational efficiency shows only the Robust Upwind scheme. Further development of the symbolic preprocessing focuses on the full automatic discretization of integrals with variable limits and additional integration variables as well as on the implementation of the Robust Upwind scheme.
Acknowledgement:
This work has been sponsored by the Deutsche Forschungsgemeinschaft (SFB 412) and by DAAD for the I.I.T. Programme 1999 ofM. Brahmadatta.
REFERENCES [1] H.M. Hulburt, S. Katz. Some problems in particle technology- a statistical mechanical formulation. Chemical Eng. Science, 19:555-574, 1964. [2] A.D. Randolph, M.A. Larson. A Theory of Particulate Processes. Academic Press, 1998. [3] W.E. Schiesser. The numerical method of lines: Integration of PDEs. San-Diego, 1991. [4] K. D. Mohl, A. Spieker, R. K6hler, E. D. Gilles, M. Zeitz. DIVA - A simulation environment for chemical engineering applications. In ICCS-97, Collected Volume of Scientific Papers, pages 8-15. State Technical University, Donetsk, Ukraine, 1997. [5] M. Oh, C. C. Pantelides. A modelling and simulation language for combined lumped and distributed parameter systems. Comp. and Chem. Eng., 20(6/7):661-633, 1996. [6] R. K6hler, A. Gerstlauer, M. Zeitz. Symbolic preprocessing for simulation of PDE models of chemical processes. Math. and Comp. in Sire., special issue "Method of Lines" (submitted), 2000. [7] C.W. Shu, S. Osher. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics, 77:439-471, 1988. [8] B. Koren. A robust upwind discretisation method for advection, diffusion and source terms. In C.B. Vreugdenhiland B. Koren, editor, Numerical Methods for Advection-Diffusion Problems, pages 117-138. Vieweg, 1993. [9] R. K6hler, S. R~iumschtissel, M. Zeitz. Code generator for implementing differential algebraic models used in the process simulation tool DIVA. In A. Sydow (ed.), 15th IMACS World Congress, vol. 3, 621-626. Wissenschaft und Technik Verlag, 1997.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
553
Heat Integration in Process Design and Retrofit -Software Tools and Data InterchangeE. Aust a, S. Scholl b, and C. Obler b a Prof. Dr.-Ing. E. Aust, Georg-Simon-Ohm University of Applied Sciences, Dept. Chemical Technology, D-90121 Ntimberg, Germany b
BASF-AG, ZAT/A-L540, D-67056 Ludwigshafen, Germany
1 INTRODUCTION Improving the economics of chemical production processes through process intensification measures is a permanent challenge in the design and retrofit of chemical processes. Heat integration may be one contributor to increase the profitability of a process. The success of the method is often determined by the starting conditions within a design or retrofit project. In some cases heat integration targets, however, cannot be attained and have to be sacrificed for aspects of plant operability and process safety e.g. Today energy integration has moved from an end of the pipe measure to an integrated aspect during design and re-engineering. The design engineer has to work with various tools, primarily for flowsheet simulation, detailed equipment design (staged unit operations, heat exchangers and others), cost estimation, and energy integration. Working in a highly integrated software environment increases efficiency as well as effectiveness of the engineer's work. This has frequently been referred to as Computer Aided Process Engineering" CAPE [1]. The advantages of an open software environment are demonstrated in a case study of an energy retrofit analysis. 2 EVALUATING POTENTIAL ECONOMICS ADVANTAGE At first, the economic potential of heat integration measures has to be addressed. Fig. 1 illustrates the economic benefit of a heat integration project for various energy and unit capital costs. The symbols represent the specific energy cost for steam heating [Euro/t] or cooling water [Cent/m3]. The lines are valid for different specific equipment costs. A value of 250 Euro/m 2 represents common carbon steel heat exchangers, while 2500 Euro/m 2 may be valid for high quality special equipment. Fig. 1 shows the annual energy cost including depreciation as a function of the heat flux. For estimating the required exchanger surface area a heat flux density of q=10 kW/m :, a temperature increase of 10 K in the cooling water and a return of investment of 3 years were assumed.
554
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---250 ~
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1000 Euro/m2
250 ~ --. 2500 Euro/m2 2 0 0 ~ . . . . - - - - = ~ = - . . . . . . . -:-:~ . . . . . . 150 100
......
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400
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! 200
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heat flux [kW]
Fig.1. Assessing economical benefits of heat integration measures Equipment costs were scaled according to cost/cOStre
f =
(area/arearef) ~
(1)
with arearer = 100 m 2 and cOStrer= 25/100/250 kEuro. The pure equipment cost were multiplied by a factor of 5 for specific equipment costs of 250 Euro/m 2, a factor of 4 for 1000 Euro/m 2 and 3 for 2500 Euro/m 2. It is obvious that heat integration projects seem promising already for heat fluxes above 500 to 1000 kW, depending on the specific energy and equipment costs. 3 TOOLS FOR HEAT INTEGRATION The computer-based tools for heat integration are mostly derived from the Pinch Analysis [2]. It provides data for the theoretical minimum demand of heating and cooling energy in a preliminary step and can be used to develop improved heat exchanger networks, select favorable utilities, and optimize overall costs. In the definition phase of a project it is required to detect savings potential, identify major limitations, and evaluate the process context. Two approaches are possible: In a first step the process topology as well as the material and energy balances are untouched. The heat sources and sinks profile of the process is not altered. It is common experience that this restriction does not allow to explore the true energy savings potential of the process. This can only by achieved if in a second step modifications of the process topology and conditions, such as column operating pressure, are allowed. Thus, the typical user of simulation and design tools is the qualified design engineer who is mostly concerned with all aspects of thermal separation techniques, such as heat exchange, distillation or other staged unit operations. Efficiency and effectiveness of the design process are increased by a common data base for physical property, phase equilibrium, as well as process data plus a facilitated interchange of generated data between various design tools. 4 HEAT INTEGRATION IN DESIGN PRACTICE Working with a heat integration problem requires four basic steps to be executed consecutively:
555 9Flowsheet simulation in order to generate a consistent set of the relevant process data; 9Heat integration analysis, from an assessment of the scope of energy savings to proposing a revised heat exchanger network or process scheme; 9Preliminary or detailed design of new or evaluation of existing heat exchangers; 9Cost estimation as basis for further economic evaluation of the proposal. For each of the above working steps different commercial as well as inhouse tools may be available. They form a computer based process engineering and design environment termed CAPE Computer Aided Process Engineering. Table 1 shows a typical situation with respect to the use of different inhouse as well as commercial software tools for the above mentioned process engineering tasks. BASF inhouse tools may be used as alternatives to corresponding commercial products. Two different work flow paths are indicated to demonstrate the interaction of the inhouse as well as commercial tools. This work flow has to be supported and facilitated through seamless communication. Table 1 Tools for selected process engineering tasks Process Engineering Tasks BASF Inhouse Tool Flow sheet simulation Chemasim Heat integration analysis WIT Heat exchanger design & KONVER evaluation Cost estimation KOSDIAS
Commercial Tool Aspen Plus, ChemCad .... Super Target, AspenPinch .... IST/HTRI, TASC/HTFS, B-Jac, ... Aspen Costing, ...
As the complexity and capability of the various design tools increase the question of the interaction of the tools gets more and more virulent. Sourcing from and feeding into a common data management system is crucial. It is desirable that all design tools derive the process stream properties from the same thermodynamic data base. When performing several design tasks with different tools the question of data transfer and interfaces is crucial. Data of preceeding design steps must be usable in a following step without any prerequesits in the previous step. Most frequently the question of using a master tool vs. a modular simulation environment arises: For example, should the heat exchanger design be checked within a heat integration program or within the flowsheet simulator, or should a separate program be used instead? It is the authors view that a modular simulation environment with an easy crossover between the different software tools provides the most efficient and effective support to the design engineer. Data transfer between the various tools should be supported by intelligent interface programs, one of which is referred to in the following example.
5 CASE STUDY OF HEAT I N T E G R A T I O N IN A R E T R O F I T DESIGN P R O J E K T The following case study is based on a process for the production of cyclohexane by catalytic hydrogenation of benzene and a separation of the product mixture in two distillation columns. The production capacity is chosen to be 28,000 t/a cyclohexane, flowsheet simulation was
556 The current analysis shall be confined to comparing the existing energy demand with the target values in order to assess the potential of heat integration. Suggestions for process modifications are then set forth to improve the energy efficiency. These process changes are implemented in the process simulation and an improved process scheme is developed, which still meets the product specifications, but requires less external heating and cooling. Fig. 3 shows the composite curve of the original process for an assumed minimum temperature difference ATmi, of 25 K. The pinch temperatures are 90 ~ for cold streams and 115 ~ for hot streams, the minimum energy requirements as taken from the composite curve are compared with the values of the existing process in Tab. 2. F
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1400
1600
Fig. 3: Composite Curves of cylohexane unit Table 2 Comparing energy demand of existing process with theoretical minimum values of pinch analysis for ATm~n= 25 K (energy target). Heating Cooling Existing Process 771 1,100 kW Energy Target 171 500 kW The figures of Tab. 2 indicate a considerable savings potential for external process heating (-78 %) and cooling (-55 %). The individual options are compiled in Tab. 3 indicating the potential energy saving of each measure. Table 3 Suggested process modifications derived from pinch analysis Measure Energy Savings Potential 299 kW Avoid mixing streams of different temperature 201 kW Integrate Flash "HP-SEP" in 2 stages 100 kW Replace preheater for benzene 4 7kW Add sidestripper or pumparound on column 1 20 kW Change operation pressure of column 2
557 The production capacity is chosen to be 28,000 t/a cyclohexane, flowsheet simulation was performed using ASPEN-PLUS V. 9.3, the PFD is depicted in Fig. 2. It is assumed that the plant is subject to a retrofit analysis using the Pinch method aiming at substantial energy savings. For this purpose the software SUPER-TARGET V. 4 [3] is used. Thus, one objective of this study was to investigate the ease of a data exchange between typical design tools and illustrate the iterative nature of energy optimization. Improvements on the energy side result in changes in the mass balance which require an interative approach between flowsheet simulation and heat integration. The major steps in performing a retrofit process integration analysis are: 9 Extracting the required data for the Pinch analysis (preferably from simulation). 9 Entering these data into the process integration software and determining the minimum energy requirement (energy targeting). 9 Identifying cross-pinch heat flow, suggest changes in the heat exchanger network. 9 Propose process changes such as column operating conditions, mixing of process streams, change process temperatures and pressures. 9 Check process changes in flowsheet simulation and decide on actual retrofit changes.
Fig. 2. PFD of cyclohexane unit prior to retrofit The first crucial step in the analysis is extracting the relevant thermal data. This procedure can be time consuming and open to erroneous results [4]. Therefore, data interfaces have been developed that retrieve thermal and topology data from process simulators and supply them to energy integration so~ware packages [5]. In the present case, the program ASPEN2ST has been used to extract process data from ASPEN-PLUS simulations. ASPEN2ST recognizes the relevant streams and unit blocks in the flowsheet results file and generates an input file for the heat integration software.
558 In this very first approach no process constraints were accounted for. Especially in retrofit cases they may reduce the accessible savings potential significantly: streams must not be matched due to safety reasons, space restrictions, heat sources and sinks may be in remote locations of the plant. In these cases contraint targetting gives a more realistic picture on saving potentials under the given conditions. In the present case it is obvious that the saving potentials decrease with every additional measure. One therefor focusses on the most efficient options in the first place. Through a remaining problem analysis the efficiency and economics of each consecutive option is evaluated and the catalogue of measures can be sorted due to economic efficiency. Decisions on further actions can then be taken by an incremental strategy. The last two suggestions in Tab. 3 require process modifications around the distillation columns. They are evaluated using the flow sheet simulation software and new mass and energy balances are generated while still meeting the product specifications. If the changes are acceptable the revised set of thermal process data can be imported again into the pinch software and re-analyzed. This iterative procedure is only possible in an effficient way with an open software environment and a seamless data exchange between different software tools. The proposed changes of Tab. 3 were implemented in the flowsheet simulation. Compared to the original design an overall reduction of energy costs by more than 80 % may be achieved. Before accepting all these modifications the economics aspects of plant operability, safety and others need to be evaluated. This, however, would go beyond the scope of this paper. 6
SUMMARY
Heat integration analysis constitutes an important element in the design and optimization of production processes in the chemical process industry. Effectiveness and efficiency are increased by an open software environment with transparent and seamless exchange of data between different software tools. These advantages were illustrated on hand of a process retrofit case study where process simulation and integration tools work hand in hand to achieve substantial energy savings. An expert software program was used to extract and transfer the thermal data for process integration from the simulator. From the heat integration analysis a number of suggested process changes resulted in modifications of the process layout and operation conditions of distillation columns. These changes were then checked in the simulator to evaluate their feasibility and effect on the product specifications.
REFERENCES
[1 ] B. Bel31ing et al.: CAPE in process design- potential and limitations. Comp. and Chem. Engng. Suppl. (Ed. S. Skogestad), Vol. 21 Suppl. (1997), pp. S 17-$21. [2] B. Linnhoff et al.: User Guide on Process Integration for the Efficient Use of Energy. Ed.: The Inst. of Chemical Engineers, Rugby/GB 1982/1994. [3] SuperTarget, V. 4.0; Linnhoff-March Ltd, Manchester (GB), 1997. [4] lJbler, C.; Diploma thesis Georg-Simon-Ohm-Fachhochschule Nilrnberg, 1997. [5] Interface-File-Format, V. 1.0, Linnhoff-March Ltd., Manchester (GB), 1997.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
559
Modelling and optimisation of polymerisation reactors in gPROMS M. Asteasuain, S.M. Tonelli, A. Brandolin and J.A. Bandoni Planta Piloto de Ingenieria Quimica (PLAPIQUI) - UNS - CONICET. Camino La Canfndanga km 7 (8000) Bahia Blanca. ARGENTINA. A dynamic model of the high pressure polymerisation of ethylene in tubular reactors is introduced and a dynamic optimisation problem is formulated for studying start-up strategies. The optimisation objectives proposed are to maximise outlet conversion and optimise the time necessary for it's stabilisation while keeping product molecular properties between commercial ranges. Results show the time responses for temperature, number-average molecular weight and conversion along the reactor axial distance for different control variable profiles. Improving in reactor productivity is achieved. The interface gOPT of the gPROMS simulator was used to resolve the optimisation problem and to perform the simulations. 1. INTRODUCTION High pressure polymerisation of ethylene to produce low-density polyethylene in tubular reactors is a widely used industrial process. It is carried out under rigorous operating conditions, and consequently a mathematical model is an attractive tool to study safely and economically the influence of the different design and operative variables on production performance and product quality. Several authors have presented different models dealing with the stationary state of this process [1-2]. However, less attention has been paid to the dynamic behavior of this reactor, despite a dynamic model may be successfully used to study the cyclic pressure pulse, start-up and shutdown strategies and production optimization. This type of tubular reactors has a main feed of ethylene monomer, solvent, inerts and oxygen initiator. In addition, there are lateral injections of initiators consisting in peroxide with or without monomer. The reactor is divided in jacket zones in order to reach an appropriate reaction temperature or to control the exothermic reaction. A pulse valve is located at the exit to control polymer build-up at reactor walls. This last feature accounts for an inherent dynamic behaviour of these reactors. Other disturbances in this process occur due to changes in temperat~e levels in the jacket fluid and concentration of impurities in the monomer feed. In industry, a single reactor is used to produce several kinds of polyethylenes of different molecular characteristics, making them appropriate for different final uses. Such type of reactor must be flexible enough to deal with a wide range of operating conditions. The main objective of our study is to analyse the start-up and shutdown strategies and perform flexibility analysis in the polymerisation reactor. For this paper, we developed a dynamic model that predict average molecular weights, monomer conversion, concentrations and temperature as function of time and axial reactor length. The simulation package gPROMS [3] was used to perform these studies. Then, the interface gOPT of gPROMS was used to resolve the start-up optimisation problem.
560 2.
MODEL DESCRIPTION
2.1. Reactor Model
The reactor configuration is displayed in Figure 1. It corresponds to a typical industrial reactor with a length to diameter ratio up to 20,000. The trait operates at very severe conditions: high axial velocities (~11 m/s), temperatures ranging from 50~ at the reactor entrance to 325~ at the peaks, and pressures between 1800 to 2800 atm. The reactor model was divided in five zones at different jacket temperatures. Jacket fluid, vapour in the first zone and liquid water in the others, flows in counter-current with respect to reaction mixture. Ethylene monomer (M), which contains inerts, oxygen and telogen (S) enter at the beginning of the reactor. Oxygen produces initiator radicals which propagates to form long macromolecules. After peroxide enters to the third and fifth zones the reaction explodes reaching a temperature peak. First, second and fourth zones are use for heating/cooling proposes. Adding telogen at the reactor entrance avoid the production of prohibitively large polymer molecules which increase viscosity interfering with heat transfer and reactor control. The initiator flow rates determines the reactor conversion, but there is a limit to the quantity to be added because of reactor runaway.
I t h y l ~ e Iii~i~
,eroxide
~]~----
""
,e oxi e ...~,,"~~ -.~ ,
'-"Z--"s )-- -
Zone.5":: ~-q d [Ethylene
!:~og:n ~
!
~olyethylene
Inerts
~
~rI:l:gSs
Fig. 1. Tubular reactor for high pressure ethylene polymerization The following basic reactions were considered in our model (Eqs. 1-6). Peroxide (I) Initiation
Oxygen Initiation
I
fkd ,,>2 RI(O )
(2)
02 + M ko> 2 R~(0) Propagation kp Ri(x) + M ...... >Ri(x + 1) Thermal Degradation
(1)
(3)
Ri(x) + Rj(y)
ktc......"Pi +j - ~(x + y) Chain Transfer to Telogen (S)
(4)
Ri(x + 1)
(5)
Ri(x) + S
(6)
ktdt ~> P i ( x ) +
RI(O)
Termination by Combination
ktrs , ~.(x)+ R~(O)
M is the ethylene monomer, l~.(x) and Pi(x) represent radical and dead polymer of chain length x and i long-chain branching, respectively. Kinetic constants (k) follow the Arrhenius law in temperature. The corresponding values may be found elsewhere [2]. Initiation radicals are generated by oxygen and organic peroxide decomposition (Eqs. 1-2) when reactor temperature reaches appropriate levels. Propagation reaction takes place when monomer (M) reacts with radicals to produce another growing macromolecule monomer (Eq. 3). This reaction accounts for most of the polymerisation heat which is 21500 cal/mol. Natural termination is produced by combination of two radicals (Eq. 4) or by thermal degradation
561 (Eq. 5). A chain transfer agent (telogen) is commonly fed at the reactor inlet in order to control molecular weight. Telogens react with radicals producing a dead polymer and an initiation radical (Eq. 6). In consequence, the presence of telogen results in shorter macromolecules. Another reactions, such as transfer to polymer and intramolecular scissions are also important in this polymerisation. They affect long and short-chain branching, and weight-average molecular weights. Up to this point, we are only concerned with conversion and number-average molecular weight predictions, so the last reactions were not included in the model. Mass and energy balances are shown in Eqs. 7 to 11. Component Concentration (Cj):
~cj
~cj
Ot - r j - v Oz
j : 1,Ncomp
Cj" [02], [M], IRa], [Pol], IS1, [I], [Inerts]
(7)
Reactor Temperature (T):
OT OT 4U(T- Tj) pCp --~=-pCpv-~z Di + kp[Ra][M](-AH)
(8)
Boundary conditions f Cj(t,O) = Cjin(t ) z = 0 [ T(t,O)=Tin(t)
Initial conditions ~Cj(O,z) = Cjo(Z) t = 0 [ T(O,z)=To(z)
(9)
Monomer Conversion
Number-average molecular weight
Ncomp
(10)
X(t,L)=I- Z[Cj]wt/wt j=l
Mn(t,L) =X(t'L)~p [Pol]
(11)
j~pol
where IRa] and [Poll: concentrations of radicals and polymer, p: mixture density, Cp: specific heat, v: axial velocity, U: heat transfer coefficient, Tj: jacket temperature, Di: inside diameter.
2.2. Dynamic Optimisation Problem Max X(t f ,L)
Objective function
Process model: Initial conditions: Boundary conditions." Time horizon bounds:
(12)
tf ,~(t,z) subject to: F(~(t,z), Vz(t,z),vt(t,z),~(t,z ), ~(t,z) ) = 0 I(F(O,z), v z (O,z), v t ( O , z ) , u ( O , z ), w ( t , z ) ) -- 0 B(~(t,O), ~z (t,O), vt (t,O), ~(t,O), ~(t, z) ) : 0
Control variable(u ) bounds: Dependent variables bounds
.
t f ,min < t f < t f ,max
Umi, <-~(t,z) <_U--max V-men <_V ( t , Z ) <_ V--max, W'--min <-- W ( t , Z ) <_ W--ma x
z~[O,L]
(13) (14) (15) (16) (17) (18) (19)
(20) t~[O, tf] where ~(t,z) : Vector of control variables; ~(t,z) : Vector of algebraic state variables, v(t,z) : Vector of differential state variables, t: Time; z: Axial length. The main objective of this work is to find optimal start-up and shut-down policies for the reactor. From the operating point of view this means reaching the maximum conversion and
562 the new steady state in a minimum time. Mathematically this situation can be represented by a dynamic optimisation problem. In a start-up policy the objective function to be maximised is the monomer conversion at the reactor exit, X(tf,L), where X is the monomer conversion, tf the final time and L the reactor length. The mathematical statement of this model is as follows.
2.3. Optimisation Problem Resolution First of all, the DAE system resulting from applying backward finite differences in the axial distance to the original model partial differential equations was solved using the general purpose modelling, simulation and optimisation package gPROMS (generalized PROcess Modelling System). In gPROMS, each operation unit model contains all the information regarding physicochemical behaviour of the equipment. On the other hand, external actions over the unit are declared independently. This allows the use of the same model when the equipment is repeated several times within the same process. Taking advantage of this property, the reactor was simulated as a series of tubular reactors. Each link of the chain represents a division in the reactor due to a different jacket temperature or a lateral feed. To check the reactor model validity, several simulation runs were performed for different initial conditions. In all cases, temperature and component's concentrations, as well as monomer conversion and number-average molecular weight evolved towards the expected values of steady state obtained with an existing rigorous reactor model [2]. The dynamic optimisation strategy within gPROMS is the DAEOPT code that implements the algorithm outlined by Vassiliadis et al. [4]. In this program the solution of the DAE system is performed using the DASOLV code while the optimisation is carried out by the SRQPD code. The interface called gOPT makes it possible to perform dynamic optimisation calculations using DAEOPT and processes defined in the gPROMS language. 3. RESULTS AND DISCUSSION Table 1 shows the design features and the steady-state base case operating conditions of the industrial reactor under analysis. The corresponding steady-state monomer conversion and number-average molecular weight are also shown. This is the steady state achieved when the start-up is accomplished setting all the fluxes at their stationary values since the beginning of the operation. The operation starts with the reactor filled only with monomer, oxygen and telogen at a temperature equal to that of the monomer feed. Table 1. Design features of the reactor and steady state conditions for the base case. Initiator 1 Initiator 2 Inert Monomer Oxygen Telogen Flow rate Flow rate Flow Rate Flow Rate Flow Rate Flow Rate (kg/s). (kg/s) (kg/s) (kg/s) (kg/s) (kg/s) 2.2 10-1 1.0 10.3 1.6 10.4 11 6.8 10-s 7.4 10.2 p (g/cm 3) Conversion (%) Mn (g/mol) L/D Tinlet (~ P inlet (atm) 0.53 30 21900 27800 76 2250 Cpi I= 1,Nzones U I= 1,Nzones Nzones Tj,,=Tj,4=Tj,s=Ta Tj,2= Tj,3= Tb (cal/g ~ (cal/cm2~ s) (oc) (~ 0.58, 0.58, 2.6 10-2,2.6 10-2,2.0 10.2, 5 168 225 0.75,0.75,0.96 1.5 102, 4.7 10.3
563 Table 2. Optimisation conditions for Policies A and B tf,min tf,m~x p
(s)
(s)
100
350
(var.s.t. bounds) [~ g/mol] Policy A:
[~
[T(t, z), Mn (tf L) ] Policy B: [T(t,z), Mn(t,L) ]
N/A
X--m= g/mol] [N/A, 21900]
U
U min
(kg/s) [Fi,~ Fi 2, F j ' '
(kg/s [5 10-5, 5 10 5, 5 1 0 -5]
Xm,x [~ g/mol] [330, 24000]
U max
(kg/s [5 10-3, 2 10 -3, 5 1 0 -1]
Policy A
Policy B
Piecewise constant u, over four different and adjustable time intervals within [0,tf]
Piecewise linear u, over five different and adjustable time intervals within [0,tf]
In order to optimise the reactor operation, we analyse different start-up policies that maximise conversion keeping the molecular weight at the same value of the base case, which corresponds to a commercial valuable product. As it was stated before, initiator and telogens fluxes were used as control variables. Table 2 details the particular variables and values employed in Eqs. 12 to 20 for two different optimisation policies. Values not consigned there remain equal to the ones of Table 1. 1.0E-3 r
~
-
~
2.0E-3
_
,,.-.,,
8.0E.4 -
-~, 6 . 0 E . 4 i~ ~ . .-~ _ LL
i
~
.'=- 4 . 0 E - 4 -
g14
-
~111I
2.0E-41 O.OE-i-O -
A,~ am
Policy A
9-
Policy B
"
0
50
100
150
200
250
start-up u~ (s)
_
_ --
Base Case
300
_-
350
400
O.OE+O
45
0
Fig. 1 First Peroxide feed flux rate 0.60
'
I
'
I
'
I
'
-;- ::::;; t
50
100
150
200 250 300 s t a r t - u p time (s)
350
400
450
Fig. 2 Second Peroxide feed flux rate
I
;
I
i
0.20 0.10
f 0
,t i
I
50
i
I
I00
i
I
150
i
I
200
J. i
I
250
,
start-up time (s)
I
300
i
I
350
l
I
400
,
"
45
Fig. 3 Telogen flux rate Figures 1 to 5 show results corresponding to the optimal start-up policies A and B detailed in Table 1. The main difference between these two policies is when molecular weight is checked (see Table 2). Figures 1 to 3 present the optimal initiator and solvent fluxes for the base case and for both optimal policies. Fluxes at the last interval were the same for both policies, since they correspond to the best combination that leads to a maximum steady-state conversion. When molecular weight is only controlled at the final start-up time, initiator
564 fluxes (Figs. 1 and 2) start at a low value in the first interval and the highest values appear in the last interval. If molecular weight is controlled all through the start-up, specifically in each interval extreme, it is necessary to start with a higher initiator flux which is almost the same as the one for the final interval. In both cases the optimum telogen flux rate was at its maximum allowed level. The optimisation model results are in agreement with the expected physical behaviour. When polymerisation starts, very high molecular weight polymer is produced; this is the situation at the beginning of the start up, as it is shown in Fig. 5. When telogen flux is high enough, low molecular weight is obtained in the range of the commercial values. Figures 4 and 5 present the evolution of conversion and number-average molecular weights at the reactor outlet during the start-up. Both start-up policies determine an increase of around 3% in conversion level with respect to the base case. Conversion reaches its highest value more rapidly in Policy B. Moreover this last policy produces polymer at the desired molecular weight value over a wider portion of the start-up time. 1.E7" 0,4
0.3~
,!
!
"~ 1.ES
~
~1.E4
0.2 . . . . . .
o!jt
O0 0
--!--
----4P-'
Base Case Policy A
PolicyB
~
tL
I,
"
i
1.E3
-&--- Base Case I--- PolicyA e-- PolicyB .......
I .E21
1oo
2OO
3OO time {sl
4OO
6OO
Fig. 4 Exit conversion vs. start-up time
I.EI
0
100
20O time
300 (s)
4O0
500
Fig. 5 Product Mn vs. start-up time
4. CONCLUSIONS The optimal start-up policies found in this work lead to maximum conversion values that are above the commonly conversions levels for this type of reactors. Reactor safe operation was guarantee by means of a restriction in maximum operating temperature and the maximum conversion is moderate enough to ensure good heat transfer avoiding undesirable increments in reactor viscosity. Moreover, reactor productivity was increased while maintaining commercial valuable molecular weight for the product.
REFERENCES 1. 2. 3. 4.
R.C.M. Zabisky, W.-M. Chan, P.E. Gloor and A.E. Hamielec, Polymer, 33, (1992) 2243. A. Brandolin, M.H. Lacunza, P.E. Ugrin and N.J. Capiati, Polym. React. Eng., 4, (1996) 193 P.I Barton,. and C.C. Pantelides, AIChE Journal, 40, (1994) 1966. V V.S. Vassiliadis, R.W. Sargent and C.C. Pantelides, Ind. Eng. Chem. Res., 33, (1994) 2111 and 2123.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
565
Modeling Particle Size Distribution (PSD) in Emulsion Copolymerization Reactions in a Continuous Loop Reactor P. H. H. Ara6jo a'b*, J. C. de la Cal b, J. M. Asua b, and J. C. Pinto a aPrograma de Engenharia Qufmica / COPPE, Universidade Federal do Rio de Janeiro, Cidade Universitfiria, CP: 68502, CEP 21945-970, Rio de Janeiro, BRAZIL. bInstitute for Polymer Materials "POLYMAT", and Grupo de Ingenierfa Qufmica, Facultad de Ciencias Qufmicas, Universidad del Pafs V a s c o - Apdo 1072, 20080, San Sebastifin, SPAIN. A detailed dynamic mathematical model that describes the evolution of particle size distributions (PSD) during emulsion copolymerization reactions in a continuous loop reactor is developed and compared with experimental data. The model is based on the assumption that two distinct particle populations exist: precursor particles and stable latex particles. Precursor particles are colloidally unstable and therefore may undergo coagulation with other precursors and be absorbed by stable latex particles. It is shown that the kinetic model is able to reproduce the rather complex dynamic behavior of the vinyl acetate / Veoval0 emulsion copolymerization in a continuous loop reactor. 1. I N T R O D U C T I O N Performing emulsion polymerization in continuous loop reactors may be advantageous in many aspects, when compared to CSTRs. Continuous loop reactors present a higher operational flexibility, smaller start-up and shut-down losses' and due to the higher surface/volume ratios, a much easier temperature control. Therefore, high conversions in short residence times can be achieved. Despite of the advantages presented by loop reactors and their industrial applications, only few works have been published about this subject tl~3]. The main commercial polymers produced in loop reactors are vinyl acetate homopolymers and copolymers for paint and adhesive industries. Under certain specific conditions, emulsion polymerization in continuous loop reactors leads to oscillatory behavior of the particle number due to intermittent particle nucleation. This often results in a multimodal PSD [81. This phenomenon resembles that found in emulsion polymerizations carried out in CSTRs [14-161. The amplitude of the oscillations diminishes along reaction time until a "pseudo" steady state is often reached. Modeling this kind of oscillatory behavior is a quite challenging aspect of emulsion polymerization engineering. Rawlings and Ray [~71 modeled emulsion polymerizations in CSTRs, but the simulated renucleations were sharper and stronger than the experimental ones. Abad et al. [4] presented a model that achieved a good agreement between experimental PSD results and model predictions. However, only limited use of the model has been carried out as the usage of a *Present address : Departamento de Engenharia Qufmica- Universidade de Silo Paulo, Av. Prof. Luciano Gualberto, travessa 3, n 380, CEP 05508-900, Silo Paulo, Brazil. The financial support from CNPq - Conselho Nacional de DesenvolvimentoCienfffico e Tecnol6gico and the Diputaci6n Foral de Gipuzkoa is gratefully appreciated.
566 more detailed nucleation mechanism is required in order to represent more general operation conditions. Coen et al. I~81 developed a quite complex model including the calculation of the PSD of precursors and stable particles, but only batch reactions, that do not present the same complex dynamic behavior as that observed in continuous reactions, were simulated. In this work, a detailed dynamic mathematical model that describes the evolution of particle size distributions (PSD) during emulsion copolymerization reactions in a continuous loop reactor is developed. The model includes the calculation of both the PSD for precursor particles, that are colloidally unstable and therefore undergo coagulation with other precursor particles or with stable particles, and stable latex particles. Only with the calculation of both PSDs (for precursors and stable particles) and considering micellar and homogeneous coagulative nucleation the model was able to reproduce the complex dynamic behavior. 2. M A T H E M A T I C A L M O D E L A mathematical model for emulsion copolymerization reactions in a continuous loop reactor was developed and then compared to experimental data. As a first attempt, a model that assumed that polymer particles were monodisperse was implemented. For the sake of brevity the kinetic model will not be presented here, but the interested readers can refer to Araujo [~91 for more details. However, the model was not able to reproduce oscillations due to renucleations, as observed experimentally. The model failed to describe the particle number oscillations probably due to its lumped characteristic as larger particles are likely to contain more radicals than smaller ones. This emphasizes the need to model the particle size distribution. In order to include the calculation of the PSD in a more detailed model of the polymerization, the following set of population balance equations were implemented.
2.1. Population Balance According to the coagulative nucleation mechanisms, radicals formed by decomposition of the initiator in the aqueous phase polymerize and form oligoradicals. When the length of an oligoradical exceeds the solubility limit, the oligoradical precipitates and gives birth to a precursor particle (homogeneous nucleation). If the emulsifier concentration is above the CMC (critical micelle concentration), then micelles are present in the reaction medium and may also give birth to precursor particles through absorption of oligoradicals (micellar nucleation). The fraction of radicals that precipitate is limited by the rate of termination with other radicals and the entry rates into micelles, stable particles and precursor. Precursors can coagulate with other precursor or be captured by the existing stable polymer particles. When the mass of a precursor reaches a critical value ( m e s t ) through propagation or coalescence with other precursor particles, a new stable particle is formed. It is assumed that stable polymer particles do not coagulate between themselves. To take the coagulative nucleation mechanisms into account it is necessary to calculate the distribution of the precursor particles and of the stable particles. For this reason, the individual population balance equations are presented separately below. 2.1.1. Precursor Particles The normalized range of variation of precursors is a m n < 1 < a m x . The normalized mass l is equal to the mass of a precursor divided by the minimum mass of a stable particle (mest). The equation that describes the population balance of the precursor particles is:
567
9
c9 (Nr, h(I)) _ _ q_q_.Nr, " h ( 1 ) c?t
-
VR
N2
[ ........
r, . (I
U
h
(l,l
')
.h(l')dl'- Nr,
) . ....... Kc,~
+Ek;:. [i]w+ k,,V,. [j]W].VW "NA "ER,c,.,,_, ]'+k
cg(u(l).h(l)) c?l
..... N,,,, 9 C [Re,,, 9 ]''
N r , . h(l). f~ ..... Kc~ ( l , m ) . f ( m ) d m + - . 1 f r,2 . f !-amn Kc, (1 - l',l').h(l') 9h(1 -Nr~ "'~R V R .....'" J~..... '
"
(1)
-
l')dl'
2
where VR is the volume of the reactor; q is the outlet flow rate; Nrt, is the total number of precursor; and h(1) is the population density of precursors, u(1) is the mass growth rate of a precursor through propagation; Vw is the volume of the aqueous phase; NA is Avogrado's number; [Rjcrit_j] w is the concentration of oligoradicals with length jcrit-1 in the aqueous phase; jcrit is the critical degree of polymerization of a oligoradical when it becomes insoluble in the aqueous phase. [Re,,t] w is the concentration of oligoradicals in the aqueous phase that can enter into a micelle; k,m is the entry rate of oligoradicals into the micelles. The number of micelles (Nmic) is calculated considering that all emulsifier is distributed between the micelles, the aqueous phase and the surface of the particles. In eq. (1) the second and sixth terms of the right-hand side member account for disappearance of precursors due to coalescence with precursors and with stable polymer particles, respectively. The third term accounts for particle growth by propagation. The fourth and fifth terms account for homogeneous and micellar nucleation rate, respectively. The seventh term accounts for the formation of precursors of mass I due to coalescence of smaller precursors. In order to calculate the coalescence rate coefficients, one may use an extension of the standard DLVO model. According to the DLVO theory ~2~ the movement of small particles suspended in a fluid may be described by a potential field that is composed of the electrical interactions among the charges distributed over the particle surfaces, the steric interactions due to the nonionic emulsifier, and the attractive van der Waals particle interactions. The rate of coalescence may be described as the rate of diffusion across the maximum in this potential field. A practical difficulty for using this approach is the large number of parameters required by the DLVO model. Most of them cannot be estimated based solely on the measurement of kinetic and PSD data. Based on the DLVO theory, the difference between the coalescence rate constants for precursors (Kce) and for stable polymer particles (Kcp) is mainly due to the size differences between the two particle populations. The coalescence constant is proportional to (ri+r92/ri rj, where ri and rj are the radius of the swollen particles subject to coalescence. As the diameters of two precursors that coagulate are much more uniform than the diameters of a large stable particle and a small precursor, the coalescence rate constant for precursor coalescence is also much lower. The homogeneous nucleation rate is calculated as the rate of formation of oligoradicals of length jcrit+l by polymerization in the aqueous phase. The rate of heterogeneous nucleation is given by the rate of radical entry into micelles. The number of micelles is calculated considering that all emulsifier is distributed between micelles, the aqueous phase and the surface of the polymer particles.
2.1.2. Stable Particles The normalized range of variation of mass of the stable polymer particles is: bran _<m < bmx. The normalized mass m is equal to the mass of a stable particle divided by the minimum mass
568 of a stable particle (mest). Nucleation of new stable polymer particle can occur through propagation of a precursor of mass amx, generating a stable particle of mass bran, and through coalescence of two precursors, generating stable particle with mass in the range between bmn and 2xbmn. Stable particles grow through propagation and coalescence with precursor. The equation that describes the population balance of the stable particles may be written as: 1
63 (NrE . f (m)) : _ q__q_.NrE . f ( m ) + _ . ,gt
+Nr, " 9s
vR
2
N2
rp
vR
. [ ........ , ......
KcE ( l , m ) . h ( l ) d l -
_
. . . . l) h(m l) h(1)dl 3,,,<~,.....,
~ " - ~.... + ......
h (amx). [k,,~.. [i] p + kpP/ 9[j]P]. 6,,,j,..... -
-Nr": "-~e " f (m).
K,. (l,m
--
3(u(m). f (m)) ~m
KcE ( l , m - l ) . h ( I ) .
f (m-l)dl"
.
(2) a if m = bran this integral is equal zero; if amn+bmn < m <2.bran, the integration limits are amn and m-bmn. where NeE is the total number of stable particles, tim) is the population density of particles and ~(amx) is the average number of radicals in a precursor of mass amx. In eq. (2), the second and third terms of the right-hand side member account for nucleation of new stable particles. The fourth and fifth terms account for the growth of stable particles of.mass m by propagation and by coalescence with precursors, respectively. The evolution of the total number of precursor and stable particles are obtained by integrating eq. (1) and (2), respectively, over the whole range of particle sizes. A finite difference scheme was used to discretize the model equations I191. The resulting set of algebraic-differential equations was integrated numerically with the code DASSL E211. 3. RESULTS AND DISCUSSION
Emulsion copolymerizations of technical grade vinyl acetate / v e o v a l 0 [22] with high solid contents (55 wt%) were carried out at 60~ and residence time of 13.3 minutes. Startup procedure was realized filling the reactor with water phase and heating until reaction temperature. Due to the high recycle ratio used (55), the residence time distribution (RTD) was similar to the RTD of a perfectly backmixed CSTR ~1. Figure 1 presents a comparison between the experimental and simulated particle number during a VAc / Veoval0 emulsion copolymerization carried out in a continuous loop reactor. It can be observed that the model is able to represent fairly well the first particle, nucleation step. The model is also able to predict the occurrence of a damped oscillatory response of particle number, as observed experimentally. Oscillations are caused by successive renucleations of decreasing intensity, before reaching a "pseudo" steady-state with an almost constant particle number. However, the model does not predict the exact experimental renucleation frequency. This is probably due to an overestimation of the number of radicals per particle. This makes the precursor particles grow too quickly, and consequently, causes renucleation to be too fast, leading to higher renucleation frequencies. The model shows that micellar nucleation is responsible for the new particles foi'med at the beginning of the reaction, but homogeneous nucleation is responsible for the oscillations observed in particle number (Figure 2). This supports what was reported by Nomura L23j, that
569 the homogeneous nucleation plays a dominant role in the mechanism for the oscillations as the oscillation phenomenon could appear more easily for water-soluble monomers such as methyl methacrylate and vinyl acetate than for water-insoluble monomers such as styrene. 1 0 1 6 [~o
]
~"...................................... 1 0 t6
r,,,--,
E _~0 ~-1 t5
O
~'~~' ~~ 1015
= o14
~ 1013
~ ' ~ Ncm Np
Nch
1012~:
==10 TM
,,,,~ 0
_13130
-"- . . . . .
.,-,~
C~
-,%,7;,-, . . . . .
1011 E" P~
"U
O
-'
"~1013 1013L
0
....
I
5
,
,
,
,
t
~
,
10
,
,
I
,
,
~
,
15
20
2
number of residence time (t/'c) Fig. 1" Evolution of particle number. (o) experimental; (--) model.
10 t~
[
4
6
8
10 12 14
number of residence time (if'c) Fig. 2: Evolution of particle number and nucleation rate. Np is the total particle number; Npp is the precursor particles number; Nl, e is the stable particles number; N c h is the homogeneous nucleation rate; N c m is the micellar nucleation rate.
Figure 3 compares experimental and simulated PSD evolution of the stable latex particles. As the model presents a faster particle renucleation frequency than experimental data, the PSDs are compared at the same nucleation state and not at the same time.
(a)
,,, 0
j
(b)
.
;;'.,~., ............. , .....
100 200 300 400 SO0 600 700 800
particle diameter (nm)
|
77:,:'.: 0
100 200 300 400 500 600 700 800
particle diameter (nm)
,.
0 100 200 300 400 500 600 700 800 particle diameter (nm)
Fig. 3" Evolution of particle size distribution in volume. ( ~ ) model' (- - -) experimental. (a) PSD before first renucleation; (b) PSD during first renucleation; (c) PSD at steady-state. It may be observed that initially a large number of polymer particles is formed' due to the presence of emulsifier in the initial charge (Figure 3a). These particles grow due to polymerization and, as they are relatively small in size, the total surface area substantially increases, adsorbing the emulsifier on their surface and depleting the system of free emulsifier. Consequently, the nucleation stopped and the number of particles decreased as
570 particles were withdrawn from the reactor. As particle withdrawal became more important than particle growth, the total surface area decreased. The probability of an oligomer to grow in the aqueous phase until reaching a critical size is higher as the total area decreases, so there was an increase in homogeneous nucleation (first renucleation, Figure 3b). After that, the total surface area increased again reducing the rate of homogeneous renucleation until the beginning of a new renucleation. This behavior proceeds until a "pseudo" steady-state is reached in which renucleations occur almost continuously resulting in a quite broad unimodal PSD (Figure 3c). 4. C O N C L U S I O N The dynamic model developed in this work was able to represent the oscillatory behavior of particle number and the evolution of PSD observed in a loop reactor. The model is able to reproduce the rather complex dynamic behavior observed during vinyl acetate / Veoval0 emulsion copolymerization reactions in a continuous loop reactor only when PSDs of two distinct particle populations (precursors and stable particles) are modeled, and micellar and homogeneous coagulative nucleation are taken into consideration simultaneously.
REFERENCES 1. Abad, C., de la Cal, J.C., Asua, J.M., Chem. Engng. Sci., 49 (1994) 5025. 2. Abad, C., de la Cal, J.C., Asua, J.M., Polymer, 36 (1995a) 4293. 3. Abad, C., de la Cal, J.C., Asua, J.M., J. Appl. Polym. Sci., 56 (1995b) 419. 4. Abad, C., de la Cal, J.C., Asua, J.M., Macromol. Symp., 92 (1995c) 195. 5. Abad, C., de la Cal, J.C., Asua, J.M., DECHEMA Monographs, 131 (I 995d) 87. 6. Abad, C., de la Cal, J.C., Asua, J.M., in Polymeric Dispersions: Principles and AppliCations, J.M. Asua (Eds.), Kluwer Academic Publishers, Dordrecht, 1997. 7. Aratljo, P.H.H., Abad, C., de la Cal, J.C., Pinto, J.C., Asua, J.M., DECHEMA Monographs, 134 (1998) 439. 8. Arafjo, P.H.H., Abad, C., de la Cal, J.C., Pinto, J.C., Asua, J.M., Polymer Reaction Engng., 7 (1999) 303. 9. Geddes, K., Chemistry and Industry, 21 (1983) 223. 10. Geddes, K., Br. Polym. J., 21 (1989) 433. 1I. Lee, D.Y., Kuo, J.F., Wang, J.H., Chen C.Y., J. Chem. Eng. Japan, 23 (1990a) 290. 12. Lee, D.Y., Kuo, J.F., Wang, J.H., Chen C.Y., J. Chem. Eng. Japan, 30 (1990b) 187. 13. Lee, D.Y., Wang, J.H., Kuo, J.F., Polym. Eng. Sci., 32 (1992) 198. 14. Kiparissides, C., Mac Gregor, J.F., Hamielec, A.E., Can. J. Chem. Eng., 58 (1981) 48. 15. Ohmura, N., Kataoka, K., Watanabe, S., Okubo, M., Chem. Engng. Sci., 53 (1998) 2129. 16. Schork, F.J., Ray, W.H., J. Appl. Polym. Sci., 34 (1987) 1259. 17. Rawlings, J.B., Ray, W.H., Polym. Eng. Sci., 28 (1988) 237. 18. Coen, E.M., Gilbert, R.G., Morrison, B.R., Leube, H., Peach, S., Polymer, 39 (1998) 7099. 19. Arafjo, P.H.H., D.Sc. Thesis, Federal University of Rio de Janeiro, 1999, in Portuguese. 20. Gilbert, R.G., "Emulsion Polymerization. A Mechanistic Approach", Acad. Press, Lonidon, 1995. 2 I. Petzold, L.R., A Description of DASSL: A Differential Algebraic System Solver. Sandia National Laboratories, Report # SAND82-8637, 1982. 22. Shell Resins, Veova Technical Manual VM 2.1, 1988. 23. Nomura, M., Chem. Engng., 39 (1994) 376, in Japanese.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
571
Process modelling of metallurgical processes- software tool and modelling concept M. Modigell a, A. Traebert a, P. Monheimb, S. Petersen c and U. Pickartz a alnstitute for Chemical Engineering, RWTH Aachen, Turmstr. 46, 52056 Aachen, Germany bSMS Demag AG, Wolfgang-Reuter-Platz, 47053 Duisburg, Germany CGTT-Technologies, Kaiserstr. 100, 52134 Herzogenrath, Germany dMannesmann Datenverarbeitung GmbH, Rehhecke 50, 40885 Ratingen, Germany A new modular process modelling tool has been developed to support the design and operation of metallurgical processes. For this purpose, a software tool has been developed and a modelling concept was formulated for the calculation of complex non-equilibrium phenomena. To test both the basic approach and software tool, the modelling and simulation of a LD converter process was undertaken. The validation of the simulation model shows good agreement of reported and calculated values for species concentration and temperature development in time. 1. THE P R O M O S Y S - PROCESS MODELLING SYSTEM The software tool ProMoSys was developed by SMS Demag AG in co-operation with Mannesmann Datenverarbeitung GmbH. The Institute for Chemical Engineering was also involved in this development. The concept of ProMoSys is influenced by traditional flowsheeting tools. But in contrast to those tools it is not a rigid software system with a proprietary interface for extension but a collection of components for the widely used programming environment Borland Delphi IDE (Inprise). This provides the whole flexibility of a 3GL programming environment connected with the ease to use point&click approach for graphical mc,delling of metallurgical (or general chemical) processes. ProMoSys consists of several components which are mainly models for unit operations and streams. Models for unit operations are e.g. stream splitter, phase splitter or equilibrium reactor. From the chemical point of view all units are connected to each other with material streams. A material stream can consist of multiple phases with different compositions. In addition its enthalpy, amount and temperature are specified. Figure 1 shows the ProMoSys components as they are installed in the Delphi Integrated Development Environment where they can be accessed by drag&drop. From these components a calculation network can be constructed without (almost) any programming. As an example, the network (flowsheet) for the LD process model constructed from the components is shown in Figure 2. The components are put on an application window per point&click and the connections between the components (e.g. a stream is input stream for an equilibrium unit) are defined by setting component properties in the Delphi object inspector. Such a calculation network can be made working by calling the standard method process of
572 the rightmost nodes in the network (usually called up from within a corresponding push button event). To make an executable WindowsNT/98 only a Delphi build (compile&link) is required. Standard editors for parameters of streams and units are predefined and callable by mouse click during program execution.
Fig. 1. ProMoSys components as available in Borland Delphi Additionally, there are programmable units of ProMoSys, which can be customized by the user according to his own needs. Using ProMoSys simple process models can be tested very quickly and without further programming knowledge. The components can easily be extended and integrated into more complex programmes. It is possible to use sub-networks as macro-components. Components to iterate complex calculations are also provided. An 'inspector' component is additionally available to analyse the network and check its formulation during design time. The models can be distributed as standalone versions without accessible source code or Borland Delphi itself.
Fig. 2. A flowsheet for the LD process model For enthalpy and equilibrium calculations, the thermochemical programmer's library ChemApp [1 ] was added to ProMoSys. Complex equilibrium calculations with ChemApp can be performed by calls to a set of interface subroutines. These routines are mainly used to
573 define the conditions for an equilibrium calculation, to execute the calculation and to retrieve information on the calculated equilibrium state (such as phase amounts or phase composition) that is needed for the process model to proceed. The equilibrium calculations in ChemApp are performed by the same Gibbs energy minimisation code as in the well-known interactive software ChemSage [2] and are thus of proven reliability. With the thermodynamic data used in the present modelling, 109 phases and altogether 202 species have been included in the calculations. The gas phase (60 species) has been treated as ideal, while the liquid Fe-phase (dilute solution approach, 14 species) and the liquid slag (GAYE-Kapoor-Frohberg model, 8 species) have been treated as non-ideal chemical solutions. 2. MODELLING CONCEPT ProMoSys was also developed to model metallurgical processes which show deviations from thermochemical equilibrium. A modelling concept was developed by the Institute for Chemical Engineering in co-operation with SMS Demag AG which enables the modelling of non-equilibrium phenomena by a specific combination of equilibrium calculations as accessible in the ProMoSys. The modelling concept is based on the general knowledge that at high temperatures reaction rates are usually high. Reaction progress can therefore be calculated assuming thermochemical equilibrium. Deviations from equilibrium, which occur in technical processes, are caused by limitations in mass and heat transfer between different parts of the reactor. In fluid systems, mass and heat transfer are dominated by macroscopic momentum transport, i.e. the flow conditions in the reactor. In solid phases, molecular transport, i.e. diffusive heat and mass transfer, has the most important influence. Based on these assumptions, the basic structure of the modelling concept is deduced. The reactor is divided into several main reaction zones. These can differ in their phase composition, being either homogenous or disperse. Inside these main reaction zones, mass and heat transfer limitations are neglected. According to the above mentioned assumption, reaction progress can thus be calculated with ChemApp assuming thermochemical equilibrium. The main reaction zones are combined by defined mass and energy transfer via streams to model the transport limitations described. With a given local discretisation into reaction zones, the mass and energy streams are defined by various models. These can be purely mechanical, e.g. to describe the macroscopic flow conditions and therefore mixing in the reactor, or specific physico-chemical, e.g. to describe complex transport phenomena close to s o l i d - fluid phase boundaries during dissolution. The mechanical models form a key part of the overall process model, as together with the choice of reaction zones they do not only describe the main flow patterns and mixing behaviour of the process, but as well the microscopic mass transfer from the bulk to the phase boundary. Additionally, as the enthalpy is a part of a stream definition, they describe the energy exchange between different reaction zones as well. 3. THE LD CONVERTER PROCESS MODEL
The LD converter process was chosen to test the modelling concept described. In the following, only a short overview of the results is given. A detailed description of the developed LD process model and all results are published elsewhere [3].
574 In the LD process, pure oxygen is blown on a molten iron bath for refining purposes. Elements dissolved in the molten iron, mainly C and also Si, Mn, P etc., and part of the molten iron are oxidised. They form a slag phase covering the hot metal. In case of C, gas bubbles containing CO and CO2 are formed. According to Figure 3, several reaction zones can be identified. In the hot spot, the oxygen directly reacts with iron and dissolved elements. Due to the impact of the oxygen jet, iron droplets are dispersed in the slag phase and slag droplets in the metal bath. The metal-slag dispersion is mixed further by CO and CO2 bubbles and serves as the main reaction zone. A third zone contains the hot metal which is not dispersed in the slag but forms the bath underneath. Droplets from the dispersion fall back into this bath. Based on these main reaction zones, the reactor is divided into sections: the hot spot, where the reaction between oxygen and iron melt takes place (hot spot reactor), the metal-slag reactor, where the conversion of the FeO from the hot spot with slag and melt droplets takes place, the metal bath (bath reactor) and additionally a slag reactor, where the slag phase is only mixed. The individual ideal reactors are combined by mass streams in a way that a circular flow through these reactors results. This models the circulation of the material in the converter and the stream of metal droplets through the slag phase as they have been observed experimentally. As discussed, mass and energy streams between the main reaction zones are defined according to process conditions. The mass flow ri'lFe,hotspot Fig. 3. The LDconverterprocessmodel between bath reactor and hot spot reactor is determined by the oxygen blow rate rio2, lance height and other geometrical parameters. Assuming the experimentally proven full conversion of oxygen in the hot spot leads to the following condition: 2. fio~-< IhF~,hotspot MFe
(1)
with MFebeing the molar mass of the iron melt. Between bath and metal-slag reactor mass transfer of metal droplets takes place. During a wide part of the process, the decarburization reaction is limited by the oxygen supply. After reaching the critical point, it is limited by the carbon transport to the reaction zone. Therefore at the critical point, Eq. (2) is valid for the mass flow rhFe,metal-slag of metal droplets:
575
2 flo2< N. V "
-
"Co =
lq. V . c o =
t'
l'i'lFe,metal-slag "co
(2)
PF~
with N being the number of droplets, V being the droplet volume, t' being the droplet residence time in the slag, Co being the critical carbon content, PFe being the density of the iron melt and lq being the droplet stream. These mass flows rilFe,hotspot and rnFe,metal-slag are a function of the oxygen blowrate and -conditions and gas production, which are constant over the main process time. Towards the end of the process, gas formation depends on carbon concentration and therefore the mass flowrates change. Not only material streams to and from ideal reactors are defined in the model but energy streams as well. Energy exchange between the reactors is calculated corresponding to mass exchange. Energy losses from the converter mouth due to radiation, radiation and convection losses through the converter walls and the temperature change of the converter wall are as well modelled. 4. SIMULATION RESULTS To validate the process model, published data from literature [4] were used for comparison. Process conditions were assumed accordingly. Based on these conditions, model parameters have been calculated from Eqs. (1) and (2). Only little further adjustment was necessary. The simulation results for hot metal composition in time are shown in Figure 4. 0,045 ~, 0,04 -I 0,035 0,030,025 .
i
2,5E-04 2,0E-04 =
1,5E-04 ~ o
0,02
1,0E-04 6
~ 0,015 0,01
5,0E-05
0,005 0
0,0E+00
-"
0
5
10 time [mini
15
--~ C-content measured [4] --~ C-content calculated --~ Si-content measured [4] --~ Si-content calculated --~ Mn-content calculated -.x- O-content calculated
Fig. 4. Simulation results- hot metal composition The decarburisation reaction is modelled very well. The amount of Si calculated in the metal phase also shows a quantitatively good agreement with measured values [4]. Additionally, the contents of Mn and O in the metal phase were calculated. Both materials show a qualitatively good agreement with measured values. As discussed above, the ideal reactors in the cell model are not only coupled by defined mass transfer but by heat transfer as well. In Figure 5, the temperature development in the hot spot and of the mean temperature of the hot metal are shown. The temperature in the hot spot is slightly overestimated. Measurements give a value of 2300 to 2400 ~ for the first 8 minutes of the process [5]. The mean temperature shows a good agreement with experimental results [4].
576
28002600o~'2400 ~ 2200 2000 1800 ~1600 14001200
hot spot calculated --~ hot metal calculated --~ hot metal measured [4]
0
5
10 time [mini
15
Fig. 5. Simulation results - temperature development 5. CONCLUSIONS The aim of the research presented in this paper is the development of a software tool and modelling concept for metallurgical processes. In the long term, the process models shall be used to improve process design. A software tool was developed which allows the easy generation and testing of process models. A modelling concept was applied according to which a process is divided into a limited number of reaction zones. Inside these reaction zones, thermochemical equilibrium is assumed. Mass and energy exchange between the zones are defined based on process conditions and physical models. Therefore the number of model parameters can be kept very small. It is the target to develop a model with a relatively "simple" structure which at the same time is able to calculate the main physical and chemical phenomena of a process. The LD converter process was modelled following this intention. Results for hot metal composition and temperature development show a good agreement with measured values. It is intended to extend the current research to obtain a tool not only for the design and simulation of established processes, but also for the computer-aided development of new process routes. REFERENCES
1. G. Eriksson, K. Hack, S. Petersen, Werkstoffwoche '96, Symposium 8: Simulation, Modellierung, Informationssysteme, J. Hirsch (ed.), DGM Informationsgesellschaft Verlag 1997, ISBN 3-88355-236-4:47 - 51 2. G. Eriksson, K. Hack, Metall. Trans. 1990: 21B: 1013 - 1023 3. M. Modigell, A. Traebert, P. Monheim, K. Hack, Development of a modelling technique for non-equilibrium metallurgical processes, Proc. of the 1st Int. Conf. on Process Development in Iron and Steelmaking, Lulea, Sweden, 1999: 2:201 - 211 4. S. Asai, I. Muchi, Transactions ISIJ 1970:10:250 - 263 5. K. Koch, W. Fix, P. Valentin, Archiv f'tir das Eisenhtittenwesen 1976: 47:583 - 588
EuropeanSymposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V.All rightsreserved.
577
Modelling High Pressure Extraction Processes Mojca Skerget, Zeljko Knez* University of Maribor, Faculty of Chemistry and Chemical Engineering, Smetanova 17, SI-2000 Maribor, Slovenia phone: +386/62/22 94 461, fax: +386/62/22 50 13, E-mail: [email protected]
The objective of this work was to study the semibatch flow extraction of oil and other active ingredients from some plant materials (Silybum marianum, pepper, paprika and cocoa) with supercritical fluids such as CO2 and n-propane at different operating conditions and to analyze the dynamic behaviour of the extraction runs by a mathematical model employed by Peker [1] and Goto [2]. 1. INTRODUCTION High pressure extraction process represents an alternative to conventional separation methods (steam distillation, extraction with organic solvents, molecular distillation,...), because of favourable properties of supercritical fluids (SCF) (solvent recovery, simple separation, favourable thermical conditions, solvent free products of high added value, etc.). Nowadays, extended research work on the application of SCF as solvents in the extraction and fractionation processes of essential oils and aromatic components from plant materials is carried out. SCF represent natural alternatives to chloro- and fluorocarbons and other ozonedepleting or smog-causing compounds. Their greatest success to date has been as replacements for chlorinated solvents in coffee decaffeination and spice-extraction processes [3]. For most supercritical applications, CO2 is typically employed. CO2 is ideal for applications in foods, beverages or pharmaceuticals because it is nontoxic, nonflammable, inexpensive and widely available [4]. For the design of high pressure extraction process, beside the knowledge of phase equilibria, the knowledge of mass-transfer rates is essential. The problem, which persists in dimensioning the SCF-processes, is that usually no physico-chemical and transport data of the investigated components are available in the literature and are difficult and timeconsuming task to measure experimentally. Therefore they are usually estimated with different group contribution or empirical methods. Further, in most cases conventional models for modelling phase equilibria and extraction rates in dependence of pressure does not fit the experimental points well because of extreme operating conditions. To solve the problems conventional models have been modified or new models have been developed. However, the problem has not been completely solved yet. *To whom correspondence shouldbe addressed.
578
2. M A T H E M A T I C A L M O D E L The model used by Peker [ 1] and Goto [2] is based on the following assumptions: 1. adsorption - desorption equilibrium of extractable component from solid tissue, 2. the diffusion of extractable component dissolved in supercritical fluid to the surface, 3. mass transfer through the external film into the bulk. The final expression for the commulative fraction of a solute extracted up to dimensionless time | is defined as [ 1,2]:
1~Ix- dO
F(|
i2ao
I
A exp(~-| = 1-a L a,
- 1)
exp(a ! _.O) - 11 a 2
(1)
J
and the final expression for dimensionless solute concentration x = -
C
in effluent is:
Co
x(o> = A[exp(a,-Ol- exp(a,.Ot]
(2)
where c is the concentration of a solute in a solvent, co is the initial concentration of a solute in material and: t 0 =(3) 1;
1(
)
(4)
4:C)
(5)
al = - ~ - b + ~ / b 2 4.C
a 2 =-~(- b - 4b 2
(1-cz).O A = LP r" + '~l - " "lJ> K 1 " Ja " a , -t a 2 )
b=
9 1 0 ( 1 - c~) + - - + ~ [3 + (1- ~).K ot ot
9
c = [13+(1-13).K].c~
(6)
(7)
(8)
F approaches unity at large values of time. a is the void fraction in bed and 13 the porosity of particle. The equilibrium adsorption coefficient K is defined by equation: % =K.ci
(9)
579 where: Cs is sorbed essential oil in the solid particle and ci is the concentration of essential oil in the pore of solid particle. For K<
(10)
where time x is the total bed volume divided by the volumetric flow rate of supercritical fluid, kp is the combined mass-transfer coefficient, given for the sphere by equation: 15.kf/R kp=
5+Bi
(11)
where kf is the external film mass-transfer coefficient and R is the sphere radius. The Biot number Bi is expressed in terms of the effective intraparticle diffusion coefficient De: Bi- kf-R De
(12)
Do =Dm'13 2
(13)
where DAB is the binary diffusion coefficient for essential oil in supercritical fluid. When Bi >> 5, intraparticle diffusion resistance would dominate over the external mass-transfer resistance [ 1].
2.1. Estimation of properties for theoretical analysis. The size of the solid particle was determined with the sieve analysis and the density of solid material was measured with helium pycnometer (multi volume pycnometer 1305, Micrometrics, USA). The bed void fraction ~ was 0.26 and the porosity of the particle [3 was calculated from solid and apparent density:J3 = 1 - p p / P s . The estimation for the initial concentration of extractable substance in the material co was obtained experimentally with the extraction run until all extractable substances were removed. The binary diffusion coefficients DAB were estimated with Takahashi method [5] in consideration of Fuller equation. For liquid propane at 40~ the binary diffusion coefficient was calculated with Wilke - Chang estimation method [5]. The external mass-transfer coefficients kf were calculated with the Wakao and Kaquei correlation [ 1,2,6]. For the calculation the FORTRAN was used and the adsorption equilibrium constant K was calculated with the regression of experimental data. 3. RESULTS Figures 1 and 2 show the comparison of experimental and calculated extraction curves and Table 1 presents average absolute relative deviation (AARD), calculated for each extraction run.
580
12]
1.2
t 40~ - - - 40~ A 40~ 40~
"
1 O
~0.8
9 9
~ 0.6
0.4
o
bar)-exp. bar-calc. bar-exp. bar-calc.
o.8
0.6 = o 0.2
150 150 400 400
bar-exp. ~'
- - 80~ i
0
..................
"~0.4 0.2
475 bar-calc.
!
!
!
10 20 30 kg CO 2 / kg material
0
t
40
i
0
I
I ........
10 20 30 kg C02 / kg material
a)
i
40
b) 65~ 9 100~ )r 100~ - - - 65~ 100~ ...... 100~
1.6 2"1.4 =o 1.2
480 bar-exp. 480bar-exp. 300 ba r-exp. 480 bar-calc. 480 bar-calc, 300 bar-calc.
! J [ [ l 1
;0.8
~ 0.6
~ 0.4 g 0.:2 0
~.~.~.;~.:~..~-" i'" ~ ,r'r--
1
0
i"l
I
f
20 40 60 kg C02 / kg material
80
c) Fig. 1. Kinetics of semicontinuous extraction of a) pepper, b) p a p r i k a [7] and c) c o c o a butter f r o m c o c o a w i t h d e n s e CO2.
I ~= 2 -] /
".~.~
[] A 9 0
25~ 40~ 60~ 80~
=
i.e
~-~--~-X X,~
--FI
_J ....... 2 5 ~
~1.5/I . . . . . 40~ /l - - 60~
~ 0.6
1 d[ - - 7 80~
[] 40~
60O exp
0.4
~0.5
[] 80~ 40~ ....... 60~ 80oC.calc.
8 0.2
0
0
0
20 40 60 kg CO 2 / kg material
a)
80
w
0
5 10 kg propane / kg material
b)
Fig. 2. Kinetics of semicontinuous extraction of seeds of Silybum marianum with a) dense CO2 at 200 bar and b) dense n-propane at 60 bar.
15
581 Table 1 Extraction conditions and estimated parameters. T (~
P (bar)
kf 105
Qv (l/h)
(m/s)
k (s ~)
DAB 109
(m2/s)
De 1011 (m2/s)
K
AARD
11.16 7.87 82.10
12.9 4.1 9.5
(%)
Carbon dioxide: Cocoa D = 0 . 0 1 6 7 m m , [3=0.2, c0=12% 65 100 100
480 480 300
34.02 38.36 46.82
0.750 1.661 3.177
2.165 4.770 9.049
1.32 2.85 5.24
Carbon dioxide: Paprika D = 0.165 mm, 13= 0.1, C0(aromaticcomponents) = 12.35%, 40 40
150 400
36.1 29.5
0.114 0.031
0.030 0.009
5.28 11.4 20.96
C0(coloringcomponents) "-
4.999 1.54
1.85%
4.999 1.54
18.77 11.85
11.9 13.0
39.08
5.70
3.8
73.2 106.4 183.5 249.4
175.49 176.56 375.92 849.52
8.4 8.1 5.9 17.7
110.9 88.0 229.5
16.41 15.87 60.42
2.1 9.9 18.3
Carbon dioxide: Pepper D = 0 . 2 5 m m , [3=0.3, co=6% 80
475
32.6
0.621
0.107
4.342
Carbon dioxide: Silybum mariannum D = 0.9 mm, [3 = 0.635, co = 23% 25 40 60 80
200 200 200 200
0.46 0.42 0.51 0.54
0.356 0.521 1.006 1.433
0.017 0.024 0.045 0.063
1.814 2.639 4.550 6.184
n-propane: Silybum mariannum D = 0.9 mm, [3 = 0.635, co = 23% 40 60 80
60 60 60
0.82 0.67 1.11
0.570 0.447 1.486
0.026 0.021 0.063
2.751 2.183 5.692
100 ~-~ ]yieldcalc - yieldexp ] AARD(%) = --N" i=z yieldexp Average absolute relative deviation (AARD), calculated for the extraction of pepper, paprika and cocoa with dense CO2 is in the range from 3.8% to 13%. In case of CO2 extraction of Silybum marianum, AARD is under 10% (from 2.3% to 8.4%) except at conditions 40~ 100 bar and 80~ 200 bar, where the yield of extraction is relatively low, adsorption constant K is high and AARD is 23.1% and 17.7%, respectively. In case of npropane, AARD is low at 40~ (between 1.8% and 3%) and with the temperature increase it varies between 1.8% and 20.2%. Due to higher errors observed for modelling extraction runs performed with n-propane at 60~ and 80~ it seems that Takahashi method (in consideration of Fuller equation) used for estimation of the binary diffusion coefficients DAB
582 is not adequate for propane gas extraction system. The errors when Wilke-Chang equation was used at 40~ are much lower. It can be concluded that the model approximates the experimental data well when CO2 is used for the extraction and when operating parameters are chosen so that the adsorption equilibrium constant is not too high and a desorption of the solute from the solid tissue is enabled. Errors could be the consequence of a fact that not only active ingredients were extracted in the process, but also some other components such as waxes, fats .... Therefore, the estimated initial concentrations presented in Table 1 are larger than the concentrations found in the literature. Table 1 presents the estimated mass transfer parameters. The adsorption equilibrium constant K changes with temperature and pressure. Generally, at constant pressure K decreases with the increase of temperature and at constant temperature K decreases with the increase of pressure. An exception can be observed for the extraction of Silybum marianum with CO2 at constant pressure 200 bar, where K increases with the increase of temperature. The values of K are generally lower when propane is used as a solvent for the extraction of
Silybum marianum. Binary diffusion coefficients in mixtures of SC gas and low volatile component calculated are in the range from 0.1 x 10-9 to 7.1 x 10-9 m 2/ s and combined mass-transfer coefficients vary from 0.001 to 0.083 s -1 for paprika, pepper and Silybum marianum extractions and are higher in the case of cocoa butter extraction from cocoa, where it varies from 2.2 to 9.0 s~.
REFERENCES H. Peker, M. P. Srinivasan, J.M. Smith and B. J. McCoy, AIChE J., 38,5(1992), 761-770. M. Goto, M. Sato and T. Hirose, J.Chem.Eng.Japan, 26,4(1993), 401-407. C. Chin, C. Crabb, G. Ondrey and T. Kamiya, Chem. Eng., October 1998, 32-41. 2;. Knez and A. Ris Some Novel Applications of Supercritical Fluids in Food Processing, Engineering & Food, Sheffield Academic Press, Part 2, (1997) pp I/5-I/8.; at ICEF 7, Sheffield, UK 5. R.C. Reid, J. M. Prausnitz and B. E. Poling, The Properties of Gases and Liquids. Fourth Edition, McGraw-Hill Inc., New York 1987, p.587. 6. G. Brunner, Ber.Bunsenges. Phys. Chem. , 88(1984), 887-891. 7. M. Skerget, 2;. Knez, Z. Novak and D. Bauman, Acta Alimentaria, 27,2(1998), 149-160.
1. 2. 3. 4.
EuropeanSymposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V.All rightsreserved.
583
Waterless wool cleaning process with supercritical carbon dioxide: extractor modeling and optimisation F. Trabelsi, J-C Luc, J. Miquel, M-A Larrayoz, M. Capilla, F.Recasens Department of Chemical Engineering. Universitat Polit~cnica de Catalunya E.T.S.E.I.B., Diagonal 647, 08028-Barcelona, Spain e-mail: [email protected] 1 INTRODUCTION A supercritical-fluid extraction plant is usually a very large investment. For example, a 10000-ton-per-year coffee decaffeination facility may well involve a capital expenditure of more than 15 MEuro [1 ]. Therefore, convenient procedures for designing certain items of the plant (i.e., desorption vessels), are necessary for both the user and the supplier. In this work, we address the problem of process synthesis for a wool-treating plant whose activity is also to produce lanolin in semi-batch operation. The solvent used is a supercritical mixture of carbon dioxide and a co-solvent [2]. Our methodology involves two stages: modelsolving and plant design. Firstly, a mathematical model for the semi-batch operation of the compressed wool bales is developed to predict solute extraction times as a function of the operating variables. From these results, two strategies are possible. The first one is based in a computer-generated, off-line, statistical correlation that predicts regeneration times as a function of bed volume or axial length and fractional recovery. Desorption rates, which depend on the type, and nature of the solutes (lanolin, and waxes) are assumed to be those studied before [3]. The other strategy is based on an on-line, direct numerical solution of the governing partial differential equations that describe desorber dynamics [4]. 2. RIGOROUS EXTRACTION PROCESS MODEL A rigorous process model is used to calculate the operating time for the semi-batch extraction of raw wool. The wool is loaded into the extractor and compressed either mechanically or by the action of the pressure differential across the bed. In either case, a model relating the compression stress to the mass balance equations is necessary. In order to develop such a model, the following assumptions are made: (a) dissolution of the lanolin and wax materials deposited on the wool fibres occurs at the fibre-fluid interface by mass transfer; fibres are distributed at random in the bed; (b) operation is isothermal; (c) the concentration of lanolin at the fibre surface equals its saturation value in the solvent; (d) wool fibres are assumed to be thin cylinders of constant diameter; (e) while axial dispersion in the bed is considered in the model, transverse dispersion is assumed to be very fast; and (f) the time for the pressure drop to establish is small compared with the extraction time, so that the mechanical energy equation is directly written for the steady state. The conservation equation of lanolin in the fluid phase can be written as follows:
OC=Dz ~--Z 02c -u~ Oc 4kf (l-t;) -E --(c Oz
df
t;
-c*)
(])
584 Some compression by the fluid on the wool during extraction was observed. Therefore, the porosity in equation 1 is allowed to change during the first minutes of extraction, due to the pressure drop. The drag force of the fluid on the fibres can be calculated using the Ergun equation for pressure drop. Neglecting the elevation and velocity changes in the bed, the change in pressure is given by: Ap 150 (1 - c) 2 #u~ + 1.75 (1 - ~) PU02 L = (p~F) = ~3 de2 - 7 de
(2)
Where de is the equivalent diameter of the fibre, for which the sphericity factor is about 0.58. In our case, Re < 20, so only the first term on the right-hand side of Equation 2 will be needed (viscous contribution). However, Equation 2 covers a wide range of Reynolds numbers. The fibre assembly is expected to deform because of the forces exerted by the fluid. The frictional force of the fluid acting on the fibres is balanced out by the frictional pressure energy lost by the fluid. If Xz is the stress based on the unit external area of the fibres, equating the force per unit bed volume to the pressure drop force gives: Lx ~;-------~) 4(1 = (-Ap)e z df
(3)
Also, the relationship between the stresses, Xx, Xy and Xz, with the volumetric deformation, is given by the generalised Hooke's law as [5]: AV_Iz-lz o _(1-2v Vo
1- ~
E
(4)
hz
where eo and e are the bed porosities at the beginning of the experiment and during the steady state respectively. In Eq. 4, v and E are Poisson's modulus and the modulus of elasticity for the wool fibre assembly. The compression coefficient appearing in Equation 4 was obtained by measuring the stress-strain behaviour in separate experiments, using a tensometer equipped with extensometric gauges. The measured coefficient is a combined constant of the elasticity and Poisson's modulus, as seen in Eq. 4. The initial and boundary conditions for the above equations are as follows: c(t=0, z)=0
c~c z=o,t = 0 - e D z ~zz
- ~ z=,,t
The operating time for the extractor is calculated by integrating the differential equation (1) with the boundary conditions (5). The breakthrough curve is obtained in terms of the concentration of lanolin at bed exit as a function of time. During a typical semi-batch extraction run, the lanolin concentration in the fluid at bed exit is about constant. In order to carry out the integration it is necessary to calculate the porosity since it appears in Eq. 1. The calculations are thus initialised with regard to porosity. The initial porosity is known as noted by Eq. 5. To solve for the steady state porosity, the pressure drop is first evaluated with the Ergun equation. Then the stress Tz is calculated using Eq. 3, and an updated value for e is obtained from Eq. 4. From this value, iteration proceeds until convergence. This calculation is done before starting the integration of Eq. 1. Later, when extraction is already in process, it is assumed that porosity and other elastic and mass transfer parameters of the wool assembly remain constant.
585
3. DEVELOPMENT
AND
DESIGN
OF THE
PROCESS
The process consists of several important stages, which take place before and after the extraction process itself, these are: - the mixture of the solvent (carbon dioxide) with the co-solvent - the separation of the solvent from the co-solvent and from the extract - the recycling of the solvent The solvent mixture composition is controlled by regulating the flows of the of carbon dioxide and co-solvent pumps. These pump the CO2 and the co-solvent in liquid state by compressing them to the required working pressure in the extractor. Before it enters the extractor, the mixture is heated in a heat exchanger. The preliminary experimental study carried out with 80% CO2 and 20% co-solvent has shown us that the extraction of the lanolin is favoured in subcritical conditions. A key stage in the extraction process is the separation of the solute from the extraction mixture and the recirculation of this mixture. It is proposed to carry out the separation according to the following outlines: In the first version presented here, two separators are used. In a first separator at the pressure of the storage tank of CO2 we obtain a gaseous phase consisting mainly of CO2 with a weak percentage of co-solvent, and a liquid phase containing co-solvent, dissolved CO2 and lanolin. The lanolin is filtered at atmospheric pressure and ambient temperature. A second separator at low temperature (~ 5 ~ is used for the recovery of the CO2 dissolved in the cosolvent. Given that the recovered carbon dioxide is at atmospheric pressure, a liquefaction unit is then necessary to restore the gas to the pumping conditions (65 bar and 25 ~ The second version proposed in this work consists in a single separator process and without liquefaction unit.
Material and energy balances: process optimisation Before establishing the material and energy balances for the processes, the following data are required: solubility of lanolin in the CO2-co-solvent mixture; production of wool per unit of time in the extractor and an equation of state to describe the liquid-vapour equilibrium of the process streams.
Solubility of lanolin in the fluid The slope at the origin of the extraction curves can be taken as the solubility of the solute in the supercritical mixture. The following Table 1 and 2 summarises the necessary data for simulating the extraction process of lanolin from wool in a medium of CO2-cosolvent at high pressure and the operating conditions in the seperators 1 and 2. With these data, the Hysys software can be used to simulate the separation of lanolin from the high-pressure CO2-Co-solvent mixture and the recirculation of the extraction fluid towards the extractor for its re-use. For optimisation purposes, the chief variables defining the problem were the operating conditions of the two separators Finally, we can sum up the process of the version 1 in the following important steps: the extraction process (similar to that of the Fig. 2, but with two separators which operating conditions are given in table 2); the filtration-distillation unit where lanolin is recovered from the co-solvent (flowsheet 200 [6]) and the liquefaction unit of the CO2 (flowsheet 300 not shown [6]).
586 Y" % lanolin extracted compared with the total lanolin contained in the raw wool '
'
i
'
i
'
i
'
100~
? //
4O
0
z
"
'
i
................... .... .
60
I
0
20
j
j
.
l
'
i
'
i
'
T
t ~
111 84
~
,~
............
9Subcritical conditions entrainer 1(20%) w/w 9subcritical conditions entrainer 1 (10%) w/w Subcritical conditions entrainer 2 (20%) w/w ............
40
60
80
i
100
'
i
120
'
i
140
9
i .........
160
t (min)
Fig. 1" Percentage of lanolin extracted in dense CO2-co-solvents, compared with the total as a function of time and % of co-solvents. Table 1 Parameters of theprocess text (min) 150 Extractor volume (L) 400 Wool density (kg/m3) 500 Quantity of lanolin/(kg ofwool) 0.1 Flow of solvent (kg/h) 1400 Quantity of lanolin extracted (kg/h) 7.56 Flow of CO2 (kg~ 1120 Flow of Co-solvent(k~qa ) ....................... 280.........
Table 2 Conditions in the flash separators Separator .... 1 T (~ 80 P (bar) 70 Q fluid, input (kg/h) 1418 Q liquid, output (m3~) 0.56 Q gas, output (m3/h) 1.17 Heat (kW) ........................ 57
2 5 1 433 0.3 0.24 9
4. INDUSTRIAL PROCESS DEVELOPMENT Next we develop alternative operating strategies that will be necessary in the scale-up of the process to an industrial plant. One of such modifications refers to the fluid recirculation rate. This is a critical variable from the viewpoint of the operating mode of the plant and its costs. From a strict economic point of view, reciculation rate increases capital costs. Consequently, depreciation charges have to be paid off by the savings in both solvent expenditure and in decontamination costs. From an environmental perspective, the net effect of using recirculation is to minimise effluent waste streams. The following factors would favour the first alternative: Pressure drop in separation units. In the f'~st alternative we proposed a process with a second flash separator operating at 1 bar. This alternative involves a downstream liquefaction plant to recover pure carbon dioxide (CO2 with 2 %wt co-solvent) for recycle. This alternative obviously increases capital costs as well as energy costs. Control of recirculation rate. At the outlet streams from the two flash drums, two gas streams are formed. This makes control difficult since the plant operates in semibatch mode, hence with variable flowrates in the two streams. Therefore the recirculation flowrate may be variable as well, hence making control difficult.
587 In order to solve the above problem, the process was simplified. The first modification is the use of a single separator operating at a pressure close to that of the CO2 return line, i.e., the line where recirculation stream is fed. The other modification is the elimination of the liquefaction plant completely. This will help reduce CO2 losses in the lanolin recovery. With these two modifications capital costs are also reduced. The last operating mode was checked experimentally in our pilot plant Separex 200 extractor. The following flowsheet (Fig. 2) would give an idea of a simplified industrial process in which investment is kept to a minimum. In this process, essentially pure lanolin is recovered by filter F. Some desorbed CO2 from the liquid is discharged to the environment, and the co-solvent is recovered for further purification. Process mass balance simulation provides the following data (for the same conditions of the alternative in Table 1). ~...
Co-solvent
l.......
Table 4 Operat!n ~ conditions in the separator 1' 2* T (~ .................90 65 P (bar) 65 65 Qin (kg~) 1418 1418 Qout liquid (Nma/h) 0.3518 0.2058 Qout gas (Nm3/h) 1.3430 1.5170 1*80% CO2+20% c0solvent 2*90% CO2+ 10% cosolvent
[co~I
~"
HL-~3
cmt~
/
ex~-solventrecite
....... C02-eo-solvent
L
.....................
....L _ lanolin T
Fig. 2: Flow diagram of the industrial plant for the supercritical fluid extraction of lanolin from row wool. Note that there is a 5%wt co-solvent in the gas stream (representing a 4% based on the 20% of the mixture), together with a loss of CO2, which is about 7% based on the inlet feed. In order to decrease CO2 losses, we propose to reduce the fraction of co-colvent to the minimum possible without affecting too much extraction kinetics and yield. In fact, the fraction of cocolvent is a variable that is subject to optimisation. The results with 10% co-solvent are very similar. Only a difference of 5% less lanolin is extracted over the same extraction period. These results can be simulated by running the rigorous model for the extraction process explained above. The results obtained with different fractions of co-solvent in the feed, provide the operating conditions for the separator, presented in the table 4. The CO2 losses are about a 3% of its inlet feed value. The cosolvent recycled in the gas stream is about 2.5% w/w (or a 2.3 % on the 10% of the mixture). It also could be stressed that in this process, 50% of heating energy in the separator, and 8% refrigeration energy in the condensation of the vapour stream from the separator are saved. These energy savings are possible because of a lower vaporisation energy for a lower cosolvent fraction in the SC solvent.
588 5. CONCLUSIONS In this work, processes for the waterless cleanup of raw wool have been presented, with emphasis on lanolin extraction. Although this was a preliminary work, some conclusions can already be drawn. Even though the conventional process of wool treatment has some inherent advantages such as the following : 1) short cycle or process time (about 1 hour including drying operations), 2) Flexibility to reach a peak production of more than 80 tons wool/day. 3) Process well fitted to downflow operations such as combing and spinning of the wool; some of its drawbacks are evident. These are: 1) While only a small fraction of the lanolin is recovered (15%), the top quality lanolin is not extracted. 2) Very large water usage. 3) Wash liquors containing a dissolved organic contaminant load of about 100 000 g/m3, high suspended solids concentration, and a concentration of chemicals derived from detergents that are impossible to dispose. 4) Increase of the sludge production in the wastewater treatment plant. The process we propose, on the other hand, cannot reach very high productivities, because a large investment would be necessary for capacities exceeding the 5-10 tons wool per day, where large volume extractors operating at high pressure would limit the plant size. However for production rates in the range of 2 ton/day (with a large cycle time) would be attractive in view of the following considerations: 1) Increase in lanolin extraction efficiency. An 85-99% based on total lanolin could be obtained, by properly adjusting process time. The extract would be of higher purity and top quality, hence with a larger market value. 2) Downstream lanolin processing would allow further refining for certain cosmetic and pharmaceutical use. 3) Higher quality of washed wool, possibly free from pesticides, could also find novel high-value added applications. Work is under way to develop the speciality-type of production associated with the use of supercritical fluids. 6. ACKNOWLEDGEMENTS The autors acknowledge the fellowships received from the Spanish Ministry of Culture and Education. Research funds were provided by the CICYT-European Regional Development Funds, 2FD97-0509-C02-02 (Madrid, Feder, Brussels) and CICYT (Madrid, Spain), QUI980482-C02-01. REFERENCES [1] Layerc, W.E., Novak, R.A., Lining, D.A., "The Economic of Supercritical Coffee Decaffeination", Proc. 2nd. Intl. Symp. on Supercrit. Fluids, Boston, 1991. [2] K. Abaroudi, F. Trabelsi, B. Calloud-Gabriel, and F. Recasens, Mass transport in modified supercritical fluid, Ind.Eng.Chem.Res., 38, 3505-3518, 1999. [3] Jones, F. W., Bateup, B. O., Dixon, D. R., Grey, S. R., Solubility of wool wax in supercritical carbon dioxide, J. Supercritical Fluids, vol. 10, p 105, 1997. [4] Akman, U. and Sunol, A., "Modeling Supercritical Desorbers with an Equation-of-statebased Isotherm", AIChE J., 37, p. 215, 1991 [5] A. J. Bonnin, Elasticidad teoria, Cpda, UPC, Barcelona, 1992. [6]J-C., Luc, L'extraction par CO2 supercritique au service du lavage de la laine de mouton: 61aboration d'un proc~d6 innovant, Intemal report, Barcelona, Spain, September, 1999.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
589
Equation Based SPYRO | Model and Solver for the Simulation of the Steam Cracking Process Marco W.M. van Goethema* , Florian I. Kleinendorst a, Cor van Leeuwent & Nils van Velzen a aTECHNIP BENELUX B.V./Pyrotec Division, P.O. Box 86, 2700 AB Zoetermeer, The Netherlands. The explicit SPYRO | model is transformed into the residual form using orthogonal collocation on finite elements. An effective and flexible system of sub-models is developed for the Open Spyro model. The Open Spyro model is solved using a fast and global converging quasi Newton method based on the update proposed by Broyden. The simulation gives the same reliable results as the original SPYRO | model. ]. INTRODUCTION SPYRO | Technip's proprietary yield prediction program for the steam cracking process, has been functionally extended and improved over the years (Dente et al. 1993, Ranzi et al. 1983). It has become a well established tool for the purpose of feedstock selection, optimal ethylene furnace operation and is for Technip one of the key instruments for the design and revamp of cracking coils. The reaction kinetics details have progressed over the years and will continue to be improved. A new kinetic scheme is being developed with an increased amount of components and reactions (Dente and Ranzi, 1999). The flexibility nowadays required of process models initiated the development of the so-called Open Spyro program. In this, all model equations are written in the residual form. It is the flexibility of the definition of the simulation problem that favours the residual form. The very same form can also be used for data reconciliation and optimisation. An easy link with other equation based models and solvers can be established. Program maintenance will be less complicated due to the clear separation of the model equations and the solver.
2. PROBLEM DEFINITION The core of the SPYRO | model is the kinetic reaction scheme. The most recently released scheme consists of 3288 reactions, involving 128 components and 20 radicals. The components vary from hydro-carbons with one C-atom up to 42 (3atoms. The heavier components are mostly lumped, e.g. all isomers of i-octane are represented by one component. * Corresponding author. Tel: +31 - (0) 79 - 3293 631, Fax: + 31 - (0) 79 - 3513 561 E-mail address: mvangoethem @technip.com. 1 Cor van Leeuwen passed away during the project on 28 February 1999.
590
The reaction scheme consists of several types of reactions: radical chain initiation, metathetical (hydrogen abstraction), radical decomposition, radical addition, radical chain termination, radical isomerisation and purely molecular. In the reaction scheme lumping has been applied as well for very fast reactions (e.g. isomerisation or decomposition of heavy radicals). In fig. 1 an example of the simplification of the hydrogen abstraction of normal-octane is given. f .
~
~
C2H4+ .
~
I
R~ + n-C8H18------~ RH +
"~'~~ ~
I
C2H4+ n-C4Hg"
'-....
Film layer
o,ymer,ayer Coke deposit
-~
C2H4 + n'C3H7"
-~
C3H6 + C2H5~
Tube wall Fouling film
I-C4H8+ n-C4H9~ <~
I Ro+n-CsH18 ~
C3H6 +
C3H6+ n-C3H7~
~
~ , C4H8+ n-C2H5 9 I-CsHIo+ CH3~
< ~ I-C6HI2+ C2H5" I-C5HI0+ n-C3H7~ I
~
'
1-C7H14+ CH3 ~
1 alC2H4 + a2C3H6 + a3C4H8 + a41-C5H10 + a51-C6H12 + a61-C7H14+/ + a7 CH3~ + a8 C2Ht + a9 n-C3H'~ + alo n-C4H'9
Fig. 1: Simplification of the metathetical of n-octane
/
Fig. 2: Schematic of layer build-up.
The plug flow reactor model is used to describe the steam cracking coil. The 148 balances for the components, the energy balance and the momentum balance are ODEs conform this model type. A detailed layer build-up, as shown in fig. 2, is used to correlate the outside tube skin temperature to the process gas temperature. Therefore several nonlinear equations are implemented, namely: equations for the heat transfer coefficients for different flow regimes, equations for the cracking that takes place in the film layer (De Blieck and Goossens, 1971), equations for the coking rate estimation (Ranzi et al., 1985), equations for the heat flow through the tube wall and equations for the estimation of the wall-friction. The complete SPYRO | model is a DAE system that contains a total of 175 ODEs and 50 AEs. 3. OPEN MODEL DESCRIPTION The model is written in the open or residual form which allows flexibility and the formulation of the basic equations in their natural form, reducing coding errors. The flexibility and the natural form are created by dividing the SPYRO | model into several sub-models. Besides the standard models (feed, mixer and sink), we created the next sub-models: 9Spyro Entrance (SPYENT): defines the inlet of a cracking coil; 9Spyro Section (SPYSEC): describes sections of the coil with the same geometric variables; 9Spyro Node (SPYNODE): connects two Spyro Sections; 9Spyro Exit (SPYEXIT): defines the outlet of a cracking coil; 9Adiabatic Zone (ADIAZONE): defines an adiabatic tube. These sub-models are connected by streams. Outside the firebox those streams are characterised by the conventional combination of temperature, pressure and composition. However, inside the firebox other variables related to the heat flow through the tube wall are added. Besides the flexible formulation of all kinds of coil
591
geometry's, it is rather simple to add new features and new sub-models for a more detailed description of coil parts. Here we present an example illustrating the flexible definition of a coil geometry. A typical scheme of a two-pass cracking coil is given in fig. 3. The first pass consists of two parallel tubes, the second of only one tube. This cracking coil contains a lot of modelling discontinuities: the coil entrance, the splitting and mixing of parallel tubes, the bends, the coil exit and the adiabatic zone. Feed
Feed
Effluent
Transfer line volume (adiabatic zone)
.....
[1
Feed
[
"~ Mixer ............. 1 Firebox -~,:,, [ SPX~ENT
Firebox
..........I
Sink
[ (Hydro Carbon) ]
(steam)
I SPYSEC l
I j
[
[ ADIAZONE ] T ]'"l sPYExIT T I"" I
[ sPYSEC2
[
L-~ SPYNODE 1
....................
Fig. 3" Schematic overview of a typical cracking coil.
Fig. 4: Straightforward configuration of coil in fig. 3.
Each pass of the cracking coil shown in fig. 3 is described by a Spyro Section. The first pass consists of two identical tubes of which only one is simulated. When desired it is possible to model both tubes separately. The most straightforward configuration contains two Spyro Sections. The Spyro Node accounts for the pressure drop due to the appendages. This configuration is shown in fig. 4. When a more detailed modelling of the geometry is required, the bend connecting the two passes can also be modelled as a Spyro Section, shown in fig. 5. I
Feed
]]
(steam)
] Fireb~
Feed
Sink
I
[(Hydro Carbon) I
~ .... i
Mixer t
SPY.ENT
[ .....
1.85
ADIAZONE ] T
I
[
SPYEXIT
I SPYSEC ~ I
I
SPYS, E,C, 3, ,, ]
[ SPYNODEI
I SPYNODE 2 I
[-~
]
SPYSEC2
. - - P(stmight)l ~P(detail) ]
1.8
T
]'"
T
T
~
1.75
" 1.7 ~'1.65
~ 1.6 1.55 1.5 1.45 1.4 0
Fig. 5: Detailed configuration of coil in fig. 3.
5
10
15
Position [m]
20
25
Fig. 6: Pressure curves related to fig. 4, fig. 5.
In fig. 6 we demonstrate the effect of the different configurations of the geometry on the pressure profile. The results of the straight forward configuration (fig. 4) are denoted by the legend 'P(straight)' and the results of the detailed configuration (fig. 5) is denoted by the legend 'P(detail)'. In both graphs a discontinuity in the pressure profile is present due to the mix junction. 4. NUMERICAL STRATEGY The Spyro model is a two point BVP. Original SPYRO | uses a shooting technique to tackle the two point BVP. The solution of the ODEs in the Open Spyro model is
592
approximated with the collocation technique (Villadsen and Michelsen, 1978; Rice and Do, 1995). The method used is the one of Orthogonal Collocation on Finite Elements (OCFE). The domain of interest (i.e. the length of a Spyro section) is divided into finite elements. Within these elements the solution is approximated with the Legendre polynomial. Typically we use third order polynomials and ten finite elements. 5. EQUATION SOLVER A damped Newton method can be used to solve the Open Spyro problem, even when no good starting values are available. The drawback of this method is that at every iteration a Jacobian matrix must be computed and a linear system must be solved, which can be time consuming for very large systems like the Open Spyro model. The implemented method reduces the computational effort by combining the damped Newton method with a Secant method. The iterative scheme we use is defined by: Xm+l
-
-
Xm,sm
(1)
Xm'o = Xm
Xm, i = Xm,i_ 1 -- 2m,i_ldm,i_ 1 for i
dm,i-l:Bm,i-lf(Xm,i-l)
=
l..s m
-1
The first update dm, o is computed using Brn, o = Jrn, resulting in a Newton iteration. The next updates are computed using a quasi Newton method with Brn, i the approximation of the Jacobian matrix at Xrn,i. Every outer iteration denoted by m will result in at least one damped Newton correction. The number of quasi Newton corrections for each outer iteration, Sr,,-1, depends on the computed damping factors /trn,*. In order to preserve the global convergence property of the damped Newton method we terminate the quasi Newton iterations when the damping factor drops below a given tolerance, 2~rn,* < /~minupdate , indicating that we are too far from the solution or that the Jacobian update is not useful enough anymore. In order to converge to the solution, even when no good initial guess is available, we need to find an appropriate damping factor/1.rn,~ at every iteration. We use a method similar to that of Deuflhard (1974). We choose/tm, i such that :
< i.,: ,
'
(=)
The more expensive choice, scaling the norm by Brn,i, resulted in the same damping factors for the tested problems and is therefore not used. 5.1. Convergence criterion A dual convergence criterion is chosen. We terminate the iterations when IDscxdmjI22 < o~x
and
]D,
(3)
Where Dscx and Dscr are two scaling matrices. There are many 'good' choices possible for Dscx and Dsct. We scale drn,~ with the vector x, resulting in a relative norm for the correction and use no scaling for the residuals f.
593
5.2. daeobian Update Method After each damped Newton correction the Jacobian is updated. We have adapted the method proposed by Broyden (1965) such that it can be used efficiently for large sparse systems as well. The update of the inverse Jacobian approximation is defined by -, Bmli : Bm,i-1 +
'
-1 ( p , _ Bm,i_lqi -1 )Pi r Bm,i-, TB-I , with Pi m,i-lqi
Pi = Xmi -- Xm,i-I
and
qi = f ( X m , i ) - - f(Xm,i-1) (4)
'
The update (4) is used efficiently without an explicit inverse update. The derived method allows us to compute the correction steps by using the known LUdecomposition of Bm, o = Jm and a number of vector products. 5.3. Sparse Linear Solver A sparse LU-decomposition is used in order to compute the solution of the linear systems. Iterative methods have been tested with various pre-conditioners but this was without any success. Similar results were found by Cofer and Stadtherr (1996). Another disadvantage is that no information can be used from previous solutions to solve a succeeding linear system with the same coefficient matrix at low cost. Such an operation occurs often in our method. Therefore a new sparse LU-decomposition algorithm is developed and implemented in Fortran 90, using a partial pivoting strategy for numerical stability. In order to compute the LU-decomposition of the Jacobian efficiently, we permute the Jacobian prior to computing the LU-decomposition. Three reordering methods are combined for an optimal result. The first step is to permute the Jacobian such that the diagonal consists of nonzero elements (Duff, 1981). The second step is to permute the matrix into a lower block triangular form (Duff et al., 1996). In the third step the diagonal blocks are reordered by the reverse CuthilI-McKee permutation (Gibbs et al., 1976). 6. SIMULATION RESULTS The results of the original SPYRO | program have proved over the years to be accurate and reliable. We have compared the simulation results of the Open Spyro program with the original SPYRO | program to validate the correctness of the implemented model and to test the solving method. Laboratory, pilot and industrial scale cases with feeds varying from gases to gasoils and cracking severity (coil outlet temperature) ranging from 750~ till 890~ have been compared. For each case many profiles along the coil have been compared, among others of ethylene, propylene and temperature. A typical naphtha feed cracked in a straight coil of seven passes at a severity of 856.0~ is chosen as an example. Table 1 summarises the comparison of the calculated data profiles along the coil and lists some results valid at the coil outlet. A good match is observed between original SPYRO | and Open Spyro for this typical case. The small differences can be attributed to the slightly different pressure drop calculations in the coil bends and of course to the different solving methods. The other cases had comparable results: an almost perfect match between original SPYRO | and Open Spyro.
594
Table 1 CH4 [wt.%] C2H4[wt.%] C3Hs[wt.%] T [K]
Statistical results of profile and coil outlet comparisons of a typical naphtha feed Profile Results Abs.prof,diff. Rel. prof.diff. Max. diff.
3.3.10-2 5.0.10-2 3.2.10.2 3.5-101
0.64 % 0.48 % 0.41%
9.1.10.2 1.1.101 -9.6.10.2 1.8
Coil outlet results SPYRO value XSpyro'XOS Rel. abs. diff.
1.0.101 2.0.101 1.1.101 1.1.103
8.9.10.2 1.1.101 -8.8.10.2 5.9.101
0,90 % 0.55 % 0.80 %
7. C O N C L U D I N G R E M A R K S
An effective and flexible system of sub-models has been implemented in the new Open Spyro program. OCFE has proven to give a good approximation of the solution of the ODEs. A fast and global converging quasi Newton method based on the update proposed by Broyden is used to solve the Open Spyro model. The same reliable results as the original SPYRO | program are obtained. ACKNOWLEDGEMENTS
Special thanks go out to Cor van Leeuwen who has guided us in spite of his illness. The authors thank Peter Verheijen for his aid and fruitful discussions. The authors are also grateful to Professor Dente and Professor Ranzi for their help. REFERENCES
Broyden, C.G. (1965). Math Comp 19, pp. 577-593. Cofer, H.N. & Stadtherr, M.A. (1996). Comp. Chem. Eng. 20(9), pp. 1123-1132. De Blieck, J.L. & Goossens A.G. (1971). Hydrocarbon Processing. March, pp. 76-80. Dente, M., Pierucci, S., Ranzi, E., Bussani, G. and Valkenburg, P. (1993). Mathematical Modeling of Steam Cracking Reactors. Chemical and Process Engineering: Abstracts, Aidic, pp. 393-397. Dente, M. & Ranzi, E. (1999). SPYRO | 2000. European SPYRO | Users Conference 1999, Technip Benelux B.V./Division Pyrotec Deuflhard, P. (1974). Numer. Math. 22, pp. 289-315. Duff, I.S. (1981). ACM Transactions on Mathematical Software, Vol. 7 No. 3, pp. 315330. Duff, I.S., Erisman, A.M. & Reid, J.K. (1996). Direct Methods for Sparse Matrices. Oxford: Science Publications. Gibbs, N.E., Poole Jr., W.G. & Stockmeyer, P.K. (1976). SIAM J. Numer. Anal.13, pp.236. Ranzi, E., Dente, M., Pierucci, S. & Biardi, G. (1983). Ind. Eng. Chem. Fundam. 22, pp.132. Ranzi, E., Dente, M., Pierucci, S., Barendregt, S. & Cronin, P. (1985). Oil and Gas Journal, September 2, pp. 49-55. Rice, R.G., & Do, D. D. (1995). Applied mathematics and modeling for chemical engineers. New York: J. Wiley & Sons. Villadsen, J., & Michelsen, M. L. (1978). Solution of differential equation models by polynomial approximation. International series in physical and chemical engineering science, Englewood Cliffs, NJ: Prentice-Hall.
European Symposiumon ComputerAidedProcess Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
595
A Shortcut Method for Design and Synthesis of Multicomponent Thermally Coupled Distillation Flowsheets Ben Guang Rong*, Andrzej Kraslawski and Lars Nystr6m Department of Chemical Technology, Lappeenranta University of Technology, P.O. Box 20, FIN-53851 Lappeenranta, Finland. * Email: benguang.rong~lut.fi The design and synthesis of thermally coupled distillation flowshees for separations of fivecomponent mixtures are studied. Four types of possible configurations are identified. A universal design procedure is developed for any types of the identified configurations. Examples demonstrated that this shortcut design method can not only be used to design any types of the proposed thermally coupled distillation flowsheets, but also can give very good initialization for rigorously simulation of such configurations. The synthesis of multicomponent complex distillation flowsheets is implemented with the developed method. 1. INTRODUCTION Among all possible new schemes for multicomponent distillation processes, the thermally coupled distillation schemes are very promising for both energy and capital cost savings (Petlyuk 1965, Smith 1995). The research on complex distillation configurations is focused on three-component mixtures (Tedder et al. 1978, King 1980, Glinos et al. 1987). Much work has been realized on some specific configurations for ternary mixtures, aiming at the performance analysis and industrial applications (Triantafyllou and Smith 1992; Wolff and Skogestad, 1995; Agrawal and Fidkowski, 1998; Mutalid and Smith, 1998). There is very little work on configurations of four or more component mixtures. Such a situation is due to the combinatorial complexity of the problem. Moreover, there is a lack of shortcut design procedure as well as modelling and synthesis methods for these types of distillation schemes. In this paper, we focus on the study of thermally coupled distillation flowsheets for the separation of five component mixtures. First, the feasible configurations of such flowsheets are analyzed. Then, a universal shortcut design method is proposed for design of any type of the feasible configurations. The examples of synthesis of complex distillation flowsheets are given as well. 2. FEASIBLE CONFIGURATIONS AND NETWORK REPRESENTATIONS
2.1. The Alternative Configurations While there are only three feasible thermally coupled configurations for ternary mixtures, there are a large number of feasible configurations with thermally coupled to separate nearlyideal five-component mixtures. When considering the feasible configurations, the number of condensers and reboilers as well as the number of column sections in a flowsheet are very important factors both for capital costs and operability (Agrawal, 1996). For this standpoint,
596 we focus on the configurations with side strippers and side rectifiers, while the configuration has the same column sections with a simple column sequence. Thus, in this work, we build a complex distillation flowsheets with the following units: (1) Main Column (MC): A main column in a complex scheme is a column with an overall condenser and a reboiler while connecting with side columns. For a complex scheme, to separate five component mixtures, there may be two such main columns of which the one with feedstock is called main column. (2) Side Stripping Column (SSC): A column with only one reboiler. (3) Side Rectifying Column (SRC): A column with only one overall condenser. (4) Simple Column (SC): A column with one feed and two product streams and with an overall condenser and a reboiler. There is a sharp separation realized in the column. With the above units, four types of complex distillation flowsheets (CDF) could be constructed as shown in Figure 1. l) Complex flowsheets with side columns connected in parallel (Fig. la, b); 2) Complex flowsheets with side columns connected in both series and parallel (Fig.lc); 3) Complex flowsheets with simple columns (Fig.ld); 4) Complex flowsheets with side columns connected in series (Fig.le). ~A
t, A
ABCDE,.~ j _ ~
~A
B
7 3
~B
~C P2 4 " .t.__.JP2 D
ABCDEI~
~E
(a)
(c)
(b) I, A ~>
~A D, B
ABCDE
4
(d)
D
ABCD
(e)
Figure 1. Feasible Configurations of Complex Distillation Flowsheets for Five-component Mixtures
597 2.2. Representations of Complex Distillation Configurations A network representation is presented for these types of complex distillation flowsheets. For example, Figure 1 (a), (c) and (d) could be represented in network as shown in Figure 2. )(~ SRCI(B) MC(A,Ey"" SSCI(B) MC(A,E SRC2(C) SRC3(D) ~.. SSC2(D)_ SRC(C)
ssc(c) SC(A, BCDE)=- MC(B,E< SRC(D)
Figure 2. Network representations of CDFs In any given network, a line represents a connection between two units. A node is one of the four types of units in part 2.1. Units of the same type in a flowsheet are distinct by their orders. Subgroups in bracket are the products of that unit corresponding to its node, to main column and simple column the former in the bracket is the distillate while the latter is the bottom product. The connected locations for the side columns in a flowsheet are identified according to their products. For example, there are three side rectifying columns SRC1, SRC2, SRC3 in flowsheet Figure 1 (a) with B, C, D being their products respectively. Thus, any alternative configurations constructed with the four types of units could be represented in such networks and this will facilitate the synthesis of such flowsheets in constructing the feasible configurations. 3. THERMODYNAMIC EQUIVALENT SCHEMES AND THE BASIC UNITS A complex scheme can be converted into a flowsheet in which each unit has only two column sections of rectifying and stripping respectively. The connections of the units are determined according to the interconnections of their streams. For example, for Figure l(b), the converted scheme could be obtained as in Figure 3. The converted configurations are called the thermodynamic equivalent simple column flowsheets (Carlberg et al. 1989). Then, three different basic units can be abstracted which are the basic units to construct any of the thermodynamic equivalent schemes. These three basic units are shown in Figure 4.
P ,A .............. i
F p p~ Pa~ 34 2
Pb2
C
................i
v
Lc
D
E (a)
Figure 3. The thermodynamic equivalent scheme of CDF of Figure 1(b)
(b)
(c)
Figure 4. Three Basic Units of Thermodynamic Equivalent Schemes of CDFs
598 4. THE DESIGN PROCEDURES OF CDFs
The distinct feature of a CDF with a simple column sequence is that the units are interconnected by the thermal coupling streams and they could not be designed separately. The design of CDF must simultaneously consider the constraints of the design variables resulted from the interconnection streams within its units. 4.1. The CDF is converted to its thermodynamic equivalent structure This conversion is based on the analysis of separation sequence and the functions of the column sections in a CDF. A thermodynamic equivalent configuration is obtained in which each unit has only one rectifying and one stripping column sections. 4.2. Shortcut designs of three basic units The three basic units in CDFs are designed based on Underwood equations. However, due to the feed and the top and bottom products in a basic unit of CDFs might be the coupling streams, thus, the design of the three basic units are different which are formed in three main steps. 1) Determine the feed qualities for each basic unit; 2) Calculate the minimum reflux ratio for each basic unit," 3) Determine the operating pressures for each basic unit. 4.3. Determination of the operating pressure for a CDF For a thermally coupled flowsheet, the relationships of the pressures for the units are restricted by the interconnections of the units through the coupling streams. For example in Figure 1 (b), the pressures at thermal coupling locations have to satisfy the following constraints.
P'I > P1; P2 > P'2 ; P3 > P'3
(1)
Meanwhile, the pressure distribution in the converted equivalent scheme must satisfy the pressure constraints of its original complex flowsheet. For example, the pressures in the thermodynamic equivalent scheme in Figure 3 must satisfy the following inequality of its original configuration of Figure 1(b). Pb3 > Pb2 > Phi > Pdl ~>Pd4
(2)
The single units must be redesigned based on the revised pressures and usually several iterations are needed to obtain the final design results which satisfy the pressure constraints. 4.4. Transfer of the designed parameters The parameters are transferred automatically by the design algorithm through the registration of the structural information of both the thermodynamic equivalent configuration and its original complex flowsheet. 5. EXAMPLE PROBLEM
The separation of a five-component mixture (A: propane, B: i-butane, C: n-butane, D: ipentane, E: n-pentane) has been widely studied for synthesis of simple column sequences
599 since Heaven (1969). Here, with the developed design procedure for thermally coupled flowsheets, we can explore the possibilities of the thermally coupled flowsheets for the separation of this mixture. The feed mole fractions of A,B,C,D,E are 0.05, 0.15, 0.25, 0.20, 0.35 respectively. Feed flow rate, F=907.2 kmol/h; Five nearly pure products are required and the recovery of each key component is 98%. The cold utility is cooling water. The design results for the selected flowsheets of Figure l(b) and (c) are shown in Table 1. The K-values, enthalpies and associated thermodynamic properties are calculated with PR EOS. Table 1 The design results of flowsheets of Figures 1(b) and (C) f~ Fi~lb col 1 col 2 col 3 col 4 Col 1 col 2 D 47.13 225.0 1 8 5 . 0 0.00 47.13 0.00 B 313.8 0.00 0.00 136.1 314.8 185.0 Tdi 45.40 101.0 1 3 7 . 3 73.14 45.40 100.3 Tbi 151.2 134.2 145.7 85.92 150.6 139.4 Pdi 14.77 15.37 1 5 . 1 6 1 4 . 8 5 14.77 15.13 Pbi 16.26 15.51 1 5 . 6 5 1 4 . 9 7 16.09 15.81 Rm 60.03 1.96 15.68 --167.3 --R/Rm 1.2 1.2 1.2 --1.2 --Nt 196 17 66 14 175 63 Nr~ 151 0 0 14 63 63 Qci 1 i .07 2.71 13.92 0.00 30.58 0.00 Qn 27.11 0.00 0.00 2.31 19.80 12.55 ZV 7862. 10172
le problem Fig.lc col 3 col 4 224.1 0.00 0.00 136.1 100.7 73.13 1 2 0 . 8 85.92 1 5 . 3 3 14.85 1 5 . 4 8 14.97 1.63 --1.2 --19 14 0 14 2.38 0.00 0.00 5.42
6. THE SYNTHESIS OF CDFs
The synthesis of the above thermally coupled distillation flowsheets is implemented based on the economic evaluation. The economic evaluation is based on the total annual cost of a flowsheet where the operating cost is calculated based on the cold and hot utility consumption, while the capital cost is a sum of columns, condensers and reboilers (Douglas 1988). The same example problem in section 5 is used to demonstrate the synthesis of CDFs. The available utilities are given in Table 2. A capital charge factor of 0.1 is used to annualise the install equipment cost, and the plant operating time is 8000 hours per year. Table 2. The available utilities Utilities Tu, K Cost, $/1000 lb Cooling water 305.15 0.06 Steam (40.0 atm) 523.15 4.52 Steam (17.0 atm) 480.15 3.72 Steam(10.0 atm) 453.15 3.4 Steam(3.4 atm) 410.15 2.8 Steam(1.0 atm) 373.15 2.28
Table 3. The synthesis results of CDFs in Fig. 1 ]EQr ZQc coc coP TAC d a b e c
28.5 27.7 29.4 36.6 37.8
26.5 26.3 27.7 31.8 33.0
62.5 82.8 86.9 74.5 89.8
303.0 378.8 395.0 474.7 496.8
365.5 461.6 481.9 549.2 586.7
600 The synthesis results of CDFs in Figure 1 are shown in Table 3. Several examples are calculated and the obtained results showed that, compared with simple column sequences, it is not a straightforward strategy by thermal coupling for costly efficient for separations of multicomponent mixtures. The detailed analysis results will be presented elsewhere. 7. CONCLUSIONS AND FUTURE WORK The thermally coupled distillation flowsheets for the separations of five-component mixtures are studied. A universal shortcut design procedure is developed for any types of the thermally coupled schemes. Example results showed that this shortcut design procedure can give all the needed equipment and operating parameters, meanwhile it presents good initial information for rigorous simulations. Thus, detailed studies of such complex flowsheets are practical by the proposed method. The shortcut procedure presented a reasonable method for the synthesis of multicomponent thermally coupled distillation flowsheets. The real five-component mixture is used for the synthesis of thermally coupled flowsheets. Some preliminary insights are obtained for thermally coupled flowsheets. The CDFs are usually favourite for atmospheric operating pressure. They are favourite for less amount of intermediate components. The detail parametric studies of these flowsheets will be presented elsewhere. NOTATIONS B-bottoms product flow rate, kmol/h COC-annual capital costs, 104$(p.a.) COP-annual operating costs, 104$(p.a.) D-column distillate flow rate, kmol/h Lc-liquid flow rate of coupling stream, kmol/h Nt-total number of theoretical trays Nrt-tray number of stripping section Pb-column bottom pressure, atm
Pd-column top pressure, atm Qc-heat duty of condenser, 106kcal/h Qr-heat duty of reboiler, 106kcal/h Rm-minimum reflux ratio TAC- total annual cost of a flowsheet, 106$(p.a.) Tb-column bottom temperature,~ Td-column top temperature,~ Vc-vapor flow rate of coupling stream, kmol/h
REFERENCES
1. Agrawal R., 1996, Ind. Eng. Chem. Res., 35, 1059-1071. 2. Agrawal R. and Fidkowski Z. T., 1998, Ind. Eng. Chem. Res. 37, 3444-3454. 3. Carlberg N. A. and A. W. Westerberg, 1989, Ind. Eng. Chem. Res. 28, 1379-1386. 4. Douglas J. M., 1988, Conceptual Design of Chemical Processes, McGraw-Hill. 5. Glinos K. and Malone M. F., 1988, Chem. Eng. Res. Des., 66, 229-240. 6. Heaven D. L., 1969, M. S. Thesis, Uni. of California, Berkeley. 7. King J., 1980, Separation Processes, 2nd, McGraw-Hill. 8. Mutalid M. I. A. and Smith R., 1998, Trans IChemE, 76, Part A, Part 1308-318. 9. Petlyuk F. B., Platonov V. M. and Slavinskij D. M., 1965, Int. Chem. Eng. 5(3), 555-561. 10. Smith R., 1995, Chemical Process Design, McGraw-Hill. 11. Tedder D. W. and Rudd D. F., 1978, AIChE Journal, 24(2), 303-315. 12. Triantafyllou C. and Smith R., 1992, Trans IChemE, 70, Part A, 118-132. 13. Wolff E. A. and Skogestad S., 1995, Ind. Chem. Eng. Res. 34(6), 2094-2103.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
601
A heating-cooling management to improve controllability of batch reactor equipped with a mono-fluid heating-cooling system H.Bouhenchir, M. Cabassud, M.V. Le Lann and G. Casamatta Laboratoire de Genie Chimique. UMR CNRS 5503 Ecole Nationale Superieure d'Ing6nieurs de G6nie Chimique 18, Chemin de la loge-31078 TOULOUSE Cedex-FRANCE T61.: (33) 5 62 25 23 62-Fax: (33) 5 62 25 23 18 E-mail: [email protected]/[email protected] In this paper, a strategy for temperature control of multipurpose batch reactors equipped with a mono-fluid heating/cooling system is presented. This strategy is based on the use of the thermal flux as manipulated variable. At each sampling time, the master controller computes the thermal flux to be exchanged between the reactor content and the fluid flowing inside the jacket. This information is then used to select the "right" thermal element to be used (electrical resistance or plate heat-exchanger) according to the physical modeling of the thermal loop. Finally the control variable is computed and applied to the selected device. 1. INTRODUCTION In fine chemical or pharmaceutical industry, the batch or fed-batch reactor functions as the heart of the transformation process. Due to the complexity of the chemical synthesis and the difficulty to perform on-line composition measurements, control of batch reactors is essentially a problem of temperature control [1], which is difficult to overcome [2]. The difficulties arise in part from the discontinuous nature of the operating modes and in part from the various uses of these reactors. The regulator must work in face of drastic changes in set points and also be adaptable to the different operating modes. Thermal control of jacketed batch reactor strongly depends on the heating/cooling system by which the temperature can be controlled. Different configurations of heating/cooling systems are commonly used in industry: the alternate or multi-fluid system and the mono-fluid system. This work is concerned with the mono-fluid system, which equips recently installed reactor. This system uses only one fluid, temperature of which is modified according to the desired reactor temperature. This temperature modification is performed by an intermediate thermal loop, which may include heat-exchangers, electrical resistance, etc [3]. Amongst the numerous advantages of this system, the following ones can be emphasized: 9 a single fluid for heating and cooling; 9 a continuous inlet jacket temperature, thus preventing thermal shock; 9 an absence of air purge, this removing dead time; 9 Ensuring quasi-constant heat transfer performances. A large number of configurations can be designed for the external thermal loop. In order to get good temperature control performances, it is important to design a system with a fast thermal dynamics and to limit the process delays. Previous studies have shown that direct mixing of hot and cold fluids is the most efficient system. Unfortunately, this solution has no
602 industrial application. The system proposed in this work is an attempt to reach this 'ideal' solution, but with a technological feasibility. An electrical resistance performs heating, while cooling is carried out by two plate heat-exchangers (one using cold water (at about 15~ and the other one using a mixture of glycol/water (50/50-weight % at a temperature of-10~ In order to avoid thermal limitations, the implemented approach consists in acting on the monofluid flow-rate, which is delivered to the plate heat-exchanger instead of acting on the flowrate of the utility fluid flowing in the exchanger. Temperature control of the reactor content during a three-stage temperature profile tracking (heating-maintenance-cooling) is achieved by modulating the temperature of the mono-fluid. The control strategy has to simultaneously perform a supervisory task (selection of the right element in the external thermal loop) and then to compute the value of the control variable. Adapting the so-called 'thermal flux limits control strategy' has done this. This control management was initially developed for a multifluid system [4]. In our case, the main controller (for example a GPC algorithm) computes the thermal flux to be exchanged between the mono-fluid flowing in the jacket reactor and the reactor content in order to reach the desired set point temperature. The thermal flux is then used in a second control loop to select the right thermal element and to compute the value of the control variable according to a physical modeling of the different devices. Practically, when a heat-exchanger is chosen, a defined percentage of the mono-fluid flow-rate is dispatch to this element, while in the case of heating the control variable corresponds to the power of the electrical resistance. In this paper, firstly a description of a bench scale reactor and its heating/cooling system used for experiments is given. The second part is devoted to the presentation of the methodology for supervisory and control of the reactor temperature. A brief presentation of the adaptive control algorithm is given. The last part is devoted to the experimental results obtained on the pilot-plant. 2. PROCESS DESCRIPTION The experimental device consists of a 11 jacketed glass reactor, fitted out with a mono-fluid heating/cooling system. The mono-fluid is a mixture of ethylene glycol and water in ratio of 50% with a flow-rate of 1000 lhr-~. A sketch of the pilot plant is given in figure (1). The reactor has the following physical specifications: internal diameter of 82 mm, reactor wall thickness of 9 mm, external jacket diameter of 125 mm, jacket wall thickness of 5 mm, reactant mixture-reactor wall heat transfer area of 0.039 m 2 and jacket volume of 0.15 1. An impeller, rotated at 260 tr/mn ensures a good agitation. Liquid reactive can be fed into the reactor. The heating-cooling system includes a 2000 W electrical resistance and two plate heat-exchangers (P.H.E), The mono-fluid flow-rate is measured by means of two flowmeters, one installed on the main thermal loop and
Fig. 1. Scheme of the pilot plant.
603 the other on the secondary thermal loop. Flow-rates of the cooling fluids are also measured. Three on-off valves allow the mono-fluid to be heated or cooled. Two other on-off valves are used to manipulate the utility fluids. A three ways air-to-open valve ensures the division of the mono-fluid in two parts during the cooling phases. A computer equipped with input/output boards provides real-time data acquisition, automatic start-up and operation control. Supervision and control programs are implemented on a PC. 3. METHODOLOGY FOR SUPERVISORY AND CONTROL OF BATCH REACTORS A strategy integrating supervisory and control is proposed. Master controller computes the thermal flux (manipulated variable). On the other hand, the maximal and minimal thermal capacities of the different apparatus (electrical resistance, heat-exchangers) are computed and used to select the "right" thermal element with a priority to the device currently used. Then, the control variable is computed and applied to the process. The limit thermal capacities of heating and cooling are computed on-line by a procedure involving reaction mixture temperature, the jacket inlet and outlet temperatures, the physical properties of the mono-fluid, cold water, glycol/water and the maximal electrical power value. 3.1, Master control loop An adaptive and predictive controller (the Generalized Predictive Controller with Model Reference: GPCMR) is used to compute the necessary thermal flux to be exchanged between the reactor content and the fluid flowing in the jacket (manipulated variable). This adaptive controller is based on the linear input-output representation of the process. The on-line estimation of the model parameters allows following the changes in the dynamics occurring during the different steps: heating, reaction and cooling. It is also a predictive controller, which computes the manipulated variable by minimizing the square errors between the future set points (reference profile) and the output predictions (temperature in the reactor) on a receding horizon. Details of this algorithm and its use for temperature control of batch reactors can be found in [5]. 3.2. Model based supervisory Thermal flux capacities are computed for the different elements of the external thermal loop tacking into account the thermal characteristics of apparatus. 3.2.a. Case of the electrical resistance The electrical resistance is characterized by a maximal power value of Pelec(max). The maximal thermal flux exchanged in the jacket corresponds to this maximal value: Q max ep = Pelec(max) (1) The minimal thermal flux is: Q min ep = 0.0 (2) 3.2. b. Case of plate heat-exchangers When the plate heat-exchangers are used, the thermal flux limits capacities of the mono-fluid are based on the model of heat exchanges between the reaction mixture and the jacket as
given by: Q1= UA, Tjin + Tj~ - Tr~ (3) 2.0 J Assuming that the jacket wall is uniformly distributed, the thermal flux delivered by the mono-fluid to the reactor is expressed by: Q2 = FCp(Tjin- Tjout) (4) By assumption that both thermal flux are equal: Q1 -- Q2, Tjoutcould be computed from eq
604
{Tjm [2FCp - UA ] + 2 Tr UA } (5) 2 FCp + UA The heat transfer coefficient U is computed on-line according to classical correlation [6]. Heat capacity Cp is expressed as a function of temperature. When the mono-fluid flows in the plate heat-exchanger, the minimum outlet temperature to be obtained is that of the inlet cooling utility fluid (cold water or glycol/water) used by this plate heat-exchanger. We consider that the maximal thermal flux is equal to zero (corresponding to Tjin=Tjout): Q max cw = 0.0 ; Q max gw = 0.0 (6) The minimal thermal flux, Qmincw and Qmingw correspond to the two plate heat-exchangers used. They are computed according to eq (3) or (4) and (5) tacking 15~ and-10~ as inlet jacket temperature for cold water and glycol/water respectively. 3. 3. Strategy of supervision The required thermal flux computed by the Temperature(~C) ....... master controller (Qcont) is compared to the limit capacities of the thermal elements, if its exceeds these limits the appropriate 66 changeover is performed. Figure (2) gives an approximate global representation of the evomuon of me" thermal nux ~' capacities as a function of the reactor mixture temperature5 111r I I I I I I I I I I U I I I a l I L I t i l e i i i a x i i i i u i i i capacities correspond to the border of the zones. Three -0.2-0.1 0 0. I 0.2 0.3 0.4 0.5 zones are considered: the first one "1" Thermal flux( kcal/s) corresponds to the limit thermal capacities of the electrical resistance, the second one "2" Electrical resistance corresponds to the limit thermal capacities of .1,,,-..-1,.~.. iJJ~ w u J u ....... w a t ~ i and ....1.~ 9 .,~.~ ( ~ ) p J a t ~ L . . . . + . . . . 1,.,,., the last one "3" corresponds to the limit P.H.E( gw ) (3) and (4) as: Tjout =
1
,-1-,I . . . . .
Lllg~
44I
,"
~..~
. . . . . . . . .
.1
IIK~aL-K;A~IIalI~K;I
,1._
Ut;~
Fig.2. Evolution of the thermal flux using glycol /water. When the thermal flux capacities for the mono-fluid takes a positive value, this means a demand of heating, and only one device is concerned: the electrical resistance. A negative value means a demand of cooling. In this case, one of the two plate heat-exchangers is used according to the value of the required flux.
3. 4. Computation of the control variable Two different control variables are computed corresponding to the devices used: the heating electrical power value (electrical resistance) or the fraction of the mono-fluid flow-rate (plate heat-exchanger). 3.4.a. Use of the electrical resistance The heating electrical power value depends on the thermal flux (Qcont) to be exchanged between the mono-fluid in the jacket and the reactor content. A linear relation is considered between (Qcont) and the maximal heating electrical power value, according to the following relationship: Qcont = fl.pelec(max) (7) 13 represents the fraction of the maximal heating electrical power value and varies in the range [0,1 ]. A saturation of the thermal flux value (Qcont) is used to respect the thermal flux
605 limits range: Qcont > Q max ep ~ Qcont = Q max ep 3.4. b. Use of the plate heat-exchangers When one of the two plate heat-exchangers is used, the problem is to compute the defined percentage of the mono-fluid flow-rate to be dispatch to the plate heat-exchanger selected. We consider that the thermal flux computed by the main controller is equal to the heat exchanged in the plate heat-exchanger, according to: t3cont _ BF(~,,[Tout _ Tin } (8) ~ - ~'~ "~ V phe "phe/ 13 represents the fraction of the mono-fluid flow-rate, which varies in the range [0,1]. Saturation is given to Qcont when we use the plate heat-exchanger using glycol/water to respect the limit thermal capacities: Qcont < Q min gw :::> Qcont = Q min gw 50
i
. . . . . . .
I
t
45 40 35
. . . .
I . . . .
I ' ' ' '
i Temperature(~
1 ' ' '
r-['
''
t
--o.--
9
11
1
45 0.6 0.4
(s~
25 .
0
.
.
.
55 P ' ' ' t ~ t ' ' ' J'''' t .... I .... J.... J,,,~ ~Temperature(*C) ! ~
40
,
30
20
r-I
Controlvariabl~
.
.
.
0.2
35 30 25 20
Time(s) i
0
0
1000 2000 3000 4000 5000 6000 7000
Fig. 3. Temperature and control variable.
1000 2000 3000 4000 5000 6000 7000
Fig. 4. Inlet and outlet jacket temperatures
4. EXPERIMENTAL RESULTS Thermal flux(kcal/s) To demonstrate the good performances of this 0.4 ~ Qcont I~ ~ Qmaxep control methodology, different experiments have |~ - Qminep j[| : Qmaxcw been carried out on the pilot plant reactor 0.3 I I .-.o.-. Qmincw previously described. In this paper we do not 0 . 2 - 1 1 ~ = Qmaxgw ~ present all experimental studies. The pilot plant 0 . 1 ~ ! i ~ A /I . . . . Qmingw j reactor was fed with 0.7 1 of water at 23~ reactor temperature control according to a four 0 steps temperature set-point profile has been -0.1 studies for various conditions:-1 st phase: heating 0 1000 2000 3000 4000 5000 6000 7000 from 23~ to 45~ in 1500 s; 2nd phase: constant temperature set-point of 45~ during 2500 s; 3rd Fig. 5. Limits and manipulated phase: cooling from 45~ to 30~ in 2500 s; and variable of the thermal flux. 4th phase: maintain at 30~ for 500 s. Figure (3) presents the time evolution of the set-point and reaction mixture temperature (Tcons) and (TR) respectively and the control variable (fl), figure (4) gives the time evolution of the inlet (Tjin) and outlet (TjouO jacket temperatures and figure (5) gives the time evolution of the limit thermal capacities for the different thermal elements used and the thermal flux required (QconO computed by the master controller (GPC). Let us recall that the control variable (/3) represents either the consummation degree's of the electrical power value 0.5
i
i
~
I
i
~1
r-'r-Tr
,
1~
i
'bJ
,
;
~ i
,
,
L i
r-r-
606 during the heating phase or either the fraction of the mono-fluid flow-rate dispatch to the plate heat-exchanger during the cooling phase. If we analyze these figures we can give the following remarks: at the beginning of the maintain phase, there is no overshooting of the setpoint due to the fact that it is possible to ensure a quick quenching of the mono-fluid by activating the plate heat-exchanger using glycol/water as utility fluid. This is performed when the required thermal flux (Qcont) overshoots (Qmincw) which presents the minimal thermal capacities of the plate heat-exchanger using cold water as utility fluid, which corresponding to zone number "3" in figure(2) [Qmin gw< Qconst < Q mincw] Let us notice that when (Qcont) Overshoots (Qminep) which represents the minimal thermal capacities of the electrical resistance, the electrical resistance is activated. This corresponds to the zone number "1" in figure(2) [Qminep
1. M.Friedrich, R.Perne. Design and control of batch reactors: an industrial viewpoint. Comput. Chem. Eng. 19, $357-$368, (1995). 2. M.R.Juba, J.W.Harmer, Progress and challenges in batch process control, in: Proceedings of the CPC III. CACHE, Elsevier, Amestrdam, (I 986). 3. H.Bouhenchir, M.Cabassud, M.V.Le Lann and G.Casamatta, Thermal control of batch reactor equipped with a mono-fluid heating/cooling system, in: Proceeding of the Second European Congress of Chemical Engineering, 5-7 October Montpelier, France (1999). 4. Z.Louleh, M.Cabassud, M.V.Le Lann, A new strategy for temperature control of batch reactors: experimental application, C.E.J, 3473, 1-10 (1999). 5. M.V.Le Lann, M.Cabassud and G.Casamatta, Adaptive model predictive control, Methods of Model Based Process Control, R. Berber (ed.), Kluwer Academic Publishers, Dordrecht, 426-458, (1995). 6. P.Fletcher, Heat transfer coefficients for stirred batch reactor design, Chem. Eng. 33-37, (April 1987)
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
607
Evaluation of time varying parameters in polymerization reactors by means of Temperature Oscillation Calorimetry Pierluigi Guemni De Luca a, Claudio Scali a, Giuseppe Maschio b a Dipartimento di Ingegneria Chimica. Universifft di Pisa. Via Diotisalvi, 2 - 56126 Pisa, Italia b Dipartimento di Chimica Industriale. Universifft di Messina. S.ta Sperone 32 CP. 2 9 - 98166 Sant'Agata di Messina, Italia Applicability of oscillation calorimetry techniques for the evaluation of time varying parameters during the course of the reaction, is analysed by referring to experimental data from a MMA polymerisation reactor. Different algorithms for the elaboration of experimental data and their sensitivity to external causes of error is evaluated in order to choose the most robust to perturbations.
1. Introduction The classical reaction calorimetry allows the evaluation of the heat of reaction and from this the rate of conversion of a chemical reaction, by means of a very simple experimental apparatus, which finds wide use in lab-scale and industrial units. The basic equation of reaction calorimetry is the heat balance of a stirred tank reactor:
cp'dTr
dt
=UA'(Tj-Tr)+Qchem
+Qloss +Pstirred
(1)
Under isothermal conditions, when the heat loss and the power dissipated by the stirrer are known or negligible, the heat transfer coefficient is known (i.e. from initial calibration of the cooling system) and constant, the heat of reaction can be computed from (1). In the cases when the heat transfer coefficient undergoes large changes during the reaction, for example in batch reactors, as a consequence of increase of viscosity due to differences between reagents and products, the heat of reaction cannot be computed from (1), unless the variation of heat transfer coefficient in time is known. This is possible, as matter of principle by adding also an energy balance of the jacket, or by adopting state estimation, but the complexity of the experimental set-up and of the software system increase [1, 2]. A case of practical relevance is that of batch polymerisation reactors, for which the changes of physical properties from the monomer to polymer are evident and the possibility of estimating the conversion during the course of reaction is very appealing, owing to difficulties of real time measurements. The temperature oscillation calorimetry allows the evaluation of the heat transfer coefficient during the course of the reaction from the analysis of reactor and jacket temperature which are forced to oscillate by an external temperature control system. By assuming that the reactor and jacket temperatures are sinusoidal oscillations at the same frequency a~ and that the variations of slow varying parameters of (1) can be neglected in the period of oscillation, the value of the heat transfer coefficients UA can be computed by different algorithms, as it will be shown in the sequel. In a second step, the reactor energy balance, without the oscillating heat contribution, is considered and the chemical heat
608 flOW Qchemis calculated. Afterward, the rate of reaction and conversion are calculated from the chemical heat flow: r=
Qchem V r 9(-AHr)
X = ~ r . ~Vr -dt no
(2)
This new technique, originally presented in [3, 4, 5], finds application in several lab-scale equipment and allows satisfactory results in many cases. In the light of possible applications to industrial size units, several points are worth to be investigated [6], including: - frequency and amplitudes of oscillations of reactor and jacket temperatures, compatible with thermal capacity and the dynamics of actuation of an industrial heat exchange system, - possibility of fulfilling the basic hypotheses for the application of the method - design of the control system able to guarantee the required specifications. In this paper different algorithms for the evaluation of the heat transfer coefficient from experimental data will be compared, starting from the general one derived from the theory of the method, to some approximated ones. Their accuracy and robustness to different sources of errors will be accounted for, by comparing results for the nominal case with experimental results obtained on an lab-scale unit, operating at the university of Messina [7], and performing a sensitivity analysis to different causes of error. 2. D i f f e r e n t a l g o r i t h m s for the evaluation of the heat transfer coefficient
The first algorithm historically developed, was presented together with the basic principles of oscillation calorimetry [3]. It allows to obtain the value of the time varying parameter UA from the measured temperatures on the basis of an integration of all terms of the balance equation (1) previously multiplied with a sine function. Integrals are calculated over one period of oscillations, in a way that the non-oscillating terms in the heat balance give no contribution to the integrated heat balance. Following these analytical developments, the heat transfer coefficient can be computed as [3]: 27~
- Cp. co. Icos(o~ - t + [3)" T r (t). d(eot)
UA = 2,~
o
(3)
fsin(co, t + 13)- (Tj (t) - Tr (t)). d(eot) 0
The phase ~ should be chosen in a way that numerator and denominator of previous equation do not go through a sign change, not to incur in numeric problems in determination of UA. General criteria to select the parameter 13 are not easy to suggest; the simplest way is to evaluate UA for different values of 13and to discard values which are not consistent. Such necessity is quite time consuming in the computer aided implementation of the method, as it requires the supervision of an expert operator and therefore is an obstacle to a complete automation of the procedure. For this reason, while the previous expression, as matter of principle, could be applied to every periodic Tr(t), it has been found more convenient to develop different relations based on the assumption that the temperature profiles is less general and can be satisfactory approximated by sinusoidal functions. The balance equation is analysed in phasorial form and the non oscillating terms are neglected. The phasorial analysis brings to two different, even if equivalent, relations, considering respectively the real and the imaginary part of
609 the balance equation itself. By considering only the real part [4], and only the imaginary part, the following expressions can be obtained: UA~ =
Cp. c o . 6 T r 9sin(g)) 8T~ 9cos(q~) - 8Tj
- Cp. co UA 2 = ~ tan(q~)
(4)
(5)
where q~ is the phase lag between the two sinusoidal signals T r and Tj. We can notice that in equation (4) appear both amplitudes (STy, 8Tj) of the oscillating temperatures and their phase lag (q~), while in (5) only the phase appears. It is so possible to eliminate (q~) from the two equations, obtaining a relation depending only on the amplitudes. According to this analytical development, the following expression has been proposed [5]: Cp.co
ua tan[acos(/] Amplitudes (ST) and phase (q~) of reactor and jacket temperatures can be calculated, for each sampled value, from the experimental measurements, by using equations for harmonic analysis with first order Fourier coefficients: 6T=~/uc 2 +us 2 ,
q~=arctan(UC / \us/
1!
;
2n
with:
(7)
1! 2~
uc = -- T(t). cos(o~, t). d(c0t) ,
us = -- T(t). sin(o~, t). d(cot)
71;
(8)
71;
All previous algorithms are theoretically equivalent. They are all derived from the balance equation by rigorous mathematical procedures, so that until the validity of the hypotheses leading to equation (1) is verified, the computation of the coefficient UA from the temperature profiles themselves is correct and univocal. This statement can be easily verified by applying the previous algorithms to the temperature profiles obtained theoretically (nominal case) under the simplest hypotheses of: perfect mixing of reactor and jacket, Proportional-Integral control acting directly on the jacket temperature as manipulated variable, perfect actuation and measurements. t~p. -dT - - r~(t) - = UA(t). [Tj (t) - T~(t)] + r(t). Vr. AM f,_, = -Kc.
d[Tr (t)
T r(t)] +
(9a) (9b)
dt From this system, the two unknown T~ and Tj can be computed, once the reaction rate r(t) and the heat coefficient UA(t) are known. The imposed set point profile (Tr~ is also of sinusoidal type. In such a way, the calculated profile of UA can be compared with the a priori imposed one. Results for the temperature profiles and the heat transfer coefficients are plotted in fig. 1 and 2. It can be immediately observed how the three calculated profiles overlap, showing no differences with respect to the imposed UA profile.
610 70 58
,' , ' '"i',
..... ,'",I':" ::::i:':i';i'
;,' ::':i:i:
60 - --
5O o
,.-.,,
, i ' ' , i , ' , : : i i ' : i : i , ' i
40
54 ....
< 30
,. . . . . , , , , : ..... ,,, : , ,,, ....
52
"
,',
'
' ,'
, , '
' '
UA imposed UA1 UA2 UA3
- - -
',
,,,
2o 10
50 0
50
100 time (min)
150
200
o;
50
1 O0 time (min)
150
200
Fig.2: Comparison of the heat transfer coefficient calculated with the three different algorithms presented and the imposed value (Nominal case).
Fig.l Reactor and jacket temperature profiles (Nominal case).
3. Analysis of Experimental Data Experimental runs were carried out at the Department of Industrial Chemistry (University of Messina- Italy), where a polymerisation reactor equipped with a Temperature Oscillation Calorimeter is operating. In this case, the hypotheses on the system seem to be very close to those assumed in previous statement of the problem, as the reactor is well stirred and the thermostat acts directly on the jacket temperature and imposes very fast changes. Measurements of the jacket inlet and outlet temperature and of the reactor temperature are available at high sampling rate. Experiments were performed for the free radical polymerisation of methyl methacrylate (MMA) in ethyl acetate, using Azoisobisbutirronitrile (AIBN) as initiator. This process is characterised by a strong increase in viscosity during the course of the reaction which makes the value of the heat transfer coefficient decreasing during reaction. The measured reactor and jacket temperature profiles are reported in fig.3. The profile of UA from experimental data has been calculated with different algorithms and the results are plotted in fig.4. It is evident that the three algorithms are not equivalent to evaluate experimental data. Apart from the beginning of the reaction, where all algorithms fail because of the inapplicability of the calorimetry itself (due to the non oscillatory profiles o f temperatures), the obtained UA2 profile presents an error which makes it sensibly different from the true value. 75~ .-. =
9- -
,,, '
'
I'"'"
~
~'f'~l"
'
I ,j
~,,
~
...... , ..... , ,,.,',,,,,,.',,,,,,,, ,, . . . . . . . , ',,,,!.,,', .... ,,',,' I i,i~,:;;,,,~:,.,~:t~.,;,i,~,,,!,,j:,~ ,,,:,,,,,:,;';~:,,:,,:,~I,,,~,,,!,,,,!!,,~:,,! .... "1 14!: .... ,,,, ;~'~'~`~``~'``~`'~'~`~'~`~'~.~!~`'~'~`~``~I```~``~``~'~`~'~`'~'~1 ,|~|llii'liIlil, I~Ill.~.~.@ i liil~l .:`~`.~:;~.ii~:;~.:";`:~.~~.~.~I~.~.:~..~~.:~r.,'lilj~ . . ' " "L" 'l|'li'r ' ..... ""' .... " " ' " " " lllll ~'ilii|lllihllljt',ll|~ ....' " ' ....'""
I--
----6%
100
Reactor Temp. Jacket Temp. i
200 300 time (min)
400
Fig. 3: Experimental temperatures profiles.
40
~
I--
[
,'
10
%
~
9
I~
,*++,,,
100
UA1 UA2J UA3}
9 *++* " * * * * ~
260 300 time (min)
+
400
Fig. 4: Comparison of time varying heat transfer coefficient computed from experimental data.
611 Also, some small differences between UA1 and UA3 algorithms can be noticed. The good agreement of UA3 algorithm with the real value is verified by the comparison of the reaction conversion, computed from (2), with the experimental values obtained by gravimetry (fig.5). Therefore it is evident that the UA2 algorithm is more sensitive to errors affecting experimental data. Several causes of uncertainty may be present and make the experimental data different from the ones deriving from theoretical calculations: differences in the basic hypotheses of the method, especially time delays between the two temperatures Tr and Tj, noise in the instrumentation, errors introduced during the calculation via Fourier transform. In next section, different kind of disturbances will be analysed in order to point out their relative importance. 4. Analysis of causes of errors
4.1. As first, a time varying error on amplitude and phase of the two nominal signals Tr, Tj is added. The maximum error on the two amplitudes is respectively: 0.5~ on Tr, 1.2 ~ on Tj, quite significant if it is considered that the amplitude of Tr varies, during the course of the reaction, from 0.65 to 1.2~ and Tj from 1.4 to 4.7~ Analogously, the error on phase is up to 60 ~. The law of variation in time is such that disturbed values can have larger/smaller amplitudes and lag/lead phase with respect to the undisturbed ones. The superimposed noise makes the measured value largely different from the nominal one. By elaborating the obtained data with the three different algorithms, different results are obtained for the heat coefficient (fig.6): UA3 is almost unaffected, UA1 shows some deviations, while UA2 gives completely wrong values. 4.2. The second simulation concerns errors only on the amplitude of the signal. The same type of perturbation given in the previous case has been superimposed on the nominal temperatures. The heat transfer coefficient computed according to the three different algorithms, is reported in Fig.7. It can be seen that in this case the differences among the three algorithms and the true value are much smaller with respect to the previous case. The worse behaviour is shown, once more, by the UA2 algorithm. Therefore the three algorithms show to be much more robust with respect to errors in amplitude of the analysed signals. 4.3. To confirm the sensitivity to errors in the phase, a third series of perturbed signals has been prepared: now the error is only on the phase value. Different laws of variation in time have been adopted: constant, linear (increasing) and sinusoidal (as previously done), while, in this case, the maximum amount of the deviation is equal to 30 ~. 80
0.8
I'.~
/&& 2 t,,.!" I
601 , ~
c0.6 0
~ I
I1
/
1 /
'
I
'"'
'
''
.................... t
---
-----
1'
-
................
UA imposed]
UA1 UA2
I ]
I
', I1
.
go.4 o 0.2 0 o
100
200 t i m e (min)
366 ....
Fig. 5" Estimation of conversion: algorithm (solid) vs. gravimetry (dots)
.......
-,
460
0t
UA3
Fig.6: Comparison of heat transfer coefficients computed by means of three different algorithms (disturbance on magnitude and phase)
o
50
,
loo
t i m e (rain)
"-
15o
! "".
612 70LI . . . . . . . . . . 60~tt~, t ~ \ ' ~ ,~ '5.~,
5oll
i~I' k~, ~, "
___ i UA imposed --UA1
---.
- --
UA2
UA3
8011
LI. 601"~'-,, I ~~.", ~, 40[, ~,~~.~.........~, -.~
t
l, j
Imposed Sinusoidal dist. Linear dist. Constant dist. -----:7-~
....
I
~~176
------,-~
~'o
~60 ........ ~,o '~N~
time (min)
Fig7: Comparison of heat transfer coefficients computed by means of three different algorithms (disturbance on amplitude only)
0
50
100 time (min)
150
Fig.8: Comparison of heat transfer coefficients computed by means of the UA2 algorithm (disturbance on phase only)
Results obtained by using algorithm UA2 are reported in fig.8, for the three different perturbations. The large deviation from the correct value of UA confirms that the disturbance in the values of the phase is the main cause of errors. Results obtained by algorithms UA1, UA3, (not shown here for brevity) confirm that UA3 is very robust, while UA1 is affected up to a certain extent. Comparing fig.8 with fig.4, where values of UA from experimental data are reported, it is evident that the perturbation affecting experimental data is very close to a linear error increasing with time. This kind of error on phase can be linked to a time delay between the reactor and the jacket temperature. This fact is in contrast with the basic hypothesis of oscillation calorimetry and can be caused by a not instantaneous actuation and measuring system or by differences with respect to a CSTR, caused by an imperfect mixing of the reacting medium or segregations which may take place in the reactor. The model of the system (1) will change from first order to higher order, approximated by a first order plus time delay. An almost linear increase of time delay with time has been confirmed by a deeper analysis of data [6]. 5. Conclusions Different algorithms developed to compute the variation of the heat transfer coefficient UA(t), from profiles of reactor Tr(t) and jacket Tj(t), show to be theoretically equivalent. Errors on the amplitudes of experimental data are well tolerated by all methods, in particular the UA1 and the UA3 algorithms. On the contrary, larger sensitivity is shown to errors on the phase of experimental data by the UA1 and UA2 algorithm, making this last not suitable. By considering that errors on experimental data have a similar effect to phase errors, due to a time varying delay between the two temperatures Tr and Tj, the UA3 algorithm can be considered the more robust and can be safely used in the elaboration of experimental data. References [1] Karlsen L.G., J. Villadsen J.; Chem. Eng. Sci., 42, (1987), pp.1153 [2] Scali C., M. Morretta, Semino D.; J. Proc. Control; 7 (5), pp.357-369 (1997). [3] R. Carloff, A. Prol3, K.H. Reichert; Chem. Eng. Technol. 17 (1994) pp. 406-413. [4] A. Tietze, A. Prol3, K-H. Reichert in: Dechema Monographs, Vol. 131, 5th P.R.E., pp.673-680 (1995). [5] A. Tietze, I. Lfidke, K-H. Reichert in: Chem. Eng. Sci. Vol. 51, No 11 (1996), pp. 3131-3137. [6] P. Guerrini De Luca, MS Thesis; Chem. Eng. Dept.- University of Pisa (2000). [7] I. Ferrara, D.G. Lister, G. Maschio; Proc. ICheaP-4 Vol. 1, 1999. pp.71-74.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
613
Integer-Programming Based Algorithms and Computational Performance for Terminal-Drop Zone Assignment Problems Ming-Teck Kong" and Nilay Shah* Centre for Process Systems Engineering Imperial College of Science, Technology and Medicine, London SW7 2BY, ENGLAND The distribution of oil products usually occurs at two levels. In primary distribution, large quantities are transported from refineries to terminals (or depots) by bulk transportation methods (e.g. rail, ship, pipeline, or virtual exchange). In secondary distribution, smaller quantities are transported from terminals (or depots) to the customer (usually by road). Finns in the oil products sector oiten have a very large number of individual customers, each with an associated demand. For logistics planning purposes, the demands of customers are commonly aggregated into 'demand zones' or 'drop zones'. This reduces complexity and also improves forecast accuracy without greatly diminishing logistics performance. The assignment problem which considers which drop zone to assign to which terminal, such that a drop zone is only assigned to one terminal, is an important part of the logistics planning in the oil products sector. The terminal-drop zone assignment is a cost minimisation problem often solved using either linear programming or heuristic methods in preference over integer programming due to perceptions of high computational costs. Both commonly used methods are undesirable, the former due to the possibility of'split-feeding' which complicates operations; whilst the latter due to non-optimal assignments. The trend of facility rationalisation within the industry has made the problem much harder to solve. As oil companies reduce the number of their terminals by factors of two or greater, the throughput capacity utilisation for terminals has grown to over 90%. In these cases, heuristics grow increasingly poor in terms of performance, and linear programming approaches with heuristic split-feeding resolution could become infeasible. In this paper, the terminal-drop zone assignment problem is solved using an integer programming formulation to avoid the undesirable aspects mentioned above. Representative data for a small problem is used to generate larger data sets in several numerical experiments. Attention is focused on the computational performance of the integer programming formulation for large problem sizes. In addition, a problem specific preprocessing procedure is considered together with its effects on performance benefits and optimality degradation. The models are solved using commercially available solvers. Problem sizes of up to 10,000 drop zones and 100 terminals are solved with a reasonable computational effort. Finally, on the basis of the results, a new algoritban is proposed for this problem.
Keywords: Assignment Problem, Oil Industry, Distribution, Optimisation, and Heuristics 1. Introduction The supply and distribution function of an oil products company seeks to supply and distribute products to retail and commercial customers from terminals and refineries. Distribution activity is classified into primary and secondary distribution. Primary distribution relates to the use of bulk modes of transportation (i.e. sea, pipeline, and rail) with transit times of several days to deliver products from refineries to terminals or depots. Whilst secondary "Email: [email protected] +Email: [email protected]
614 distribution relates to the distribution of products from terminals to customer using road tankers as the main mode of transportation with transit times of hours. Assigning each customer an individual terminal leads to decomposed tactical and operational scheduling. Often, due to number of customers that have to be served, some level of aggregation is required for customers who are often clustered into "drop zones'. Drop zones represent the aggregated customer demand and can be determined quantitatively using some mixed demand/spatial metric. In practice, many companies utilise a geographical aggregation where a geographical catchment is partitioned into grids of a certain size. Terminals are linked to drop zones (at the planning level) and assigned to individual customers (at the operational level). Terminals receive large bulk quantities of product from a refinery, store it, and distribute it to customers. The terminal drop-zone assignment problem is the one that considers which drop zone to assign to which terminal such that each drop zone is only assigned to one terminal. It is solved by associating a cost to every potential linkage between terminal and drop zone, and subsequently performing a cost minimisation optimisation. Two solution methods are commonly used in industry for this problem: heuristics, and linear programming. A common heuristic would be of the greedy type that matches terminals to drop zones taking into account constraints on the terminal capacity. Heuristic algorithms have the benefit of speed, but suffer from non-optimal results. Linear programming methods offer close to optimal results, but suffer from the problem of having 'split-feeding' in the mathematical solution. Split feeding occurs when a drop zone's demand is distributed over two or more terminals. This is undesirable as it complicates automated vehicle routing and scheduling, but perhaps more dangerously, blurs the line of operational responsibility for a drop zone between two or more terminals. When split feeding occurs, it is usually either resolved manually, or through some heuristic. Facility rationalisation within the industry has made these problems much harder to solve, because as the capacity utilisation for terminals has grown to over 90% heuristics grow increasingly poor in terms of performance, and linear programming approaches with heuristic 'split-feeding' resolution could become infeasible. As such, in this paper the terminal assignment problem is solved using an integer programming formulation to overcome the undesirable aspects mentioned above. Attention is focused on the computational performance of the integer programming formulations for large problem sizes both in terms of speed and quality. In addition, a problem specific preprocessing scheme and its effects on solution speed and quality are considered. 2. Literature Mehring and Gutterman[1] describe a linear programming approach to this problem, but do not constrain each drop zone to being served by only one terminal. The problem is also considered by Sear[2] who also uses a linear programming approach, describes some commercial aspects of the problem, and provide a cost model for road/trip transportation. Van der Bruggen et aL[3] solve the assignment problem using a Lagrangian relaxation to a mixed-integer formulation at the first level of a three level decomposition algorithm which was manually tested for consistency between levels. For decentralised companies that designate customers to individual terminals, it is essential to solve the assignment problem. Firms that adopt a more centralised view sometimes allow the dynamic assignment of orders to a selection of terminals at a day to day dispatching level (for example, Bausch et al. [4]). The assignment problem can be viewed as a special instance of the NP-hard Generalised Assignment Problem (GAP) which seeks to determine the minimum cost assignment of n tasks to m agents such that each task is only assigned to one agent subject to capacity constraints on the agents. As reviewed by Cattrysse and van Wassenhove[5] there are many applications of the GAP problem such as assigning software development tasks to programmers. Most exact algorithms for GAP depend on forms of tree search. Examples of exact algorithms include those from Fisher et al.[6], Guignard and Rosewein[7] and more
615 recently from Savelsbergh[8] and Narciso and Lorena[9]. These authors use a variety of methods such as Lagrangian relaxation, combined Lagrangian/surrogate approaches, and column generation. Heuristics are also widely used: Benders and van Nunen[10] prove an important result concerning the number of fractional jobs from a LP relaxation to GAP. Trick[l l] proposed an approximation algorithm based on a linear relaxation heuristic. Shmoys and Tardos[12] propose a form of rounding procedure to restore integrality. Metaheuristics for GAP have also been widely used: for example, Osman[13] compares simulated annealing and tabu search whilst Beasley and Chu[ 14] use genetic algorithm. 3. Problem Statement A set of drop zones (c=1,2.. .NC) are served by a set of terminals (d=1,2.. .ND). Further, each drop zone has a demand DEM. Each terminal has negotiated a minimum throughput level, THR~" in its contract with its main suppliers. Further, there are upper bounds on the amount of throughput, THR','" which a terminal can handle due to limitations in its storage capacities and loading facilities. The costs of the system include variable costs arising from transportation and handling of product, together with fixed terminal and transportation costs. The assignment of terminal to drop zone is unique, such that a drop zone is served by only one terminal. The overall objective is to find the network of terminals and drop zones that minimises the total costs. To aid development of the mathematical model, all costs (fixed and variable) are assumed to be lumped into a single 'linkage cost' which determines the cost of serving a drop zone c from a terminal d per unit of product. The problem is modelled as a mathematical programming problem, specifically, one that corresponds to an integer programming problem (IP). To develop the mathematical form, we firstly define the decision variable Xce= 1 if and only if drop zone c s demand is fully served by terminal d, otherwise X~ = 0. The objective function and constraints are:
min C = ~_,E x,,ac cuDEMc c
d
~.Xcd =IVc d
r H R ~ <_y_, Sea DEMr <_rHR2 ~x V d d
where all costs are lumped into a single linkage cost cce denoting the cost of serving drop zone c from terminal d per unit of product. The first expression is the objective function. The second expression represents the 'no-split feeding condition forcing each drop zone to be served by a solitary terminal. The third expression above expresses the capacity constraint for each terminal. For this particular problem, THRT", the minimum throughput level is set to zero reducing the capacity constraints to a one sided upper bound constraint only. The linkage cost per unit of product demanded, c., is measured according to the costing data and methodologies of an individual firm but at a minimum should encapsulate the costs mentioned above. Sear[2] develops a suitable expression for this cost where road transportation cost per unit product between a particular terminal and drop zone is derived from distance travelled and time required. Some other costs are not included, specifically terminal supply costs (for own terminal or competitor terminal through exchange agreement) and operating costs, hence for this problem the following is used: cca-- VCS + VCH + RD + ~: (Own Terminal Use) c,a -- VCS + CCH + RD + ~; (Exchange Agreement) where ~: is the road transportation cost per unit product from Sear[2], and where VCS, VCH, CCH and RD are the unit variable cost of supply to a particular terminal, the unit cost of product handling at a firm owned terminal, the unit cost of product handling at an exchange agreement terminal, and the refinery differential respectively.
616
4. Solution Methodology Two main solution techniques are used to solve the assignment problem: formal optimisation and heuristics, with the latter only to provide basis of comparison to commonly used heuristic methods. Preprocessing rules for the formal optimisation are also considered. The formal optimisation approaches requires the solution of a problem which falls in the class of mixed integer linear programming (MILP) problems. The GAMS modelling and optimisation tool is used to access IBM s OSL branch and bound (B&B) solver for MILP. Heuristic methods can be used to assign a drop zone to a terminal. A typical heuristic used in industry, similar to nearest neighbour heuristics for the travelling salesman problem, was implemented and tested to provide an indication heuristic for this problem. It considers each drop zone in turn (in order of descending demand) and assigns it to the least diseconomy (i.e. minimum cost) terminal subject to the terminal not being full. Two types of pre-processing rules were used for the MILP approach to the problem under consideration. The first type of rule attempts to narrow down the solution space by considering the LP results of the relaxed integer formulation result. The second type of rule attempts to use the cost parameters of the problem to reduce the combinatorial complexity. Table 1. Description of Preprocessing Rules Relaxed IP Preprocessing Rule 1 (RPR1): Worst Cost Removal Rule 1 (WCR): 1. Solve the relaxed integer programming 1. Set w terminals out of ND to remove problem as an LP. from the problem consideration. 2. Fix all zero values of Xca Methods by which this could be 3. Resolve problem using branch & bound as determined include historical data, a MILP. trained neural net, or ranking of Relaxed IP Preprocessing Rule 2 (RPR2): matches from problem s LP relaxation. 1. Solve the relaxed integer programming 2. For each drop zone, set the values of problem as an LP. Xca=O corresponding to the w worst 2. For all c where Xcd=l, Set Xcd =0 where cost arcs (i.e. the most costly)between d ;~d. terminal and drop zone. 3. Resolve problem using branch & bound as 3. Resolve problem using branch & MILP. bound as MILP. The motivation of these rules is that at the integer solution of the problem, precisely only I/ND of the total number of integer variables are non-zero. Thus, it is sensible to attempt to fix as many variables to zero as possible before attempting the B&B procedure. Rules RPR1 and RPR2 make use of the LP relaxation to fix variables to zero. The minimum fraction of variables which can be fixed after the first LP step is 1-'/NC (by application of the result of Benders and van Nunen[10]). Rule RPR1 is a form of formalised rounding where all zero values from the LP are fixed hence allowing any split-fed drop zone to be resolved in the integer B&B. Rule RPR2 considers each drop zone, and for the case when there is no split feeding in a drop zone (from the LP) proceeds to fix any zero values in the LP solution for the possible terminals that can be assigned to the drop zone under consideration. Rule WCR is based on the observation that it is unlikely that for a given drop zone, all terminal matches are equally desirable. Thus, for each drop zone the rule removes a number w of worst cost terminal assignments a priori. This does not impact solution quality as long as in the true optimal each drop zone has matches near the individual best cost match with no capacity constraint. The value of w/ND appropriate to speed the B&B algorithm without impacting solution quality will depend on the specific parameters of a problem instance.
5. Some Computational Results and Discussion A set of data from an industrial case study (Set Ia below) was used to generate randomised sets of much larger problems. Space limitations preclude a complete description of the data generation procedure. Briefly, it comprised random demands similar to the original data set, with costs randomly uniformly distributed in the cost range of the original data set.
617 The solution quality achieved by the formal optimisation (denoted as Full below) is of the order of a few percentage points better than the heuristic, which as expected is very fast. Table 2. Summary of Test Data and Optimisation Results ....
Set Ia II a II b II c II d IIe III a IV a IV b * Results
No. Zones
No. Term.
No. Arcs
Term. Utilisation
349 7 2443 93.3% 2443 20 48860 92.7% 2443 20 48860 93.8% 2443 20 48860 92.8% 2443 20 48860 92.9% 2443 20 48860 93.7% 4886 40 195440 92.9% 10000 100 1000000 93.6% 10000 100 1000000 97.7% using WCR preprocessing rule.
Obj. Heur. 1.71 8.52 8.97 9.55 6.20 6.58 8.16 9.15 9.52
CPU(s) Heur. <1 <1 <1 <1 <1 <1 <10 <30 <30
Obj. Full
CPU(s) Full
1.49 7.93 8.26 8.80 5.79 5.96 7.44 8.46 8.60
13 1372 1878 1914 2193 2159 573* 4416" 6107"
The performance of the rules RPR1 & RPR2 was disappointing in that infeasibility arose for all cases. This can probably be attributed to the high terminal capacity utilisations involved. Hence, the computational statistics for these two heuristics are not reported. One disadvantage of the RPR family though, is that they require solving at least one complete LP, which for extremely large problems could take a significant amount of time in itself. Rule WCR performed significantly better and is considered in three areas: effect on solution times, effect on optimality, and onset of infeasibilities. Effect on solution times of increasing the fraction of worst cost terminals removed is tabulated below. Surprisingly, solution times decreased in near linear fashion with increasing fraction removed w/ND, until an infeasibility region was hit. For Set II results this was w/ND>0.9, implying having the choice of one terminal per drop zone which is infeasible given capacity constraints. Solution quality did not degrade significantly, though the network structure for solutions to the optimal relaxed LP (but not optimal integer) changed slightly when nearing the infeasibility region. Table 3. Effects of Worst Cost Removal Preprocessing Rule Opt. Gap (to Relaxed Full problem) .cPU Required (compared to Full) .... Mean Min Max Mean Min Max 0.0 0.34% 0.08% 0.76% 100.0% 100.0% 100.0% 0.25 0.25% 0.07% 0.35% 70.4% 36.2% 92.6% 0.5 0.24% 0.08% 0.29% 45.1% 27.1% 63.6% 0.75 0.30% 0.08% 0.43% 18.5% 8.1% 30.0% 0.8 0.33% 0.10% 0.44% 14.7% 7.0% 27.3% 0.85 0.43% 0.12% 1.13% 11.7% 5.1% 19.8% 0.9 0.29% 0.12% 0.40% 6.7% 3.8% 11.3%
Set IIResults w/ND
From the results of rule WCR above, three even larger problems were solved using removal fractions of fractions of 0.9 and 0.95 in Sets III and IV respectively. Without the preprocessing rules it is unlikely that an MILP approach would be successful within 24 hours on the Sun workstation used (given a million binary variables in the Set IV problem).
6. Concluding Remarks The computation experiments performed demonstrate that an integer formulation for the terminal-drop zone assignment problem is feasible, and it is desirable for its benefit of 'nosplit feed' over the linear programming approach. Several sizes of problems were solved in reasonable computation time. Without preprocessing rules, problems with up to 50,000 binary decision variables were solved in under one hour, which is equivalent to the problem size for a country such as the U.K. The use of preprocessing rules was examined. Fixing the binary decision variables of a certain number of worst cost arcs was found to be an effective
618 means of increasing solution speed without sacrificing optimality to any large extent. A combined preprocessing rule/integer formulation allowed problems of up to 10000 drop zones and 100 terminals to be solved within reasonable time. This is a problem size equivalent to a country such as the U.S.A. or to a country such as the U.K. at the client level. On the basis of the results obtained, the following algorithm is proposed. Let us first define WCR(ct) and GH to be the worst cost removal preprocessing rule with branch and bound algorithm with fraction removed ~--w/ND and greedy heuristic respectively. Then: 1. Define desired y~ percent savings over heuristic (greedy) algorithm, i.e. y=100 [c(xWCR)-
C(X")]/C(X""). . Solve GH to find C(X~) where C( ) is the objective function. 9 Set 0 close to unity, e:~. ~=~v=0.95. 4. Solve WCR(~) with X as best feasible solution initially. 5. IF WCR(~) reports Infeasible THEN Relax ~ i.e. ~=~-A~, where As is some small value e.g. 0.05 and GOTO Step 4. [Note: Infeasible above means either LP infeasibility or CPU time greater than some limit] 6. IF WCR(~) reports Optimal found OR Percentage Savings exceeds desired minimum amount, Yd< Y THEN Stop and report solution XwcR. Some of the areas in which further work would be beneficial include: (a) The testing of algorithm and preprocessing rules on large sets of commercial data; (b) The interaction between drop zone partitioning and the assignment costs/problem; (c) The effects of parallelisation of the solution procedure for computational speed. References
[1] J. Mehring and M.M. Gutterman. Supply and Distribution Planning Support at Amoco (U.K.)Limited. Interfaces, 20(4):95 -104,1990. [2] T.N. Sear. Logistics Planning in the Downstream Oil Industry. Journal of Operational Research Society, 44(1):9 -17,1993. [3] L. van der Bruggen, R.Gruson, and M. Salomon. Reconsidering the distribution structure of gasoline products for a large oil company, European Journal of Operational Research, 81:460-473, 1995. [4] D. O. Bausch, G.G. Brown, and D.Ronen. Consolidating and Dispatching Truck Shipments of Mobil Heavy Petroleum Products. Interfaces, 25(2): 1-17,1995. [5] D.G. Cattrysse and L.N. Van Wassenhove. A survey of algorithms for the generalized assignment problem. European Journal of Operational Research, 60:260 -272,1992. [6] M.L. Fisher, R. Jaikumar, and L.N. Van Wassenhove. A multiplier adjustment method for the generalized assignment problem. Management Science, 32:1095 -1103,1986. [7] M. Guignard and M. Rosenwein. An improved dual-based algorithm for the generalized assignment problem. Operations Research, 37(4):658 -663,1989. [8] M.W.P. Savelsbergh. A Branch-and-Price Algorithm for the Generalized Assignment Problem. Operations Research, 45(6):831 -841,1997. [9] M.G. Narciso and L.A.N. Lorena. Lagrangean/surrogate relaxation for generalised assignment problems. European Journal of Operational Research, 114:165 -177,1999. [ 10] J.F. Benders and J.A.E.E van Nunen. A property of assignment type mixed integer linear programming problems. Operations Research Letters, (2):47-52,1983. [ 11] M.A. Trick. A linear relaxation heuristic for the generalized assignment problem. Naval Research Logistics, 39:137-152,1992. [12] D.B.Shmoys and E.Tardos. An improved approximation algorithm for the generalized assignment problem. MathematicalProgramming, 62:461-474,1993. [13] I.H. Osman. Heuristics for the generalised assignment problem: simulated annealing and tabu search approaches. OR Spektrum, 17:211-225,1995. [14] P.C. Chu and J.E. Beasley. A Genetic Algorithm for the Generalized Assignment Problem. Computers and Operations Research, 24(1): 17 -23,1997.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
AUTOMATIC GENERATION CHEMICAL PROCESSES
OF
SWITCHING
619
START-UP
SCHEMES
FOR
E b e r h a r d K l e i n a, A l e x a n d e r I t i g i n a, J S r g R a i s c h b, A c h i m K i e n l e b
aInstitut ffir Systemdynamik und Regelungstechnik, Universits Stuttgart, Pfaffenwaldring 9, D-70550 Stuttgart, F R Germany, email: {klein, itigin}@isr.uni-stuttgart.de bMax-Planck-Institut fiir Dynamik komplexer technischer Systeme, Leipziger Str. 44, D-39120 Magdeburg, F R Germany, email: {raisch, kienle}~mpi-magdeburg.mpg.de
A b s t r a c t . This paper explains how modern techniques from the areas of hybrid and discrete event systems theory can be used to automatize the start-up procedure of chemical processes. As an application example we synthesize a discrete feedback strategy for the automatic start-up of a distillation column.
1. I N T R O D U C T I O N This contribution addresses the problem of synthesizing a discrete supervisory control scheme for the start-up procedure of chemical processes. We switch between discrete values of the control input to speed up the start-up procedure and, during start-up, rely on highly quantized measurement information. Once the system is considered to be "close" to the desired operating point, one switches to "conventional" continuous feedback. Hence, start-up of a chemical process can be interpreted as a special hybrid control p r o b l e m - the plant state "lives" in ~ n , whereas control inputs and measurement signals are discrete-valued, or symbolic. Our approach is based on "translating" the hybrid problem into a pure DES (discrete event systems) problem by approximating a given continuous plant model M by a nondeterministic finite state machine Ml which captures the behaviour of the underlying continuous plant dynamics. If specifications for start-up can also be represented by finite automata, standard methods from DES theory (e.g. [4]) can be used to synthesize a supervisory control scheme for the automaton model Ml . We will show that any controller forcing the approximation to obey the specification will also make the underlying continuous plant satisfy the specification. In Section 2, it is explained how to build an approximating automaton M1 for a given continuous plant model M. Section 3 addresses the problem of how to synthesize a discrete supervisor forcing the approximation M1 to obey the specification. It also explains why the discrete controller is guaranteed to work for the underlying continuous plant model. In Section 4, we use our approach to design a discrete supervisory feedback strategy for the automatic start-up procedure of a binary separation process in a pilot plant scale distillation column. This contribution is based on a previous conference paper [2]. The main difference is a much more efficient computational technique- we can therefore construct a more accurate discrete approximation and hence enforce stricter specifications.
620
2. A P P R O X I M A T I N G
AUTOMATA
FOR A CONTINUOUS
PLANT
MODEL
The continuous "base" model of the plant is described by the (vector) differential equation
~(t)
- f(~(t), ~(t)),
(1)
t C ~ + denotes time and x(t) C X C_ ~ n is the continuous state at time t. ud(t) C Ud is the control input, with Ud being a finite set of symbols- Ud -- {U(dl),..., U(d~)}. During start-up we are going to work in a discrete-time framework with fixed equidistant sampling grid T - {to, t l , t 2 , . . . } C ~+. Hence the control signal is piecewise constant and can only change values for t - tk, k - 0, 1,.... Likewise, measurement information is only available at the sampling instants:
y~(tk) = q~(x(tk)).
(2)
(1) and (2) will in the following be referred to as the continuous plant model M. Note that, in this terminology, a continuous model can be defined on continuous or discrete time, and a discrete model is characterized by a discrete-valued state variable. It is assumed, that equation (1) can be solved for any initial condition x(tk) and any control input ud(t) = ud(tk) in the interval tk <_ t <_ tk+l. The measurement signal Yd: T -+ Yd "lives" on a finite set of symbols Y d - (y(1),..., y(7)}; qu: X -+ Yd is the measurement map. To build a discrete approximation Ml for the continuous plant model M, we use the approach suggested in [3]: it generates Mt as a nondeterministic finite Moore automaton. The state Xd of the automaton at time ta is defined as a string of measurement and control symbols"
Xd(tk) --
([yd(tO),...,yd(tk)],[Ud(tO),...,Ud(tk-1)]), if k = 1 , . . . , 1 ([yd(tk-1), .. ,yd(tk)], [Ud(tk-l),..., Ud(tk-1)]), if k _> l,
1, (3)
i.e. Xd(tk) contains the present measurement yd(tk), and if available, the last 1 measurement and control symbols, l is a design parameter. It determines, how much information about the past is captured by the state of the automaton Ml. The measurement and control symbols in (3) take values from the sets Yd and Ud, respectively. As both sets are finite, the state set of the automaton, Xd, is also finite: an upper bound for the number of automaton states is given by the number of different strings (3) one gets by exhaustive permutation of measurement and control symbols. It is obvious, however, that the continuous plant M cannot generate all these strings. Strings which are not compatible with M are to be eliminated. Hence, for 1 _< p < l, the string ([y(Jk-p),..., y(Jk)], [U(dik-p),...,Ud_(ik-~)l~jj is an element
(ik_p) ,.. . , u (ik-~)] of control symbols of the automaton state set Xd if applying the string F LUd can make M respond with the string [y(Jk-p) Ld , ' " , y(djk)] of measurement symbols. Checking compatibility is the computational core problem. It can either be performed by using the feasibility part of optimization packages, or by cell-to-cell mapping methods. The same methods can be used to compute the transition structure for Ml (see [3]). It is intuitively clear that approximation accuracy is monotonically increasing with the design parameter 1. This fact can be formally expressed within the framework of Willems' "behavioural" system theory, see e.g. [5]. Denote the set of all functions from T into (Ud • Yd) by (Ud • Yd) T. Then, the behaviours B ( M ) C_ (Ud • Yd) T and B(Mt) C_ (Ud • Yd) T are the sets of all pairs of control and measurement signals which are compatible with the continuous plant model M and the automaton model 3///, and it can be proven, [3], that
B(Mo) ~_ B(M1) _~ . . . B ( M I ) . . . ~_ B ( M ) .
(4)
621
Relation (4) shows that every sequence of control/measurement symbols that the continuous plant model can generate, can also be produced by the automata models Ml, 1 = 0, 1,.... The "size" of the difference B(M1)\B(M) is an indicator for the accuracy of the automaton model. Increasing 1 decreases B(MI)\B(M), and hence the loss in "prediction power" when replacing the continuous plant model M by an approximating automaton Ml. 3. S Y N T H E S I S
OF SUPERVISORY
CONTROL
SCHEME
We first illustrate the idea of approximation-based control synthesis in the behavioural framework. Suppose, specifications are given as a set Bspec of legal pairs of i n p u t / o u t p u t signals, i.e. Bspec C_ (Ua x Ya) T. We want to synthesize a causal feedback controller (with input signal ua and output signal Ya) which enforces the specifications for the plant model M. However instead of dealing directly with M, we want to perform controller synthesis on the basis of the discrete approximation Ml. Obviously, the controller behaviour Bcontr is also a subset of (Ua x Ya) T. The feedback loop consisting of controller and automaton model exhibits behaviour B(Ml) A Bcontr - only those pairs of input/output signals "survive" that are compatible with both Ml and controller. From Fig.I, it is immediately clear that
(5)
B(Ml) N Bcontr C_ Bspec ==~ B(M) N Bcontr C_ Bspec;
in other words: if the controller forces the automaton model Ml to obey the specification, the feedback loop consisting of controller and continuous plant model M will also meet the specification. Within the proposed framework blocking (B(M)NBcontr = 0) cannot occur: as we work on a fixed sampling grid, "blocking" would imply that connecting controller and continuous plant model would, literally, stop time. It can of course happen that a synthesis problem cannot be solved for the automaton model Ml. This implies that either the approximating automaton M1 is too coarse (we need to provide a finer approximation Ma, k > l), or the specifications are too strict (they cannot be met no matter how accurate our approximation is) and need to be relaxed. It is obvious that increasing 1 implies increasing complexity, i.e. increasing the number of states and transitions. Hence, approximation of continuous plants for purposes of control synthesis always involves a trade-off between the desire for simplicity and the need to maintain a certain level of accuracy, or "prediction power". If specifications can also be formulated as finite automata, we can use a straightforward extension, see [3], of Ramadge's and Wonham's supervisory control philosophy [4]. It checks whether a suitable controller exists for Ml. If a solution exists, a least restrictive controller is computed. It can be interpreted as another automaton that tracks the strings of measurement and control symbols and, at each sampling instant, disables all control symbols that might allow Ml to "escape" the specifications. All other control symbols remain enabled, any of them can be picked without violating the specifications. Figure 1: Control and behaviours. 4. S T A R T - U P
PROCEDURE
FOR A DISTILLATION
COLUMN
We consider a distillation column in pilot plant scale which is operated at the Institut fiir Systemdynamik und Regelungstechnik in Stuttgart. It is about 10m high, and consists of 40 bubble
622
cap trays (consecutively numbered by z = 2,... ,41 from bottom to top), a reboiler (z = 1) and a condenser (z = 42), see Fig.2. Feed is supplied on tray 21. Our application example is the separation of methanol and propanol. [ ~i ,,@,," The following steps can be distinguished during conventional column start-up: initially, the column trays are partially filled with liquid mixture from the previous experimental run. Further feed is added, and the column is heated up until boiling point conditions are Ps, established in the whole column. During this start-up step, the column is operated at total reflux and reboil. waste At the end of this step, a single concentration front is established. The position of this front depends on the initial concentration and varies from experiment Figure 2: Distillation column. to experiment. In a second step, the feed F, reboil V and distillate flow rate D are adjusted to their desired steady state values, and the initial front splits into two fronts. Now, in a third step, the two fronts move very slowly towards their steady state position. This is illustrated by the simulation results shown in Fig.3, which have been obtained with the continuous plant model of the distillation column to be described subsequently. Start-up 40 is considered to be finished once the plant is "close 35 enough" to the desired steady state. ,T ,30 We try to speed up the third step of the start-up procedure by introducing a suitable supervisory control strategy. The starting point for our approximation based controller synthesis is a continuous distillation column plant model which incorporates the following assumptions: (1) constant molar overflows, (2) 0 0.2 0.4 0.6 0.8 constant molar liquid holdups, (3) negligible vapour mole fraction methanol [ - ] holdups, (4) total condenser, (5) constant relative Figure 3: Third step of start-up with- volatilities, (6) a tray efficiency of one. Therefore, the out control action; time interval be- model is based on material balances only and consists tween shown profiles: A t = l h . of one nonlinear first-order ODE for each tray, the reboiler, and the condenser: nZL'Xz
Y~
=
F~ +1 xz+l - F[~xz + F~ -1 y z - 1 - F ~ y z +
=
i+x~(~-i)"
Xz~
F x F if z = 21 0 else '
(6) (7)
FL denotes the liquid molar flow rate, P v the vapour flow rate and nL the molar liquid holdup;
x and y are the methanol mole fractions in the liquid and in the vapour; a=2.867 is the relative volatility, and xF=0.32 is the methanol mole fraction in the feed. The z-indexing denotes the tray number. The flow rates and liquid holdups on each tray in (6) are to be computed according to Table 1. Although this model is fairly complex from a control synthesis point of view, a chemical engineer might still consider it to be too simple to adequately describe a distillation column
623
during start-up. However because we will concentrate our efforts on the third step of startup, where thermal start-up has already been finished and boiling point conditions have been established in the whole column, the assumptions stated above are justified. As can be seen from Fig.3, during the third start-up step, the state variables X z , z = 1 , . . . , 42, are not arbitrarily distributed, but are "glued" to a three-dimensional manifold in i7~42. It is parameterized by the methanol mole fraction on the feed tray, x21, and the front positions, Ss and s t , of the wave profiles in the rectifying and the stripping section of the column. The two profiles can be represented by (see [1]): P2s -- P l s X z = P l s + 1 + eP~(Z-S~) ' z --
1,...,21;
Xz = P l r
P2r -- P l r + 1 + epr(z-s~) '
z = 21,...,42,
(8)
where Pl and p2 are the asymptotic values of the methanol mole fraction at the b o t t o m and at the top of the wave and p represents the slope of the front. The indices "r" and "s" denote variables in the stripping and the Table 1: Flow rates and liquid holdups. .......... rectifying section of z .....F~+I F~ F~ -1 F~ n~. [moll the column. The condenser 42 0 V V 0 1.922 two waves are cou22-41 V D V D V V 1.922 pled by the joint feed tray 21 V D F + V D V V 1.922 boundary condition 2-20 F+V D F t V D V V 1.922 given by t h e m e t h a reboiler 1 F+ V D F D 0 V 135 nol mole fraction on the feed tray x21. With respect to the characteristic process scenario shown in Fig.3, we further assume constant methanol mole fractions in the reboiler and condenser, xl = 0, x42 = 1 , as well as constant slopes, Ps = 0.465, Pr = 0.5717 (these are the slopes of the waves in the steady state). W i t h these assumptions the asymptotic values can be eliminated from equation (8) and one gets z = 2 , . . . ,20;
Xz = X z ( S s , x 2 1 ) ,
Xz -
Xz(Sr,X21),
z -
22,...,41.
(9)
(9) establishes an one-to-one mapping between the triple (Ss,Sr,X21) and Xz. Instead of working with 42 state variables Xz we now can work with three parameters Ss, s r and x21. This drastically reduces the computational effort - we can determine more accurate approximating a u t o m a t a and hence enforce stricter specifications. The measurement map qy implements a Table 2: Control symbols ( W s and wr are gistraightforward quantization of the three ven in [trays/10 min]) variables Ss, s t , and x21, resulting in 245 D [mol/h] V [mol/h] symbol Ws Wr measurement symbols yO) ' " " " ' Yd ~ (245) . Each -3 -3 35.8070 188.2433 u~ ) symbol represents a box in the space spanned -3 0 59.3318 158.6412 u~ ) by s s , s r and x21. For the case where s s , Sr or x21 is not in any of these boxes, an -3 3 82.8566 129.0391 u~ ) additional symbol y(d) is introduced, hence 0 -3 46.8782 217.8455 u~ ) Y d -- {Y(dI ) _ (245), y(dd) }" 0 0 70.4030 188.2433 u~ ) ,''" ,Yd 0
3
93.9278
158.6412
u~ )
3
-3
57.9494
247.4476
u~ )
3
0
81.4742
217.8455
u~ )
3
-3
104.999
188.2433
u~ )
.
.
.
.
.
.
The control signal can switch between 9 discrete values, corresponding to 3 values for both the distillate flow rate D and the vapour flow rate V (Table 2). They have been selected such that the propagation velocities
624
in the stripping and the rectifying sections Ws and wr are 0 ~-, +3 trays per sampling interval At=10min (see [2]). Start-up is considered to be finished if 8 < sr < 12, 29 < ss < 33 and 0.315 < x21 < 0.325, (123) i.e. if measurement symbol Yd occurs. Within the selected framework, it is reasonable to expect that the task can be completed within 20 minutes (i.e. two sampling intervals). This can be formally expressed through Yd~ the specification automaton shown in Fig.4: we first allow two ~ 9 Yd ~ 9 Yd 9 arbitrary measurements symbols to occur, but subsequently only (123) ~d is deemed to be acceptable. Figure 4: Specification. We now apply the controller synthesis procedure described in Section 3. It turns out that for the coarsest possible approximation, the Moore automaton M0, no adequate controller exists. Hence, we compute the more accurate approximation 3/1. For this a suitable controller does exist. It enforces the specifications not only for M1, but also for the underlying continuous plant model M. Namely, it guarantees that the third step of start-up can be finished within 20 minutes (Fig.5) for any initial condition corresponding to the measurement symbols y(d1) , ' " , ~ d~ (245) . Clearly, this represents an enormous 40 improvement compared to the open-loop behaviour 35 shown in Fig.3. Z 3o
'
~2o
1
5. C O N C L U S I O N
lO 5 o
oi2
o'.4 ....................o'.6
o'.8
mole fraction methanol [-]
1
In this paper, we have outlined how to synthesize a discrete supervisory control strategy for the startup procedure of chemical processes. The suggested method has been applied to automatize the start-up of a distillation column.
Figure 5: Third step of start-up: simulation of continuous model under discrete control; time interval between shown profiles: A t = 10rain. REFERENCES
1. KIENLE, A.: Low-order dynamic models for ideal multicomponent distillation processes using nonlinear wave propagation theory. CES, vol. 55, 2000, pp. 1817-1828. 2. KLEIN, E., KIENLE, t . , RAISCH, J.: Synthesizing a Supervisory Control Scheme for the Start-up Procedure of a Distillation Column- an Approach based on Approximating Continuous Dynamics by DES Models. Proc. LSS'98, Patras, 1998, pp. 716-721. 3. RAISCH, J. and S. D. O'YOUNG: A Totally Ordered Set of Discrete Abstractions for a given Hybrid or Continuous System. In: Hybrid Systems IV, (P. Antsaklis, W. Kohn, A. Nerode and S. Sastry, Eds.), LNCS, vol. 1273, 1997, Springer-Verlag, pp. 342-360. 4. RAMADGE, P. J. and W. M. WONHAM: Supervisory control of a class of discrete event systems. SIAM J. Contr. Optimization, vol. 25, 1987, pp. 206-230. 5. WILLEMS, J. C.: Paradigms and Puzzles in the Theory of Dynamical Systems. IEEE Transactions on Automatic Control, 36(3), 1991, pp. 259-294.
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
625
Creative Design of Distillation Flowsheets Based on Theory of Solving Inventive Problems Ben Guang Rong, Andrzej Kraslawski* and Lars Nystr6m Department of Chemical Technology, Lappeenranta University of Technology, P.O. Box 20, FIN-53851, Lappeenranta, Finland. *Email: [email protected] A new methodology of distillation process design is presented based on TRIZ (Theory of Solving Inventive Problems). In the paper, there are identified the characteristics that represent the distillation flowsheets, as well as the principles used for the improvement of distillation process designs. Finally Contradiction Matrix is constructed. Two examples are presented to illustrate the proposed methodology. 1. MOTIVATION Design in process engineering is a very complex activity. It requires the decisions to be taken at the different levels of abstraction and use of qualitative, semi-quantitative, and quantitative information. Moreover, the majority of the decisions is of multiobjective nature. They have to take into account simultaneously economic, safety, environmental, controllability, operability and other aspects of the processes. The challenges of design motivated a lot of research aiming at understanding, systematization and improvement of design process. However, the basic issue of engineering design research (EDR)- its methodology- is not often discussed. The research community usually concentrates on the so-called "received scientific methodology" (Reich, 1994). It leads to the applied research and the practical impact but not to the advancement of EDR as it is claimed. In order to better handle the engineering design research, several approaches were proposed, aiming at more scientific studies of design process (Braha and Maimon, 1998). The need for innovative as well as creative design has been also stressed by process engineering community many times. However, the results of research are not very impressive (Han and Stephanopoulos, 1995). A new method of engineering design is proposed by the Theory of Solving Inventive Problems (TRIZ). In this paper, a new approach for creativity in design of distillation processes is introduced. 2. TRIZ METHODOLOGY In TRIZ (Altshuller 1998), the solving of inventive problem is based on the resolving of the contradictions encountered when improving of some characteristics of a technical system. 40 universal principles for any technical systems are extracted based on the analysis of 200,000 patents. 39 universal characteristics of technical systems that generate contradictions are identified as well. The principles and characteristics create Contradiction Matrix. The contradictions generated by any technical system could be overcome using this matrix. It is obviously that the creative chemical process design follows the same paradigm as TRIZ based Systematic Innovation. Therefore, we can identify all the characteristics of chemical
626 processes, and extract the principles that govern the decision-making in process design. After identifying the characteristics and the contradictions, one can next search all the available principles useful in solving the conflicts. It allows to overcome the limitation due to the discrete or inadequate knowledge of the design problems. Keeping all the aspects of a design problem in mind as well as all the available principles allows to produce the nearly-optimal alternatives. 3. T ~ Z - B A S E D DISTILLATION PROCESS DESIGN
Synthesis of optimal distillation process is a combinatorial and hierarchical problem. There is no a systematic method to consider all of the possible heat integration strategies. The creativity in design process is needed to consider all the possible heat integration strategies and complex distillation schemes to obtain a global optimal solution in a reasonable time. Although there are some available heuristics for distillation process design (Nishida et al. 1981, Isla et al. 1988), these heuristics are either too general or too specific. They are usually used to construct the knowledge base in knowledge-based systems as indicated by Han, Stephanopoulos (1995). They are difficult to be combined into the existing design methodologies. The presented TRIZ-based methodology allows designers to consider all heat integration strategies through examination of all characteristics and careful consideration of the principles of distillation systems design. The promising structures and schemes of distillation flowsheets are identified first. Next, the work is focused on the computational part of design process. 3.1. Characteristics of distillation processes The characteristics of distillation processes are identified as presented in Table 1. These characteristics (parameters) are used to describe the physical states and performance of distillation systems. Some of them are very specific, while others are common to process engineering operation. Because of space limitation, the detailed descriptions of the characteristics are not presented here.
Table 1. Characteristics (Parameters) of Distillation Systems 2 separation specifications 1 feed conditions 3 constraints of components in the mixture 4 type of separation agent 6 reflux ratios 5 relative volatilities 8 amount and compositions of purges 7 number of trays 10 pressure profile 9 temperature profile 12 mass flow profile 11 energy flow profile 13 number and types of distillation columns 14 number of condensers 16 number and types of heat exchangers 15 number of reboilers 18 hot utility levels of a system 17 number and types of compressors 20 number of recycle streams in a system 19 cold utility levels of a system 22 uncertainties of a system 21 number of purge streams in a system 24 capital cost 23 complexity of a system 26 environmental impact 25 operating cost 28 flexibility 27 safety and relief 30 capacity/productivity 29 controllability 31 lifetime .cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
627 3.2. Principles of distillation process design Design of distillation process is a multi-level decision-making problem. It needs principles at the different levels to support the decision-making. However at this stage the principles are not the shallow or deep knowledge as in knowledge-based approaches for problem solving. By the principles, we understand the generic suggestions for performing an activity to, and within, a distillation system to improve some of its characteristics. For this purpose, some principles, as presented in Table 2, are very specific for designers to perform a specific activity. While others, as presented in Table 3, are very general suggestions. Because of space limitation, the detailed descriptions of the principles and their functionality are not presented here. Table 2. The SpecificPrinciples for Creative Distillation Process Design ......... 1 Change the separation method 2 Change the components(products) separation sequence 3 Change the separation agents(for mass separation agent based methods) 4 Heat Integration principle 5 Mass Integration principle 6 Change the temperature and pressure of separators 7 Change the reflux ratio of column 8 Change the number of trays of columns 9 Consider the use of multifunctional units( e.g. reactive distillation) 10 Change the number of recycle streams, their recycle flow rates and compositions 11 Change the interconnections between the units 12 Change the streams compositions 13 Increase or decrease the number of units in a system 14 Change the utility levels to match the heat flow profile of a system 15 Consider a complex distillation scheme with sidestream columns 16 Consider heat integration strategy of intermediate heat exchanging 17 Consider heat integration strategy of heat matching among simple columns 18 Consider heat integration strategy of heat pump distillation schemes 19 Consider heat integration strategy of multieffect distillation schemes 20 Consider heat integration strategy of thermally coupled distillation schemes Table 3. The General Principles for Creative Distillation Process Design 21 Identify bottlenecks frstly principle 22 Decomposition & Independence principle 23 Design process following the evolution from simple column to complex column 24 Simplification Principle(Complexity reduction principle) 25 Sequential and hierarchical process design 26 Combination of qualitative and quantitative information principle 27 Maximal utilization of available knowledge and experience principle 28 Information content balance principle ..29 Balancethe role of computer and human in design process ....................................
628 4. THE DESIGN METHODOLOGY AND EXAMPLES The given design is determined by the specific set of the parameters. The values of the parameters directly affect the system performance. Usually, designers perceive that some characteristics should be changed to improve the performance of the flowsheet. The simultaneous improvement of some characteristics usually causes the deterioration of the others. It is a source of conflicts. The basic idea of TRIZ-based method for process design is to identify the emerging conflicts. Next, the principles appropriate for resolving of those conflicts must be found. Contradiction Matrix is used to solve this problem.
4.1. Contradiction matrix and the methodology A Contradiction Matrix is constructed using the 31 generic characteristics and 29 principles which is used to identify promising principles for identified conflicts. The suggested principles can generate the most promising concepts for tackling a conflict. Here, we do not intend to provide an automatic implementation procedure of the proposed concepts for distillation process design. Our goal is to point out the possible decision-making of process design. The main function of the proposed principles is for designers to make the right decisions to direct the work for the detailed calculations or analysis towards an optimal design. The proposed approach will not be able to carry out an innovation automatically, but it could certainly suggest what parameters should be changed. There are two ways for using the presented approach: Method 1: Use the Contradiction Matrix to identify the most effective principles, Method 2: Analyze every principle and choose the most appropriate one.
4.2. Example one In the production of ethylene, the distillation column for the separation of propylene is a very large energy consumer. The typical column for propylene distillation is presented in Figure 1. The objective of design is to reduce the energy consumption. The preexisting conditions for this problem are: two components separation with one column, the available operating parameters of feed, top and bottom streams. Information for the calculation of operating and capital costs is available as well. To reduce the operating cost, the main conflicts for this problem are identified as shown in Table 4. Table 4. The identified principles for example one Conflicts Coordinates in matrix 1. operating cost/capital cost 25 • 24 2. operating cost/complexity 25 • 23 .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Suggested Principles 9,4,15,7 23, 22, 24, 28
From the suggested principles, the most appropriate is principle 4 - heat integration. Among heat integration strategies for single column, the available ones are intermediate heat exchanging (principle 16), heat pump schemes (principle 18), multieffect schemes (principle 19). For this case, based on the available information, the most promising one is Heat Pump distillation Column (HPC). For this column, the operating pressure is relatively high, we suggest the using of bottom liquid flashing heat pump scheme. Based on the above decisions, the calculations are performed and the obtained flowsheet is shown in Figure 2 (Rong 1996).
629
D, Pd, Td, Xl, Qc
Per, Tcr, Qcr Qca Pd, Td
I
F, Xl, x2, q, Pf, Tf
[
wf I
I
Pf, Tf PD, TD Qr~ "1...x2, Qr
Figure 1. The conventional distillation column for example one
Figure 2. The proposed heat pump distillation scheme of example one
The obtained results are as follows. Operating Cost Saving = 572843.0 $/yr, Capital Cost Increasing = 169151.0 $/yr, Total Annual Cost Saving = 555928.0 S/yr. In this case, it is also easy to identify the appropriate principles by using Method 2 (analysis of all the principles one by one). As mentioned before, several solutions may be applicable. In this example the double-effect distillation scheme is also a promising alternative. 4.3. Example two The separation of the effluent mixture produced by crude oil refining is studied in this example. The components and the mole fractions of the mixture are as follows. A: ethane(0.0263) , B: propylene(0.2796), C: propane(0.0926), D: i-butane(0.2064), E: ibutylene(0.1140), F: butylene-l(0.0638), G: n-butane(0.0528), H: butylene-2(0.1504), I: npentane(0.0141). Feed conditions: F-343.13 kmol/hr, Tf=40.0~ Pf=20.50 kg/cm 2, q-l.0. The separated products are AB, C, DEF, GH, and I. The existing operating flowsheet is shown in Figure 3. Its Total Annual Cost TAC= 127.46 • 104 S/yr.
AB
DEF
G
AB
~c I "q- ..I t "-l- d
I "T-..I I "T..c
Figure 3. Existing operating distillation process of example problem two
I PI
..I
v
Figure 4. The proposed heat-integrated distillation process for example two
630 The imposed constraints on proposed retrofit process are efficient energy reduction, less complexity and lower capital investment. By using method 2, the most appropriate principles are principles 23 and 4. And based on the shortcut calculation and analysis of the operating temperature and pressure, as well as heat duties in condensers and reboilers, a heat-integrated flowsheet is obtained as illustrated in Figure 4. Its Total Annual Cost TAC=88.08 x 104 S/yr. The saved Total Annual Cost is ATAC=39.39 x 104 S/yr. This problem demonstrated that the proposed principles are simple but useful and powerful aid for the designers. 5. SUMMARY In this paper, a TRIZ-based methodology for inventive distillation process design is proposed. 31 general characteristics are identified. Moreover, 20 specific principles and 9 general principles for the improvement of distillation flowsheets are proposed. A Contradiction Matrix is constructed for identification of the appropriate principles. This systematic methodology can support the decision-making at the early stages of design. Two examples demonstrated the applicability of the proposed methodology in design of distillation processes. ACKNOWLEDGEMENT
Ben-Guang Rong is grateful to Center for Intemational Mobility (CIMO) of Finland for the support during the preparation of this publication. NOTATIONS B = bottom product flow rate D = column distillate flow rate F = feed flow rate PD = column bottom pressure Pcr = pressure of consenser/reboiler of HPC Pd = column top pressure Pf = pressure of feed stream q = quality of feed stream Qc = heat duty of condenser Qca = heat duty of auxiliary condenser
Qcr = heat duty of consenser/reboiler of HPC Qr = heat duty of reboiler Qra = heat duty of auxiliary reboiler of HPC TAC = total annual cost of a flowsheet Tb = column bottom temperature Tcr = temperature of consenser/reboiler Td = column top temperature Tf = temperature of feed stream Wf = shaft work of compressor xi = mole fraction of component i
REFERENCES 1. Altshuller G., 40 Principles: TRIZ Keys to Technical Innovation, Technical Innovation Center, Inc. MA, 1998. 2. Braha D., O. Maimon, A Mathematical Theory of Design: Foundations, Algorithms and Applications, Kluwer Academic Publishers, 1998. 3. Han C. H. and G. Stephanopoulos, 1st Inter. Conference on Intelligent Systems in Process Engineering, Snowmass, Colorado, July, 9-14, 1995. 4. Isla M. A. and J. Cerda, The Chem. Eng. Journal, 38, 16 l-177, 1988. 5. Nishida N., G. Stephanopoulos and A. W: Westerberg, AIChE J., 27, 321-351, 1981. 6. Reich Y., AI EDAM, 8, 263-274, 1994 7. Rong B. G., Study on the Synthesis of Optimum Heat-Integrated Distillation Flowsheets, Ph.D. Thesis, University of Petroleum Beijing, 1996.
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
Technological aromatics
change
by
system
design-
631
the
industrial
production
of
G.P.J. Dijkemd ~and J. Grievink b :~Department of Technology, Policy and Management, Delft University of Technology, Jaffalaan 5, 2628 BX Delft, The Netherlands; [email protected] bDepartment of Chemical Technology, Delft University of Technology, Julianalaan 132, 2628 LX Delft; The Netherlands; j.grievink @stm.tudelft.nl The objective of the presently reported work was to investigate the usefulness of functional modelling to explore technological change within, at and above the single process level: process system innovation-. An iterative decomposition procedure for complex production systems was developed which is completed when a set of functional elements is selected that meets tile criterion of functional cohesion. In the application to the industrial aromatics system, it is demonstrated that functional modelling provides a structured mechanism for abstraction that can yield novel directions for R&D. 1. I N T R O D U C T I O N We conjecture that a part of the research and design-space in the chemical industry has been under-utilised: process development that implies or relies on a change of network structure. The objective of the presently reported work was to investigate the usefulness of functional modelling [1] to explore this R&D-space. We used functional modelling as a basis for tile assessment of a large process network in the chemical industry, and for the identification of long-term options for process at the network and process level, an open problem. The production of aromatics was addressed as a test case. In tile paper, firstly functional modelling is introduced. Secondly the modelling of the aromatics system is reported and the assessment results are summarised. Thirdly, by functional modelling options for process system innovation are identified. Finally, conclusions are drawn with respect to the suitability of functional modelling to address open problems in process system engineering. 2. PROCESS SYSTEM INNOVATION Where unit operations constitute the building blocks of an individual plant design, individual process plants are the building blocks of the chemical industry. Thus, we expect that there is scope to improve the chemical industry by rearranging or reselecting its buildingblocks, whilst the development of new blocks remains an option, and technology change within existing building-blocks is taken into account. This combined set of technological change -within, at and above the single process level- we label process system innovation. Whilst the evaluation and selection of options for process system innovation comprise a closed problem that can be solved e.g. by optimisation, the exploration and identification of
632 options represents an open problem. In process system innovation the search is for promising but yet unknown R&D directions, for which the solution may involve a change in technology and system boundary. We therefore adapted and applied functional modelling and decomposition [1, 2] to address the identification of a portfolio for long-term technological developments in the chemical industry.
2.1. Functional modelling In technical objects, a function is fairly synonymous with a black box that is filled with operations leading to the fulfilment of an objective. Hence, a black-box with function X may serve to achieve objective Y. The function is generally concerned with what should be achieved, not how, and can be expressed as a general means of action. An objective def7ned /Traction is defined as a sub-set of the operations of a system that consist only of those operations directly necessary to the achievement of the same specific objective, or set of specific objectives [2]. A physical system, the black box, is completely defined by its o[!iective-defined functions. Its particular means of action is not described nor prescribed, i.e. its specification is completely technology-free [3]. 2.2. Towards a decomposition procedure When we are confionted with the task of assessing and improving an existing system, we would prefer some decomposition procedure that includes a stop criterion. The concept of objcctive-defined functions opens the way to a functional decomposition of systems by apl?ropriate selection, as each objective-defined function must relate to a sub-set of the operations of a system, or a subsystem. Since the subsystem' objectives serve to fulfil the general o[!jective of the system a hierarchy of functions appears (see Fig. 1). In computer system modelling, cohesion methods are used to arrive at an adequate degree of loose coupling between subsystems or system elements. In case the target decomposition-depth has been selected, achieving cohesion provides a stop-criterion for the functional decomposition.
ective Defi, edFunc .
Fig. I. A hierarchy of functions
633 A model is considered functionally cohesive in case the set of objective-defined functions is mutually contingent. A set of objectives S is said to be mutually contingent in case V (X,Y)e S S = (X AND Y) OR S = ((X OR Y) AND NOT (X AND Y)) OR S = (X- +Y)
(both objectives are achieved always in parallel) (mutual exclusion, either one is achieved, but not both.) (achievement of X leads to achievement of Y).
When a system is decomposed in functional elements F (FI..F,) with respective objectives S (S~..S.,,), where in> n, for each pair of objectives (X,Y) ~ S, the decomposition is complete when there exist instances of the system only that meet the above criterion. 2.3. E x a m p l e Consider a system, for example that is decomposed in three elements with functional ol!jectives, X, Y and Z. Table l" States of an exalnpl e syste m ...................... Objective Possible states X True Instance 1 Instance 2 Instance 3
Y True
z
True True
Table 1 lists all possible states of the system, its instances. In this case, the set (X,Y,Z) is not mutually contingent, as instance 1 and 3 violate the condition of mutual exclusion. A subsystem described by the subset (Y,Z), however, does meet this criterion. A list of approximate steps to follow can now be given (See Fig. 2). The procedure is iterative, and domain specific knowledge is required to describe the functions of systems, subsystems and system elements. The method implies some 'trial-and-error', because the execution of step 1 already assumes some consideration of the systems objectives and possible formulation of system intermediate objective. To start the iteration, it is suggested to use an initial guess based on present knowledge of the decomposition of a particular system. {~i
) Identity parent objective(s) b) List elements of initial decomposition
|
Eliminate unsufficiently related elements ~ _ from groups
7
5
I' 2 ~ I Identify inte,mediate objectives .........
~
y
_ _
.
Group elements that are closely related per intermediate objective 9
, .........
~_~ 6
Fig. 2: Decomposition Procedure
Form separate groups, of elements a) previously eliminated - b) common ... 9c) tightly coupled ... ... to a number of different groups
i iii ....... Check for mutual contingency
634 3. THE INDUSTRIAL AROMATICS SYSTEM Benzene, toluene, and ortho- and para-xylene (B, T, oX and pX respectively) originally were produced flom cokes-oven benzole. Today, in Europe these are mainly obtained fiom pyrolysis gasoline, one of the by-products of the steamcracker. Alternatively, these compounds can be isolated from intermediate product streams of a petroleum refinery. B, T oX and pX are used for the production of a variety of commodity chemicals, notably monomers such as styrene, caprolactam, terephtalic acid and phthalic anhydride. In the Netherlands alone the aromatics system comprises some 40 different plants. To tormulate options for an R&D portfolio, we first need a functional model of this system that allows suitable assessment. Our model of the 'industrial aromatics system' starts with the production of BTX-rich feedstock from fossil resources, and it ends where the monomers derived from B, T oX or pX respectively are converted into polymers, or where the commodity chemicals are sold for other final use. At the top-level of abstraction, one can distinguish a production system for pure B, T, oX and pX, and a system where consumption of these aromatics is the common denominator. This is a functional cohesive decomposition as these objectives lneet the criterion of mutual contingency when we relate the objective to a mole of B, T, oX and pX respectively (Figure 3).
Lprocess
I polymerisation I PS ABS Nylon etc.
I process i
] polymerisation]
Orl'de~ t Re,,ner, f --f
st. Cracker
BTX pro!
~176 t CokesovenI BTXProd
oX pX
BTXCon
Fig. 3. Functional model of the aromatics system Further decomposition of BTXCon is straightforward, because we can define intermediate and final f\lnctional objectives as the intermediates and polymers produced. Decomposition of the BTX production system, however, is not straightforward. After isolation of the aromatics, somewhere in the production system BTX is transformed to essentially pure B, T, oX and pX, In 'present art' there exist a variety of system configuration, that depend on feedstock selection and required product-mix. Functional modelling provides a method for abstraction to define a model that encompasses all possible realisation of the conversion of complex hydrocarbon mixtures to the required product-mix of B, T oX and pX (Figure 3) The definition of the following objective-defined functions suffices: 1. produce BTX-rich stream 2. pretreat and precondition 3. separate aromatics / non-aromatics 4. produced pure B, 5. produce pure T
6. produce mixed-X 7. produce pX 8. produce oX 9. produce B X-mixture 10. produce TX mixture
635 By suitable aggregation, models can be constructed for the assessment of existing industrial systems, for instance those that include the production of B by hydrodealkylation of T and/or X, or the production of B and X by the disproportionation of T.
3. I. Pertormance Assessment Mass- and energy balances were used to compute (sub)system efficiency and losses. The o[!iective of the analysis is to develop a quantitative image of the production system, and to determine the position of processes relative to each other, in order to obtain a global priority for the search for improvements. Since it is neither the objective nor possible to judge the actual situation in individual production locations, the data used relate to the situation in or before 1990 are considered to be suitable. In the assessment of the processes four simple criteria are used, namely mass loss (ML), mass efficiency(ME), energy loss (EL), and energy efficiency (EE). Mass or energy loss ,fi'om the system is in the form of unwanted by-products or combustion products (CO2, H20). I-llaSs]
energy"
quantity main product plus required by-products per quantity of resources energy-content of main products and required by-products per energy-contents of resources, catalysts and 'utilities'
3.2. Analysis total production system The overall energy efficiency of the Dutch aromatics system is some 77%, and annually some total 110 PJ is 'lost' by degradation to inferior, unwanted by-products. The associated energy is removed from the system as waste-heat by cooling, via flue gases or in the form of unwanted by-product flows. The results for the whole case are presented in table 2. The total losses ot: the analysed aromatics system BTX-Tot are evenly spread over the two parts. Table 2 Results for the aromatics system Process ME [%]
ML [Mt/yr]
EE [%]
EL [PJ/yr]
BTX-Prod
93
0.65
86
53
BTX-Con
86
0.69
68
63
BTX-Tot
89
1.34
77
116
3.3.Detailed analysis of BTXProd and BTXCon Since the results do merit priority attention to one of the two functional elements, both were analysed in more detail using the formula introduced above. The selection between wanted and unwanted by-products is essential to interpret the results. The caprolactam process, for example, has a low mass efficiency because a large quantity ammoniumsulphate is formed in this process. On the basis of the used literature, e.g. [4], this stream is seen as an unwanted by-product in the analysis. In case ammoniumsulphate is a wanted by-product, then the mass loss reduces dramatically. From a detailed analysis of the BTX system it was concluded that the stabilisation of pyrolysis gasoline and hydrodealkylation of toluene/xylene to benzene cause the largest losses in the production system, while styrene and caprolactam create the losses in BTXCon.
636 4. T O W A R D S AN R & D P O R T F O L I O In the search for R&D directions, it is common to look at (incremental) improvement of currently employed processes. Using functional modelling, the same can be done. In the process for of the stabilisation of pyrolysis gasoline, for example, the efficiency can be increased by replacement of the selective hydrogenation of 'impurities' in the feedstock. It appears that these 'impurities' can be recovered by enhanced extraction as valuable products. Whilst this involves a change in the realisation of function 2., pre-treatment, actually a new functional element is introduced: 'recover impurities X as products', which will have an effect beyond the particular process studied. Further to intra-process innovations, we can search for process system innovations using the abstraction by functional modelling. One might argue, for example, that the primary ti~nction of hydrodealkylation is to upgrade toluene that would otherwise be sent to the gasoline-pool. As has been stated, this process involves a relatively high energy-loss, so would it be possible to construct a production system where this function is not economic anymore? The continued development of applications of toluene and xylene that replace the use o1: benzene can lead to such process system innovations. A new intermediate function can be introduced in BTXCon for example the conversion of toluene into para-methyl-styrene (PMS) [5]. The polymer, poly-para-methyl-styrene (PPMS) can replace ordinary polystyrene in a range of applications, i.e. its functions. As a consequence, the actual realisation of BTXProd will be affected, because the operation of the dealkylation process can be avoided, as well as its associated loss. Its functional model, however, remains the same because dealkylation is only one option for realisation of function 4, produce pure B.
5. CONCLUSION The results and examples given demonstrate the usefulness of functional modelling for assessment of existing industrial systems, and the associated identification of novel directions lk~r technological change. Functional modelling provides a structured method for abstraction, th:~t allows one to fleely consider novel realisations of existing functions of the chemical industry at every system level of interest, and thereby address the open problem of process system innovation. REFERENCES I. G.P.J. Dijkema and M.A. Reuter, Dealing with complexity in material cycles, Comp. Chem. Eng. 23, $795-798, 1999 2. E.N. Baylin, Functional Modelling of systems, Studies in Cybernetics 21, Gordon and Breach, New York, 1990 3. O.A. As[!jornsen, System Engineering Principles, Skarpodd, Houston, 1992 4. A. Chauvel and G. Lefebvre, Petrochemical Processes, Technical and Economic Characteristics, (vol. 1 and 2.), Gulf. PuN. Houston, Texas, 1989 5. W.W. Keading, L.B. Young and G. Prapas, Para-methylstyrene, ChemTech, sept. 1982, 556-562
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
637
Symmetric multiprocessing algorithm for conceptual process design Eric S Fraga Department of Chemical Engineering, University College London, Torrington Place, London, United Kingdom WC1E 7JE, e . f r a g a @ u c l , a c . uk Automated process synthesis can be a computationally demanding application. However, the recent introduction of affordable personal computers based on the symmetric multiprocessing architecture makes it feasible to consider parallel applications for everyday use. This paper describes a new algorithm for process synthesis which is based on multithreading and shared memory. The new algorithm has been implemented in Java and results for the computationally intensive problem of looking at dynamic behaviour in synthesis are presented.
1. INTRODUCTION Automated process synthesis is based on the solution of mixed integer nonlinear programming problems. Although a number of approaches have been taken [5], one approach that has been shown to be particularly successful for a large class of problems is the use of implicit enumeration with dynamic programming and branch and bound [ 1,2]. In particular, implicit enumeration techniques are efficient when implemented on distributed memory multicomputers [3,4]. Distributed memory multicomputers are either not commonly available or are difficult to manage. Recently, however, we have seen the introduction of affordable symmetric multiprocessing (SMP) personal computers. These systems have initially been made available in dual processor configurations although 4 and 8 processor systems are now appearing. The main advantage of SMP systems is the shared memory paradigm they implement. Shared memory encourages the development of new parallel algorithms. This paper describes one such algorithm for automated process synthesis. The implementation makes good use of the resources provided by SMP-PCs and therefore provides an affordable route for the use of parallel computing in industrial applications.
1.1. The Jacaranda system for automated synthesis The Jacaranda system is the software implementation of an implicit enumeration algorithm for automated process synthesis. The use of implicit enumeration, combined with dynamic programming and branch and bound, together with the underlying discrete programming approach, yields the following advantages: 9 Problem formulation consists of the list of processing technologies available, the list of available raw materials, the desired product specifications and a set of ranking criteria for process selection.
638 class
Problem
method
init()
best
+- n e w
end method
method
List()
solve()
while
hasMoreElements()
node
~
nextElement()
do
node.evaluate()
best.insert(node)
end while
end method
method
method inner
hasMoreElements()
class
method
end
end
nextElement()
class
Node
evaluate()
...
...
returns
returns
boolean
boolean
...
class
Fig. 1. O~ect oriented framework ~ r search procedure. 9 The use of discrete programming makes it possible to generate efficiently and simultaneously a ranked list of solutions for each of the ranking criteria specified [6]. Jacaranda is ideal as an exploration tool in early design. The results of Jacaranda can subsequently be used as input for more rigorous and detailed evaluation although there are no limitations on the rigour or detail of the models used in Jacaranda. Figure 1 presents the pseudo-code for the search procedure. The problem object represents a particular sub-problem in the search graph. A sub-problem is defined by a set of streams and the desired or required properties of the solution. The solution procedure consists of generating a set of nodes corresponding to unit designs. The h a s M o r e E l e m e n t s and n e x t E l e m e n t methods together implement the implicit enumeration aspects of the algorithm through the use of the unit models available for the problem. Unit designs may generate output streams and each output stream is used to define a new sub-problem. The procedure is therefore recursive and is based on a depth-first pre-order generation and traversal of the search graph. The use of discretization ensures a finite search graph and enables the use of dynamic programming. Jacaranda is written in Java and provides a generic object oriented framework for automated synthesis. The genetic nature of the framework provides a basis for the implementation of a parallel version which should inherit the positive attributes of the sequential version. 2. THE IE-SMP ALGORITHM
This section describes a new algorithm, based on an implicit enumeration approach, for implementation on SMP systems. The aim is to retain all the positive attributes of the sequential approach while enhancing efficiency through multiprocessing. Shared memory makes it possible to implement a parallel algorithm which mimicks the behaviour of the sequential algorithm. Previous parallel approaches have been based on designing a new procedure for generating and traversing the search graph. Fraga & McKinnon [3] described a dynamic programming
639 class Problem m e t h o d init() b e s t +- n e w List() end m e t h o d method doStep() w h i l e n o d e s a v a i l a b l e do if h a s M o r e E l e m e n t s ( ) then jobs.put(nextElement()) e n d if end while
class Node method doStep() e v a l u a t e () for e a c h u n i t o u t p u t o do j o b s . p u t (new P r o b l e m ( o ) ) e n d f or e n d me t h o d
end
class
end method
end
class
Fig. 2. IE-SMP algorithm: Problem and implicit enumeration node classes. approach. Although scalable to large numbers of processors, the search graph generated was larger than required by the sequential procedure. An improved method was subsequently developed based on a two step procedure in which a tighter search graph was generated before traversing it [4]. However, this method still suffered from the loss of the effect of pruning by the branch and bound procedure used in the sequential method. For small numbers of processors, the parallel implementation was less effective. The motivation for the new approach, therefore, is to preserve both the dynamic programming and branch and bound aspects of the sequential approach and be efficient for small numbers of processors. The new parallel algorithm is based on the same branch & bound depth-first pre-order traversal of the search graph. Parallelism is inherent in this procedure because there are often several units which may be considered for processing the streams associated with any subproblem and because each unit often has a set of design alternatives for the available feed streams. Both aspects will be exploited as neither is sufficient. There are problems in which either there is only one type of processing unit (e.g. distillation sequence synthesis) or where units have few processing alternatives (e.g. in biochemical processes). To ensure load balance, an asynchronous multithreaded job queue scheme is used. A set of threads is created and each is responsible for both enumerating nodes in the search graph and evaluating specific nodes. Each thread retrieves a job from the job queue and executes one step of the job (via the d o S t e p method). A step may create new jobs; these jobs are placed in the job queue and a link between the new jobs and the current job is created so that the current job can be notified when the new jobs have been completed. There are two types ofjobs: ones that correspond to the enumeration of nodes and those which evaluate specific nodes. Figure 2 presents the pseudo-code for the new parallel algorithm. There is one parameter which controls the overall behaviour of the new algorithm: each new problem object has associated with it a set of implicit enumeration nodes. When the enumeration procedure identifies a new node in the search graph, a node object is retrieved from the set. In the sequential algorithm, there is essentially one node available for each problem object. By increasing this number, the degree of parallelism is increased. By default, in the parallel implementation, the set consists of two enumeration nodes.
640
P2 P1
P2
H
@
Pl
Seq
|
Seq 0
1O0 200 300 Elapsed Time (s)
400
(a) Linux version 2.2.5-15smp with IBM's JDK, version 1.1.8.
0
1O0 200 300 Elapsed Time (s)
400
(b) MS Windows NT 4.0 with Sun's JDK, version 1.2.2.
Fig. 3. Timings for 3 component separation problem on a dual 450 MHz Pentium II Xeon system. 3. RESULTS The primary motivation for the use of parallel computing is to reduce computational times. Historically, in process synthesis, simple models have been used to alleviate computational resource problems. Recently, however, the need for high fidelity modelling has become apparent, especially for design for controllability, operability and reliability. One of our interests is the design of operable processes and the application of synthesis procedures to this problem. Therefore, the sample problem comes from that area. We are interested in generating process flowsheets that exhibit good behaviour in a variety of aspects: economic performance (capital and operating costs), maximum deviation from steady state due to disturbances in the feed stream, and the time to steady state from start-up. Using the multicriteria feature of Jacaranda, all three aspects can be considered simultaneously. The need for dynamic models poses a significant impact on resources. Therefore, this problem is ideal for parallel computation. The implementation of the SMP version of Jacaranda is based on version 1.1 of Java. Although the majority of development has been undertaken on a Linux system, the resulting code is indeed write once, run anywhere, as Sun Microsystems claims. The Java language provides the basic requirements for defining, using and manipulating multiple threads within a single program. The basics for synchronisation are also provided. However, performance can vary between different architectures. To demonstrate the algorithm, we first consider a small three component separation problem. Separation is based on distillation which is modelled using a rigorous tray by tray procedure. Figure 3(a) shows the task allocation for the problem both sequentially (the bottom of the three time graphs) and in parallel (the two top graphs, each of which corresponds to one processor on a dual processor Pentium II Xeon 450 MHz system). These timing results were generated using Linux 2.2.5-15smp as distributed with Red Hat 6.0. The parallel implementation is not particularly effective at reducing the computational time involved. There are four instances of unit model designs in this problem: 1. separation of component
641 A from BC, 2. separation of AB from C, 3. separation of A from B, and 4. separation of B from C, as indicated on the figure. The same task takes longer in the parallel version than in the sequential case. Analysis of the processor usage during the parallel run shows that the system seldom reaches 100% utilisation. It is important to note that the times recorded are wall-clock or elapsed times as opposed to actual CPU time. At first glance, one could assume that the reduction in computational efficiency (on a per processor basis) is due to the new algorithm's implementation. For instance, the new algorithm does result in an increase in object management, purely on the basis of having a queue of jobs to handle. However, using the same hardware, we have solved the same problem using MS Windows NT and the results are shown in Figure 3(b). The sequential version is about 10% faster on Windows NT as compared with Linux. Version 1.2 of Java from Sun was used in Windows NT and version 1.1.8 from IBM on Linux. The interesting result is that the parallel version on Windows does make efficient use of the two processors. The elapsed time reduces from 350 seconds to a little over 200. We do not expect 100% efficiency (i.e. a reduction to 180 or so seconds) for a problem with so few tasks due to the coarse granularity of the task definition. The labelling on the graph shows that each task takes approximately the same amount of time either sequentially or in parallel. One possible conclusion is that the handling of threads is better on Windows NT than on Linux. This can be due to either the operating system or the actual Java run-time system. Indications from the current work on the Linux kernel are that it is the former that is responsible for the loss of efficiency on Linux. The next version of the kernel for Linux is promised to be more efficient for symmetric multiprocessing systems. In fact, some tests with the latest development Linux kernel (version 2.3.33) show an improvement of approximately 15-20% over the current stable version. Further improvements are expected so it will be interesting to re-visit these results in the coming year. An interesting side-effect noticed on Windows NT is that one processor appears to be faster than the other. In particular, tasks on the "first" processor (middle time line in the figure) are completed in approximately the same amount of time as in the sequential approach. Tasks on the second processor, however, take a little longer (see tasks 2 and 3). There are two possible explanations for this and both are based on how the scheduler in Windows NT works: The scheduler may be biased to the first processor or the garbage collecting thread in Java may be allocated to one processor and remains there for the life of the virtual machine. It should also be noted that the sequential approach does make use of the multiple processors available on the system. The Java runtime system uses multiple threads and so the garbage collection thread, for instance, will typically use the idle processor. A larger 5 component separation problem has also been solved. The timings for the NT version are shown in Figure 4(a). Again, the total elapsed time has been reduced significantly. However, it can be seen that the second processor is idle for a significant amount of time at around 1500 seconds into the problem. This is because the job queue has emptied and no new jobs are generated until the job currently being processed by the first processor finishes. One parameter that can affect this behaviour is the number of simultaneous implicit enumeration nodes to generate for any given sub-problem. Figure 4(a) corresponds to the generation of just 2 nodes for each sub-problem. If we increase this number, this will have the effect of increasing the number of jobs sitting in the job queue at any moment. Figure 4(b) shows what happens if we increase the number of available implicit enumeration nodes to 4.
642
P2 P1 Seq
I ! IIIII II IIil II IIII II Illl II IIII I
0
|
1000
i
2000
I
3000
P2 P1 Seq
I III IIIIk III IIIII Ill ! I IIII II I III 11Illl |
4000
Elapsed Time (s) (a) With 2 simultaneous implicit enumerationnodes.
0
i
1000
i
2000
i
3000
4000
Elapsed Time (s) (b) With 4 simultaneous implicit enumerationnodes.
Fig. 4. Timings for 5 component separation problem on Windows NT. The gap disappears and the overall elapsed time decreases accordingly. In general, increasing the number of nodes leads to a broader front through the search graph and thereby reduces the amount of pruning that may be possible. For the problem discussed in this paper, this is not an issue as the search is essentially exhaustive due to the multi-criteria nature of the ranking. 4. CONCLUSIONS Symmetric multiprocessing is increasingly affordable and provides an easy route for increasing the computational resource available on engineers' desks. The use of Java, with its built-in support for multithreading, enables the use of multiple processors in a shared memory architecture. Together the result is an easy to use parallel computing resource. Synthesis can be a computationally demanding application and the Jacaranda system has shown how it is possible to use SMP systems for interesting synthesis problems. Symmetric multiprocessing provides a shared memory architecture which makes it possible to implement parallel algorithms which inherit the positive features of the sequential algorithm. Java provides a good framework for multithreaded applications. The promise of write once, run anywhere is achieved in terms of code portability but not necessarily in terms of performance. Improvements in the underlying thread support in some systems is needed before multithreaded applications can be truly portable across the wide range of systems used in practice. REFERENCES 1. Fraga, E S, 1998, Chem Eng Res Des 76(A1) 45-54. 2. Fraga, E S & K I M McKinnon, 1994, Chem Eng Res Des 72(A3) 389-394. 3. Fraga, E S & K I M McKinnon, 1994, Computers chem. Engng 18(1) 1-13. 4. Fraga, E S & K I M McKinnon, 1995, Computers chem. Engng, 19(6/7) 759-773. 5. Grossmann, I E, J A Caballero & H Yeomans, 1999, Korean J Chem Eng 16(4) 407-426. 6. Steffens, M A, E S Fraga, & I D L Bogle, 1999, Computers chem. Engng 23(10) 14551467.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
Optimisation
of distillation and pervaporation
643
system for ethanol
dehydration Z. Szitkai, Z. Lelkes, E. Rev, Z. Fonyo Chemical Engineering Department, Technical University of Budapest, H-1521 Budapest, Hungary Ethanol dehydration systems consisting of a distillation column and pervaporation modules are rigorously 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 deterlnined. A method for radically decreasing the number of equivalent structures covered in the superstructure is suggested and applied. Computational experiences with GAMS DICOPT are presented. With full structural multiplicity, the solver was not able to determine the optimal structure; but with reduced multiplicity the optimal structure has been deterlnined. Optimal structures with and without distillation column are presented. 1. I N T R O D U C T I O N FOUl" ethanol dehydration process classes are known in the literature, namely adsorption, distillation, pervaporation, and hybrid procedures. Distillation processes are the most widespread in industrial practice. Either a pressure-swing distillation is used, with the consequences of extra costs; or a third component as entrainer, with its unfavourable side effects, is applied in the extractive and azeotrope distillation [2,7]. Pervaporation is an emerging membrane separation technology with the merit of low operation costs. The disadvantages of the pervaporation are the low maximal capacity and the high capital costs [3]. The most promising technologies are the hybrid processes, especially the distillationpervaporation system. Although these hybrid methods are most economical, the design and optimisation of these systems are difficult. In this article the hybrid distillation-pervaporation process is dealt with. In our MINLP formulation both the distillation column and the membrane modules are rigorously modelled. An optimisation of the pervaporation system is already presented by Srinivas and El-Halwagi [5]. They used a state space model, which was optimised by MINLP, but did not address the multiplicity issue. Viswanathan and Grossmann optimised the distillation column with rigorous MINLP modelling [6]. On the other hand, they did not consider capital and oper,ttion costs but optimised for the number of theoretical trays at minimal reflux ratio. Smlder and Soukup [4] experimentally determined the permeate concentration and flux in the li~nction of feed concentration at different temperatures in a pilot plant of ethanol dehydration.
644
2. P R O B L E M S T A T E M E N T The problem is identical to one of those given by Sander and Soukup [4]. The flow rate ol: an ethanol-water mixture from fermentation is 408.7 kg/h; its composition is 90 mass % (this corresponds approx. 12000 litre/day absolute ethanol product). The minimum purity of the ethanol product is specified as 99.7 mass %. The ethanol is concentrated in a column lhen Ihe distillate enters the membrane system where the water is separated from ethanol. The l~elmeate water is either recycled to the distillation column or is taken as waste. The column works at 1 bar; the feed is in liquid state at its normal boiling point. The task is to determine the optimal design and operating parameters including number of trays, teed location, reflux ratio, number of membrane sections in series, and the numbers of parallel membrane modules in each section of the membrane train. 3. P R O B L E M M O D E L L I N G Two systems are modelled and optimised with MINLP. The first one is a pervaporation system consisting of a membrane network with pump, vacuum pump, heat exchangers and membrane modules. The second system consists of a distillation column for approaching the elhanol/water azeotrope and a pervaporation system for producing pure ethanol. Our aim is to find the optimal structure and operational parameters for the above systems with MINLP, so we have to face the following problems: Construction of a superstructure; representation of the superstructure in an equation oriented model (mixed integer and non linear equations); solving this MINLP problem.
3.1 Construction of the superstructure The superstructure applied for the membrane network is presented in Figure l a. The top product of the column (in cases 2) or the feed (in case 1) is pumped in the first section of the membrane train. In each membrane section the retentate is collected and fed to a heat exchanger for reheating. At the permeate side of the membranes there is a vacuum pump. The permeate is withdrawn as a product stream. N
- 'y,);J:~,g~a~,,~
2-,E
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i I
membranes p
PI
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Figure l a: Superstructure of the membrane network
Figure l b: Superstructure of the distillation column
Ill the second case there is also distillation column in the system. The superstructure and
645 model developed by Viswanathan and Grossmann [6] is adopted for our system. The superstructure of a distillation column is shown in figure lb. The maximum number of theoretical trays was taken 30.
3.2 Representation of the superstructure in equation oriented model The MINLP model of Viswanathan and Grossmann [6] has been used for the d i s t i l l a t i o n except that the cost function was modified. Margules equation, with parameters taken from DECHEMA data bank, is used for calculating activity coefficients [9,10]. Column cost functions were taken from Z. Novak et al.[8], although the column shell and valve tray costing equations were used in the following different form: column,
Cs/,,, , = M & S. 937,6___~1.D].o(,<, . Ho.~o2
( 1)
280 C,,,,, = M &S-136,14 .DL.55. H 280
(2)
For the total annual investment cost calculations ten years linear depreciation was taken. The number of trays, teed location and reflux ratio are the optimisation variables. Tile representation of the m e m b r a n e n e t w o r k is more complicated, because of applying uniform modules, and the difficulty of modelling the pervaporation. Using the experimentally determined characteristic pervaporation functions of Sander and Soul
646 four different variations of the binary variables, but only three different structures as solution. This because the structures assigned by the existence arrays (1,0) and (0,1) are CClUivzllcnt: i.e. these arrays denote the same structure. Generally, in case of n possible ~n()(ltlles irl parallel, the number of different structures is only n+l of the 2" variations. Tlais type of structural multiplicity can be eliminated by always selecting the first n modules of the possible maximum number. This can be achieved by additional logical constraints; however this would lead to increase the number of binary variables. A simple and more efficient way is to define the actual number n in a binary number system, by explicitly using binary variables for the digits. This definition also involves a decrease in the Ilumber of binary variables. Dependence of the number of binary variables on the number of modules is shown in Figure 2b for both the direct definition and the modified one. II: the binary variables mean the existence of the same type of modules (direct definition), and if the sections (with the same number of module) are in series the multiplicity increases cxp(~neniially with the number of sections. Figure 2c shows an example where a section has one module and there are two sections. In Ibis case there are four different variations of the binary variables, but only three different structures as solution. This type of structural multiplicity can be eliminated by using monotonic constraint for the number of modules in section. The monotonic constraint means that the actual number of modules in the sections of the series may not increase from the feed to the end. The advantage of this monotonic constraint is that we do not need to change the definition of series, only to use that extra constraint. 25
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i direct definition modified defini~on
~15
i
t
i
i
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Figure 2c: Illustration of multiplicity arising from sequential arrangement
i
.a
,i
.......
0
5
10
15
20
,i,i
25
number of modules
Figure 2b: Number of binary variables in the case of direct definition and the modified one
Figure 2a: Illustration of multiplicity arising from parallel arrangement
In our exainple in the membrane superstructure there are 16 modules in parallel and 16 sections in series. With direct definition this means 256 binary variables, but with our method the nulnber of binary variables is only 80. To determine the membrane network costing we assumed that the total capital investment ot ihe nlembrane network is linearly proportional to the overall membrane surface, and the f~roportionality constant is 1616.1 USD/m 2. This includes all the pumps, vacuum vessels, contrc~t cquipment etc. Equations for the variable cost calculations were taken from Srinavas ci ztl.I51, except that the PVA membrane replacement cost was taken 775 USD/m 2 based on in(tusirial practice. The objective function to optimise the whole network is either the total annual cost (TAC) or the total cost per unit product (TCP).
647
3.3 Computational experiences in solving the MINLP problem GAMS package with DICOPT++/Conopt/Osl is used to solve the MINLP problem [1]. The opt imisation problems were solved on a SUN Ultrasparc-1 workstation. Solution times around 30 minutes were typical. The following parameters are found to have significant effect on the search efficiency: the number of variables, especially the number of binary variables, scaling of variables, bounds applied to continuous variables, initial values and realistic extra constraints that can help in enhancing the convergence. It can be stated in general that the solution time increases steeply with the number of binary variables used. This is the main reason why the number of binary variables has to be drastically decreased by eliminating structural multiplicity. Trying to find a solution of problem given by the direct definition (with the possibility of structural multiplicity) always led to inconvergence. Both the high number of binary variables and the presence of multiplicity may cause the inconvergence. In some cases convergence can be reached by eliminating structural multiplicity even if the number of variables is not decreased. Since the range of the numerical values of all variables is an important parameter in MINLP, dimensions are selected in a way that numerical values of the variables and parameters belong to the range of 0.01 to 1000. Tight bounding of the continuous variables is also important. For the column variables positive lower bounds on liquid flow rates below the feed tray, and on vapour flow rates over the teed tray are applied. Additional bounds to the mole fractions on the trays are set by Mcabe-Thiele (tray-by-tray) calculations performed at extreme reflux ratios and purity parameters. Bounds on the streams and concentrations around the membrane modules are set according to the validity of the membrane model. Initial values for the column variables are determined by CHEMCAD 4.0 flow sheet simulator (Copyright 9 Chemstations Inc.); initial values for the membrane subsystem are determined with help of the model equations. Extra (additional) constraints are applied as follow: Concentrations in the column cannot step over the azeotropic point; when permeate recycling is not applied a lower bound is set to the recovery ratio of the ethanol in the distillation column; monotonic concentration profile is predicted in the membrane train; the number of modules decreases also monotonically along the train of the membrane sections. With these extra constraints the computational time can be radically decreased. However, the computational time is strongly dependent on the applied initial values. 4. R E S U L T S The solution for the system of only pervaporation membrane modules with 90% alcohol in the teed can be seen in Figure 3a. (90% is the minimum alcohol concentration that the membrane can tolerate because any inlet containing more water would drastically shorten Ihe PVA membrane's life time.) The numbers in brackets show the current number of parallel modules in the actual section. The multipliers in front of them show how many sections of this type follow each other in series. The decreasing number of modules in the sections is due to the decreacisng rctentate flow rates. The structure we got can serve as basis for the actual apparatus plannig.
648
The same problem is also solved with TCP as objective function. The resulted structure is shown in Figure 3b.
li..cd: I(i,'-r I.:~ h,,ms Q ( ) ill;lSs (1, clJl;-ItlO]
product: 295.8 kg/homs
feed: 408.': kg/hours o0 ma.~s % ethmlol
product: 301.0 kg/homs OQ.7 lllass oo ethanol
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Figures 3a,3b: Membrane system optimised for TAC (on the left) and TCP (on the right) The use of TCP as objective results in a slightly different structure with different costs. The TCP belonging to the first case is 0.0428 USD/kg" the TAC belonging to the second case is 102 653 USD/yr. rc l']tt>, r',lI io: 3 1
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retentate 'product'l: 218.2 kg/tu ~).7 mass % EtOH
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= r ' - -II 1--'-1
12 i ! 12 p4 of 1 in-'lllelnbrltllelllOdllles
o[!iecti\'e f u n c t i o n is T A C ,
i -
' -
-
i
~" . . . .
t)ermeate:
:_'.3k~]. 21.8 llla[gs ~o water 151 582 U S D y r
Figure 4: Optimised system with distillation column
Since 90% is the minimum alcohol concentration that the membrane can tolerate, a distillation column is inserted to cope with the problem of working up a feed of 80% alcohol. The resulted system is shown in Figure 4. In this case the alcohol recovery is 70 %. Our model can be easily extended to handle permeate recycle in order to increase the recovery. Optimisation results of systems with permeate recycle are presented by Lelkes et al. [I1].
REFERENCES I. A. Brook, et al: "GAMS A User's Guide", Release 2.25., boyd & fraser, USA, 1992. 2. Z. Lelkes et al., AIChE J., 44, 810-822, (1998). 3. ,I. Neel, Membrane Separation Technology. Principles and Applic., Ch. 5, Elsevier, 1995. 4. U. Smlder and P. Soukup, Journal of Membrane Science, 36, 463-475 (1988). 5. B. K. Srinivas and M.M. El-Halwagi., Computers Chem. Engng., 17, 957-970 (1993). 6. ,l. Viswanathan and I. E. Grossmann, Computers Chem. Engng., 17,949-955 (1993). 7. S. \Vidagdo, W.D. Seider, AIChE J., 42, 96-130 (1996). ~. Z. Novak et al. Computers Chem. Engng., 20, 1425-1440 (1996). 9. V.N.Stabnikov et al. Pishch. Prom. (Kiev) 15, 49 (1972). I(). ,I. Gmehling, U. Onken: Vapor-Liquid equilibrium data collection, Vol. I., Part 1, Vcrlag+Druckerei Friedrich Bischoff, Frankthrt, 1977. 11. Z. Lclkes et al. Rigorous MINLP model for ethanol dehydration system, PSE-2000 abstract ( stl b nlitted)
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
649
Shape and Terminal Velocity of Single Bubble Motion" a Novel A p p r o a c h G. Bozzano a and M. Dente a aCIIC Department, Politecnico di Milano, Piazza L. da Vinci, 32, 20131 Milano, Italy
1. ABSTRACT The terminal relative rising velocity of a single gas bubble, moving into a liquid phase, is determined by its size, by the interfacial tension, by the density and viscosity of the surrounding liquid. Both shape and velocity are strongly interacting. Several methods have been presented along the years, for solving the problem of bubble deformation and relative rising velocity, at least in connection with some specific regimes of motion and/or shape of bubble. In this work a new approach is proposed. 2. INTRODUCTION The understanding of bubble motion mechanism is essential for many gas-liquid operations, not only related to chemical processes applications. Even if in practical applications the overall motion regards bubbles swarms, the behavior of the single bubble (i.e. spoiled of the interactions with the other surrounding bubbles) can support a better knowledge of the overall. Some theories have been presented in the literature covering specific aspects of the problem (ref. 1 to 5). This work has been concentrated on the single bubble behavior: by itself it is a quite complex problem, particularly because the purpose has been to cover a wide range o f properties. The first aspect of the proposed approach is to consider that the approximate shape assumed by the rising bubble is that one that minimizes the total energy associated with the bubble The second one is constituted by the (approximated) generalization of the drag coefficient
3. APPROXIMATED SHAPE AND RELATED TOTAL ENERGY The selected shape is constituted by the superposition of two oblate semi-spheroids (figure 1) having in common the larger semi-axis. This shape asymptotically can degenerate towards something resembling a spherical-cap. The total energy associated to the bubble is the following: (1)
Eto t - Epo t + Esu p + Eki n
where
Epot
-
potential
energy
with
(PL--Pc~)'g" V, "(3" bl + 5" b2)/8
reference
to
the
upper
pole
=
650 Esup = surface energy = cy SB =
Esup -
(7
(
1 b~ 2 rt a 2 + - r c - - l n 2 e1
I I b2 ;l+e ll l+e~ 1-el
1 "eln +-re " 2 e2 \l-e~
= kinetic energy of the virtual mass of adherent liquid displaced by the bubble motion. It is, from a theoretical point of view, an extremely complex function to be evaluated (because of the bubble shape). However, and as a first approximation, it can be obtained by extending the expression with a correction for spherical bubbles. At sufficiently high Reynolds numbers Ekin
3
it can be estimated (in a first approximation) as: EKi n - 2/3. rc0La U
2
Of course the bubble volume is given by: V B - 4/3. rcR 3 - 2/3. rca 2 (b 1 + b2)
b2 = 0
Fig. 1 - Basic shape of the bubble and asymptotic degenerations
The total energy has to be minimized as a function of the two geometrical parameters bl/a and bJa.
4. D R A G C O E F F I C I E N T AND E X T E N D E D E X P R E S S I O N When the bubble reaches the steady state motion the balance of the forces gives: . . . . . . (PL. Pa)gVl~
pLU2
. r~a2 . . .2
f
(2)
By neglecting the gas density in comparison with that of the liquid, the previous equation gives Ur
4 g-D o
-S cD
(3)
651 With a good approximation, and as a consequence of interpolating the minimization procedure, the generalized friction factor that is proposed is the following (taking into account the effect of the bubble deformation): 48 ( 1 + 12. Mo 1/3] Eo f - Re 1+36 ~ J +085 1.8. (1 + 30. Mo 1/6) + Eo
deformation factor =
I~0_l 2 = 3.4(l+30.Mol/6)+3.1-Eo .4(1 + 3 30. Mo 1/6) + Eo
(4)
(5)
So that the drag coefficient is given by the product of equations 4 and 5, and therefore it is an explicit expression ofEo, Mo and Re numbers:
C D - f.
(<12a
(6)
By substituting equation 6 into equation 3 a simple second order equation is generated. 5. WALL EFFECT Most of the experiments on bubble rising velocity have been performed inside limited diameter tubes. For relatively large bubbles (D0/Dt >0.2+0.3) the presence of thew tube wall has the effect of reducing the absolute rising velocity (compared to that attainable in an infinite environment). A first approximation of this {u effect can be estimated as follows. The flowrate entrained by the bubble wake is about CD/2(rc/4. D2) 9U. For continuity, an equal flowrate has to descend crossing the restricted section between the tube wall and the bubble equator. This (negative) contribution gives place to a maximum absolute descending velocity (VD, see figure 2) equivalent to: Fig.2 CD DO VD= 2 D 2 - D 2 U
(7)
Therefore the relative velocity (at the equator of the bubble) is:
uo- u + VD ~ U- U~,'""I
CD/2 l 1+ (Dt/Do)2 _ DEF
U0 is assumed equal to that one of the same bubble rising in infinite environment.
(8)
U U
U
ouozuoqo.~l!N u! solqqn~[ -qV t7 s ! d (tua) .~olotue!( I lUOleA!nt)~l 001,
01,0
I, U U
I
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""J ,D ~.,.,,.
0~ _
< 9
00~
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9
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o
poluoso~ao~ IlOA~ pue op!~ s! 'lopotu oql Xq p0.IOAO3 S0ttlI~'O.I o!tueuXp-p!nlJ ,,Io 'oouonbosuoo e se 'pue 'uo!suotu!p olqqn q jo o~ue.~ oql leql osaosqo ol olq!ssod s! 1! s v (/, u!seo iopolAI .~OlXe• p!AeG oql) u!seq oz!s o:&IeI e s e ~ snle.mdde ieluotugodxo oqk Iopom poluoso.ld oql jo sllnso.~ oql pue '.tole~ m solqqn q .he ol OA!lelO-~ '(9) uolaolAI pue uetu.~oqeH jo elep Ieluotu!-~odxo ~UOU113 uostJt~dllIo3 ottl slAodoJ s oJn~!~ opl~tlI uooq OAl~tI 1131.]1suostJedttlo3 fdolol~ots!ll3s 113101
oql jo oldtues e oJe Xoqz suo!l!puo3 jo oo~ueJ Op!A~ e ~IIIJOAO3 'elep lelUOtU!Jodxo oJnleJOl!l ql!st poJedtuoo 'lopotu posodoJd oql jo luotuooJ~e pou!mqo oql A~oqs soJn~ U ~U!A~OIIOJ 0 q i ~dL l l l ~ ~ l l ~
9
653
~, r]3
9 Experimental
Theoretical j
9 Experimental
100
10o r
Theoretical I
.... ! :
>.-, +..a .,..~ O
9
.,--,
o >
10
10
> E
.,-.~
E
r
1
~D
~"
I _
0.01
0.10
1.00
0.10
1.00
Equivalent Diameter (cm)
Equivalent Diameter (cm)
Fig. 5' Air Bubbles in Pyridine
Fig. 6: Air Bubbles in Cottonseed Oil
Figures 4 to 6, are relative to the experiment of Peebles and Garber (8) obtained in a 2.62 cm diameter tube. Also in this case the agreement is satisfactory.
100
9 Experimental
Theoretical}
%-,
E o o
.,..~
;>
_
-: 9~ , , ~
o o
9 Experimental
100
Theoretical I I-++
_
i -:7, , :
I
--~
J
10
-VFT:~ , .
.,-~
"'=
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r
~-
r
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1.00
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1
i Z
0.10
4
i
4
.
.
.
.
+-+.
'" '
r , l
1.00
10.00
Equivalent Diameter (cm)
Equivalent Diameter (cm)
Fig. 7 Air Bubbles in Glycerol 90.6%
Fig. 8' Air Bubbles in Glycerol 99%
Finally Fig. 7 and 8 show the comparison of simulation results with the data of Calderbank (9). The system is constituted respectively by 90.6 Glycerol in Water and 99% Glycerol in Water. The tube diameter is 10.6 cm. The physical data of liquids that have been used for the experiments are reported in Table 1. A wide range of viscosity has been covered. Table 1 Pyridine Density (g/cm 3) 0.987 Surface Tension (g/s z) 36.6 Viscosity (g/cm/s) 0.0085
Nitrobenz. 0.987 42.5 0.0167
Cot. seed Oil 0.910 35.5 0.59
Glycerol 90.6 1.235 64.0 1.8
Glycerol 99 1.260 63.0 7.75
654 7. C O N C L U S I O N S The presented new approach has allowed to obtain a unified model for the description, in extended fluid-dynamic conditions, of a single bubble motion. Both terminal velocity and shape of the bubble can be determined. The comparison with different literature experimental data, covering a wide range of physical properties and bubble sizes, are satisfactory. NOMENCLATURE Reynolds number: Re -
9LDc)Uo/btL
EOtvosnumber:Eo-(gL-gg) gD2/cs Morton number: Mo - g bt~/9LCS3 el, e2 " eccentricity of the two semi-spheroids defined as e - -~/1- b : / a : f = friction factor U = absolute bubble terminal velocity U0 = bubble terminal velocity in infinite environment VD = absolute descending velocity CD = drag coefficient a = bubble major semi-axis bl, b2 = bubble minor semi-axes L - liquid, g = gas Do = diameter of the equivalent spherical bubble D = equator diameter of the spheroid Dt = tube diameter Ro = radius of the equivalent spherical bubble (s = surface tension ACKNOWLEDGEMENT
The authors wish to thank Rita Bizzozzero and Christina Ktihlwetter for their contributions to the computational efforts of this work. REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9.
Batchelor, G. K., Cambridge University Press, (1970) Bhaga D. and Weber, M. E., J. Fluid Mech., 105 (1981) 61-85 Grace, J. R., Trans. Instn. Chem. Engrs., 51 (1973) 116-120 Mendelson, H. D.,A.I.Ch.E. Journal, 13 (1960) 250-253 Sadhal, S. S., Ayyaswamy, P. S. and Chung, J. N., Springer New York (1997) Haberman, W. L. and Morton, R. K., Soc. Civil Eng. Trans., 121 (1956) 227-251 Bryn, T., David Taylor Model Basin Transl., Rep. No 132, 1949 Peebles, F. N., Garber, H. J., Chem. Eng. Progr., 49 (1953) 88-97 Calderbank, P. H., Johnson, D. S. L. and Loudon, J., Chem. Eng. Sci., 25 (1970) 235-256
European Symposiumon Computer Aided Process EngineeringS. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights
10
reserved.
655
The Myth of Decomposition Padmanaban Kesavan and Paul I Barton* Department of Chemical Engineering Massachusetts Institute of Technology Cambridge, MA 02139 Abstract Branch and Bound (BB) and Decomposition algorithms are the two main deterministic approaches that have been applied successfully to solve Mixed-Integer optimization problems. A Generalized Branch and Cut (GBC) algorithm is presented here and it is shown that all current decomposition and BB algorithms are essentially specific instances of GBC with different sets of heuristics. First it will be shown that Decomposition algorithm based on Outer Approximation for solving convex MINLPs and INLPs [5, 6] is a BC algorithm with a certain set of heuristics. A different set of heuristics for the BC algorithm that may be potentially more efficient (on average) to solve convex MINLPs will then be presented. Finally, numerical results from several example problems is compared to identify an efficient set of heuristics to be employed in the GBC algorithm that will reduce the computational time (on average).
1
Introduction
Many problems in a simultaneous approach to the design and operation of complex systerns can be formulated as Mixed-Integer Nonlinear Programs (MINLPs) in which functions of the continuous and the integer variables are usually nonlinear and either convex or nonconvex. Mixed-Integer optimization problems belong to one of the most difficult classes of problems with respect to the computational complexity involved. All deterministic algorithms currently available to solve Mixed-Integer problems of generic type exhibit exponential complexity and no polynomial time algorithms are known at present to solve these problems. Three classes of exponential time algorithms, namely Branch and Bound (BB) [3, 8], Branch and Cut (BC) [12] and Decomposition algorithms [5, 6, 7], have been proposed and applied successfully to solve convex Mixed-Integer Nonlinear Problems. BB algorithms are developed on the basis of a divide and conquer strategy. Decomposition algorithms are based on solving a sequence of subproblems to generate lower and upper bounds. The sequence of nondecreasing lower bounds is generated by solving a so called Relaxed Master Problem which is easier to solve (with respect to the existence of algorithms that yield the global optimum) compared to the original problem. The upper bound is obtained by solving the original problem with the integer variables fixed to the current solution of the Relaxed Master Problem. In this work, a Generalized *to whom all correspondence should be addressed. Email:[email protected], Ph: (617)253-6526
656 Branch and Cut (GBC) algorithm is presented. It is shown that the Decomposition algorithms (Generalized Benders Decomposition (GBD) and Outer Approximation(OA)), BB algorithms and BC algorithms proposed for MINLPs, are in fact specific instances of the GBC algorithm with a certain set of heuristics. This provides a unified framework for comparing all current algorithms. In addition, this framework illuminates the heuristics inherent in each of these algorithms, which aids in identifying superior sets of heuristics for improved computational performance compared to all the currently available algorithms.
2
High Level Description of the GBC Algorithm
A Branch and Bound tree is considered for the integer variables. Each node corresponds to fixed values for a subset of the integer variables. The set of active nodes is initialized with the root node (no integer variable is fixed). The GBC algorithm to find the global solution of the convex MINLP or INLP is based on the derivation of the following subproblems: R e l a x e d P r o b l e m the global solution of which represents a valid lower bound to that subset of the integer combinations (correspond to the leaf nodes of the BB tree) not yet explored by the algorithm. Several alternatives exist to derive the Relaxed Problem. For example, the Relaxed Problem can be derived from the Outer Approximation [5, 6] Master Problem of the original problem, followed by deletion of some of the constraints of the Master Problem. The Relaxed Problem thus obtained is a MILP with finite number of constraints. Lower Bounding Problem ( L B P ) a continuous optimization problem which can be solved to guaranteed global optimality. The solution of the LBP will yield a valid lower bound to the solution of the original problem at the current node and all its descendant nodes of the BB tree, and an upper bound to the solution of the original problem under certain conditions. For example, if the Relaxed Problem is obtained by OA, then the LBP solved at any node is a LP (obtained by fixing some subset of the integer variables in the Relaxed Problem). Upper Bounding Problem the solution of which represents a valid upper bound to the global solution of the original problem. If the Relaxed Problem is an MILP, then the upper bounding problem can be obtained by fixing the integer variables in the original problem. LOOP
1. Node Selection a. Check termination criteria. If satisfied, E X I T . b. Choose an active node ni based on a heuristic. 2. Should the Lower Bounding Problem be solved? If yes, solve the lower bounding problem and update the lower bound. 3. Should the Upper Bounding Problem be solved? If yes, solve the upper bounding problem and update the upper bound. 4. Should cuts be generated? If yes, derive cuts valid for the entire BB tree. 5. Branching vs Refathoming Should the problem be resolved with the new Relaxed Problem (augmented with cuts)? If yes, refathom the BB tree with the new Relaxed Problem and the current UBD.
657
6. Branching Step If the current node is not a leaf node, choose a branching variable (based on a heuristic), create children nodes and add them to current set of active nodes. Delete ni from the current set of active nodes. E N D LOOP There are several decisions which need to be made in the GBC algorithm. These are listed below:
1. Node Selection The active node to be considered may be selected in several different ways. An example is to choose the node with the lowest lower bound. 2. Lower and Upper Bounding Problem.s Several choices exist for the problems to be solved for the lower and upper bounds. An example is to solve a LP for the lower bound and an NLP for the upper bound. Decisions also have to be made whether to solve for the lower bound or the upper bound or both at a particular node. 3. Cut Generation There are several alternatives to generate cuts and the nodes at which the cuts are derived. For example, cuts may be derived from the solution of the upper and/or lower (e.g., Gomory cuts) bounding problems, an integer cut [1] excluding the integer combination corresponding to the current node, or by solving an additional problem. 4. Branching and Refathoming The children nodes may be created in several different ways (branching rule). Instead of branching, the nodes may be refathomed starting at any node (not necessarily the root node) once the cuts are generated. The conditions under which each of the six steps in the GBC algorithm is executed also have to be specified. In general, there is no rigorous procedure to make these choices for the GBC algorithm. In other words, the choices made define the set of heuristics employed.
3
Heuristics employed by the Outer Approximation algorithm to solve Convex M I N L P s
The set of heuristics inherent in the decomposition algorithm based on outer approximation [5, 6] are listed below:
1. Relaxed Problem an MILP (the Relaxed Master Problem of OA), the solution of which yields a valid lower bound to the global solution of the convex MINLP (referred to as P hereafter). 2. Lower Bounding Problem an LP obtained by fixing a subset of the integer variables (represents a node in the BB tree) in the Relaxed Problem and relaxing its complement to be continuous. 3. Upper Bounding Problem an NLP, obtained by fixing all the integer variables in P. Solve the NLP at the node corresponding to the solution of the Relaxed Problem. 4. Node and Branching Variable Selection Use the heuristic as employed by OA algorithm to solve the Relaxed Master Problem. 5. Cut Generation Derive cuts at the node corresponding to the solution of the MILP. The cuts are derived by linearization of nonlinear constraints and the objective function of P about the solution of the corresponding NLP. If the NLP is infeasible, derive cuts by solving a feasibility problem [6].
658
6. Branching versus Refathorning Branch and create child nodes until the solution of the Relaxed Problem is obtained. A u g m e n t the Relaxed P r o b l e m with the cuts derived (valid for all nodes of the BB tree) and resolve the Relaxed P r o b l e m (i.e., r e f a t h o m all the nodes s t a r t i n g at the root node of the BB tree) with the current UBD. The nodes are r e f a t h o m e d starting at the root node as soon as cuts are derived r a t h e r t h a n branching and continuing the u n f a t h o m e d active nodes. The set of heuristics presented above when applied to the G B C algorithm of section 2 yields the decomposition algorithm proposed by D u r a n and G r o s s m a n n [5] and later modified by Fletcher and Leyffer [6]. A set of heuristics similar to the above except for Step 6, where the nodes are not r e f a t h o m e d once the cuts are a u g m e n t e d to the Relaxed Problem, can also be employed. There are several other heuristics which may be derived based on the outline presented in section 2. A detailed discussion is presented by Kesavan and B a r t o n [9].
4
Numerical Results and Discussion
Five example problems were solved employing the Decomposition heuristic as presented in section 3 and with the heuristic t h a t the nodes are not refathomed once the Relaxed P r o b l e m is a u g m e n t e d with cuts. The first four example problems are discussed Table 1' Test Prob.
Results for the test problems 1-4. Decomp. Heuristic with integer cuts
No R e f a t h o m i n g Heuristic with cuts
Algorithm a
Algorithm b
Algorithm c
nodes visited = 11 LPs feasible = 6 LPs infeas. = 5 NLPs solved = 3 C P U time = 0.28s nodes visited = 44 LPs feasible = 29 LPs infeas. = 15 NLPs solved = 4 C P U time = 0.96s nodes visited = 86 LPs feasible = 64 LPs infeas. = 22 N L P s solved = 4 C P U time = 1.8s nodes visited = 2341 LPs feasible = 1213 LPs infeas. = 1128 N L P s solved = 3 C P U time = 64.48s
nodes visited = 5 LPs feasible = 3 LPs infeas. = 2 NLPs solved = 3 C P U time = 0.16s nodes visited = 34 LPs feasible = 24 LPs infeas. = 10 NLPs solved = 4 C P U time = 0.76s nodes visited = 70 LPs feasible = 56 LPs infeas. = 14 NLPs solved = 4 C P U time = 1.49s nodes v i s i t e d - 637 LPs feasible = 364 LPs infeas. = 273 N L P s solved = 3 C P U time = 21.07s
nodes visited = 9 LPs feasible = 4 LPs infeas. = 5 NLPs solved = 3 C P U time = 0.2s nodes visited = 21 LPs feasible = 10 LPs infeas. = 11 NLPs s o l v e d = 4 C P U time = 0.46s nodes visited = 67 LPs feasible = 33 LPs infeas. = 34 NLPs solved = 4 C P U time = 1.18s nodes visited = 217 LPs feasible = 118 LPs infeas. = 99 NLPs solved = 3 C P U time = 5.07s
Decomposition Heuristic
.
.
.
.
.
.
659 Table 2: Results for test problem 5. Algorithm a
Nodes Visited 43840
CPU Time (s) 3112
13711
>>3112 > > 789 789
LPs LPs feasible =24744 LPs infeas = 19096
NLPs NLPs feas. = 3 NLPs infeas. = 3
LPs feasible = 6855 LPs infeas. = 6856
NLPs leas. = 3 NLPs infeas. = 3
by Duran and Grossmann [5]. Problem five is a modest extension of problem four to one hundred binary variables. The relevant input data for problem five is available at http://yoric.mit.edu/paddi. The results are summarized in Tables 1 and 2. In test problems 1-4, algorithm a refers to the OA algorithm described by Fletcher and Leyffer [6]. Algorithm b augments the Relaxed Problem of the OA algorithm with integer cuts [1] corresponding to infeasible lower bounding problems and integer combinations at which the upper bounding problem is solved. Algorithms c and d are similar to Algorithms b and a respectively, except that the nodes of the BB tree are not refathomed once the Relaxed Problem is augmented with cuts. The algorithms were implemented on a HP-C160 dual processor machine with 256 MB RAM and running HP-UX B.10.20. The LPs were solved using CPLEX via GAMS [4] and the NLPs were solved using Minos 5.4 [101 via GAMS [4]. The CPU times in Tables 1 and 2 are the sum of the solver times alone.
Refathorning versus no refathoming: The computational performance of the GBC algorithm with the no refathoming heuristic (Algorithms c and d) is superior on average as compared to the refathoming heuristic (Algorithms b and a). Implication of the addition of cuts The cuts when augmented to the Relaxed Problem performs better computationally (Algorithm b performs better than Algorithm a on average). However, if all the cuts that are derived at every node of the BB tree are augmented to the Relaxed Problem, then the size of the LP to be solved at the nodes of the BB tree will increase. This was validated by test problem 5. In particular, the CPU time increased dramatically for Algorithms b and c as compared to Algorithms a and d respectively. An instance of the GBC algorithm similar to Algorithm d was proposed by Quesada and Grossmann [11]. However, the upper bounding problems were solved corresponding to the nodes with an integer feasible solution of the LBP. From the discussion presented here, it is clear that this instance of GBC will also perform poorly as compared to Algorithm d. In summary, while cuts improve the quality of the lower bounds obtained at the nodes of the BB tree, augmenting all the cuts derived to the Relaxed Problem will increase the CPU time dramatically. It is proposed that empirical constraint dropping strategies similar to that for BC algorithm for MILPs [2] be developed.
5
Conclusions
In this paper, a Generalized Branch and Cut (GBC) algorithm has been presented to solve Mixed-Integer Optimization problems. This provides a unified framework for comparing
660 all currently available deterministic algorithms to solve these problems. The decomposition algorithm based on outer approximation is shown to be a specific instance of the GBC algorithm with a certain set of heuristics. A set of heuristics which may be computationally more efficient on average is also discussed. Finally, numerical results have been presented for five example problems. The results indicate that there is a trade off between solving a computationally more intensive problem which yields a tighter lower bound to the global solution of the original problem at the nodes of the BB tree and computationally less intensive problem which yields a looser lower bound at the nodes of the BB tree (which will therefore visit more nodes of the BB tree on average).
Acknowledgements This work was supported by the National Science Foundation under grant CTS-9703623.
References [1] E. Balas and R. Jeroslow, "Canonical Cuts on the Unit Hypercube," SIAM Journal of Applied Mathematics 23 (1972) 61-79. [2] E. Balas and S. Ceria and G. Cornu~jols, "Mixed 0-1 Programming by Lift-andProject in a Branch-and-Cut Framework," Mgmt. Sci. 42 (1996) 1229-1246. [3] B. Borchers and J. E. Mitchell, "An Improved Branch and Bound Algorithm for Mixed Integer Nonlinear Programs," Comp. Ops. Res. 21 (1994) 359-367. [4] A. Brooke, D. Kendrick and A. Meeraus, GAMS: A Users Guide (Scientific Press, California, 1988). [5] M. A. Duran and I. E. Grossmann, "An Outer-Approximation Algorithm for a Class of Mixed-Integer Nonlinear Programs," Mathematical Programming 36 (1986) 307339. [6] R. Fletcher and S. Leyffer, "Solving Mixed Integer Nonlinear Programs by Outer Approximation," Mathematical Programming 66 (1994) 327-349. [7] A. M. Geoffrion, "Generalized Benders Decomposition," Journal of Optimization Theory and Applications 10 (1972) 237-262. [8] O. K. Gupta and A. Ravindran, "Branch and Bound Experiments in Convex Nonlinear Integer Programming," Management Science 31 (1985) 1533-1546. [9] P. Kesavan and P. I. Barton, "The Decomposition Heuristic. Part I: Convex Problems," Operations Research (1999) submitted.
[10]
B. A. Murtagh and M. A. Saunders, "MINOS 5.1 Users Guide," Technical Report SOL 83-20R, Systems Optimization Laboratory, Department of Operations Research, Stanford University (1987).
[11]
I. Quesada and I. E. Grossmann, "An LP/NLP Based Branch and Bound Algorithm for Convex MINLP Optimization Problems," Computers Chem. Engng. 16 (1992) 937-947.
[12]
R. A. Stubbs and S. Mehrotra, "A Branch-and-Cut Method for 0-1 Mixed Convex Programming," Mathematical Programming 86 (1999) 515-532.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
661
Parameter Analysis and Optimization of Ideal Heat Integrated Distillation Columns (HIDiC) Masaru Nakaiwa a, Kejin Huang a, Kiyoshi Naito a, Akira Endo a, Masaru Owa a, Takaji Akiya a, Takashi Nakane a and Takeichiro Takamatsu b aNational Institute of Materials and Chemical Research, Tsukuba 305-8565, Japan [email protected] p blnstitute of Industrial Technology, Kansai University, Suita 564-8680, Japan [email protected] Parametric analysis is performed for ideal heat integrated distillation columns (HIDiC) and heuristics are provided for the effective process design. A better process configuration is suggested, which is demonstrated to have both higher energy efficiency and higher flexibilities than its original configuration. Simulation results confirm the conclusion. 1. INTRODUCTION Ideal heat integrated distillation column (HIDiC) was created by chasing reduction of energy consumption in distillation columns. The way to pursue energy reduction is to adopt heat integration between rectifying and stripping sections and this results in a seemingly far different counterpart of conventional distillation columns. Figure 1 shows a representation of the process and Table 1 lists a representative operating condition. The ideal HIDiC is such a process that its stripping section and rectifying section are separated into two columns, while connected through a number of internal heat exchangers. To accomplish internal heat transfer from the rectifying section to the stripping section, the rectifying section is operated at a higher pressure and a higher temperature than those of the stripping section. For adjusting the pressures a compressor and a throttling valve have to be installed between the two sections. Owing to the heat integration, a certain amount of heat is transferred from the rectifying section to the stripping section and generates the reflux flow for the rectifying section and vapor flow for the stripping section. Thus the condenser and reboiler are, in principle, not needed, as a result, both fixed and operating cost could be reduced. The synthesis and analysis of the ideal HIDiC were already discussed thoroughly by Takamatsu and his coworkers [1]. The examination of process dynamics and operation were conducted by Nakaiwa et al. [2, 3] recently. Although these studies indicated the ideal HIDiC is certainly more energy efficient than conventional distillation columns and operation feasible, one remained question to be answered is the process flexibility. Does the ideal HIDiC hold its superiority in a large operation region over conventional distillation columns? If not, what measures should be taken to enhance the process flexibilities? These questions are very essential to the applicability of the process to the chemical process industry. In spite of sharp difference in process configurations and operation, there exist similarities in concept between the ideal HIDiC and conventional distillation colullms. Clarifying these similarities is very useful to understanding the principle of the ideal HIDiC and this constitutes another purpose of this work. In this work we will focus on investigating the static characteristics and synthesizing the optimum process configuration for the ideal HIDiC. Special emphasis will be paid to the enhancement of process flexibility to operating condition changes.
662 Table 1 Steady-state operating conditions for three processes Items Conventional 22 No. of stages 12 Feed stage Vapor feed Liquid feed 12 1.0 Stage holdup (kmol) 0.1013 Rectifying section pressure (MPa) 0.1013 Stripping section pressure (MPa) 100 Feed flow rate (kmol/h) Feed composition (Benzene) 0.5 (Toluene) 0.5 Feed thermal condition 0.5 2.4 Relative volatility Vaporization heat (kJ/kmol) 30001.1 9803 Heat transfer rate (W/K) 0.995 Overhead product composition 0.005 Bottom product composition
Ideal HIDiC 20 11 11 1.0 0.2963 0.1013 100 0.5 0.5 0.5 2.4 30001.1 9803 0.995 0.005
5.0
Better HIDiC 22 11 12 1.0 0.2026 0.1013 100 0.5 0.5 0.5 2.4 30001.1 9803 0.995 0.005
4.0 3.5
,..-., 4 0
3.0 3.0
Compressor
2,O 099
-
I 0,992
I 0994
I 0,996
I 0998
2.O
(a) 3.0
5.0
92.9
4.5
35 2.7
~ 3.0
26 2.5
Throttling valve
2,5 0.I
0,2 0.3 0.4 0,5 0 6 zf[-]
I 100 F [kmol/h]
I 125
150
(b)
i 4.0
Fig. 1. Schematic representation of an ideal HIDiC
i 75
50
yl [-]
F, zf
\---)
>.i
2.5
0.7 0.8
0,9
20
L.. 6
18
;
'/2
I 24
I 26
218
30
nl-] (c) (d) Fig.2. Functions of pressure difference, Pr-Ps
2. P A R A M E T R I C S T U D I E S F O R T H E I D E A L H I D i C 2.1 Functions of pressure difference, Pr'Ps The pressure difference, Pr-Ps, plays an important role in heat integration between the rectifying and the stripping sections. When process design has been finished, the purer the specifications for the overhead and bottom products become, the higher the pressure difference, Pr-Ps, will be needed (Fig. 2a). Up to a certain high product specifications, the advantage of the ideal HIDiC is expected to be totally lost, because electricity is generally several times more expensive than heating steams. It is therefore extremely necessary to assess the flexibility of the ideal HIDiC. 2.2 Functions of feed thermal condition, q The mass balance equation for the ideal HIDiC yields: zf : (1-q)yl+qxn
(1)
663 It is readily to understand that the feed thermal condition q influences only the material balance of the process. It is therefore reasonable to call it a variable for material balance control, although it is an energy term that reflects the thermodynamic state of the feed. 2.3 Influences of feed flow rates, F
Figure 2b illustrates the relations between feed flow rate and the pressure difference between the rectifying and the stripping sections, when the both end products have been kept on their specifications, respectively. The larger the feed flow rate is, the higher the pressure difference, pr-ps, will be. As the pressure difference is enhanced, the superiority of the ideal HIDiC will diminish because electricity is usually several times more expensive than heating steam. It is anticipated that up to a certain flow rate the potential of energy saving will be totally lost. On the other hand, when the feed flow rate becomes too small, the necessary pressure difference will go down drastically, and so does the process energy efficiency. Thus, after an ideal HIDiC has been constructed, only within a certain range of the feed flow rate, can the energy efficiency of the process be justified. It is, therefore, imperative to guarantee high process flexibility by intensive process design. 2.4 Influences of feed composition, zf
Figure 2c illustrates the necessary value of the pressure difference, Pr-Ps, between the rectifying and the stripping sections in order to keep both top and bottom products on their specifications, when the ideal HIDiC is fed with mixtures of different compositions. Around the region of feed composition equaling 0.5, the pressure difference, Pr-Ps, reaches its maximum value. Away from 0.5 it gradually decreases. As the variations of pressure difference is quite limited in magnitude (in this case, around 0.4 atm), it is reasonable to consider that feed composition will not influence the process energy efficiency substantially, and thus it will not impose strict requirements to the process flexibility, either. 2.5 Effect of feed location
As the feed is a vapor/liquid mixture (0
Figure 2d shows the relation between the total number of stages and the pressure difference, Pr-Ps, between the rectifying and the stripping sections. The illustration was obtained when the both end products are kept on their specifications, respectively. As the number of stages has been increased, the necessary pressure difference, Pr-Ps, becomes lower. In other words, the operating cost is reduced with the expense of fixed investment. When the total number of stages reaches a certain value, the direction of heat transfer from the rectifying to the stripping sections becomes difficult to be maintained, especially when extemal disturbances occur. Inverse heat transfer is detrimental to process energy efficiency and adds difficulties to process operation, thus it should be avoided. As the number of stages is further increased, the pressure difference reaches the minimum value and involves almost no changes. Under this circumstance, inverse heat transfer occurs and the net heat transfer remains the same as before, namely, Q = Q r s - Qsr = c o n s t
(2)
664
3. SUMMARY Pressure difference is the most important variable that influences the design of the ideal HIDiC. It is the way of energy input to the process. It is imperative to assess the economical operating region because electricity is generally several times more expensive than heating steams. Production specifications and feed flow rate are the main factors that can substantially affect the process energy efficiency. For pursuing consistent process design, not only should pressure difference be carefully chosen, but also effective measures have to be taken to guarantee the process with high flexibility, as drastic changes in operation conditions might occur due to market requirements and process retrofit. 3.1 A Better Process Configuration A better configuration of the ideal HIDiC is shown in Fig. 3, which is created with the aid of the above parametric studies. The vapor and liquid portions of the feed are divided and fed into the column at different locations. The vapor portion of the feed is introduced at a lower location than n/2+ 1 and liquid portion of the feed is introduced at a higher location than n/2+1. The addition of an overhead partial condenser and a bottom partial reboiler provides two extra degrees of freedom for process optimization. It is also extremely effective in improving flexibilities of the ideal HIDiC, because they can effectively overcome the influences introduced by feed flow rates and product specifications. For example, when the pressure difference is too high for reaching a separation, the condenser and reboiler can generate certain external reflux and reboil flows and move the process to an operating condition that is suitable for heat integration between the rectifying and stripping sections. In the sequel we will refer the new configuration as better HIDiC. Comer
... .......V' .....Yl .
~ ,
Q~~ '~1Lt
~ "-.. .........
o i ~
~ F(1-~ F, zf ;::
HeatT
,a~ ._'_'2....
Fq "-"
Ln,x n _ Throttlingvalve Fig.3. A suggested configuration for ideal HIDiC
3.2 Flexibility Comparisons After conceptual design of the ideal HIDiC has been completed, it is necessary to examine its optimal operating region. In this work it is undertaken by comparisons of profits among different process configurations, namely, a conventional distillation column, an ideal HIDiC and the better HIDiC. For the ideal HIDiC, the decision variables are the feed thermal condition q and the pressure difference, Pr-Ps, between the rectifying and the stripping sections.
J1(pr-ps, q): V1Cd+L,,Cb+Q1C1-FCf-QjCq-EpCp
(3)
The term Ep is the electric power of compressor, which is dependent closely on the pressure difference between the rectifying and the stripping sections.
665 For the better HIDiC, it is assumed to have a partial condenser, a partial reboiler and the same number of stages as the ideal HIDiC. The decision variables are the feed thermal condition q, the pressure difference, pr-ps, between the rectifying and the stripping sections, reflux and reboil ratios, R and S. Here, we fix, Pr-Ps, at 1 atm, and q at 0.5 arbitrarily for simplification. (4) Jg(P,.-Ps,q,R, S) = V1Cd+LnCb +QIC1-FCf-'Qj~q-QcCc-QbCb-EpCp For the conventional distillation column, it is assumed to have a partial condenser, a partial reboiler and the same number of stages as the ideal HIDiC, but without heat integration between its rectifying and stripping sections. The decision variables are the reflux ratio, R, and reboil ratio, S.
(5)
J~(R, S) = VICd+ LnCb-FCf-QjCq-QcCc-QaC6
First, the influences of feed flow rate are examined. Figure 4 shows comparisons of profits: J1, J2, and J3 with feed flow rate as a varying parameter, when the both end products are kept on their specifications, respectively. It is clearly shown that the ideal HIDiC is, only within some region, namely, F<230kmol/h, more economical than its conventional counterparts. Beyond this region the ideal HIDiC will lose its advantages in energy utilization. The general HIDiC is always more energy efficient than the conventional distillation column and this demonstrates its higher flexibility than the ideal HIDiC. In fact, with the addition of a bottom trim-reboiler and overhead trim-condenser, the general HIDiC can be designed to be more energy efficient than the ideal HIDiC through optimization of the extemal reflux and reboil flows, irrespective to any changes in operating conditions. For example, when F<150 kmol/h, it is not necessary to employ extemal reflux and reboil flows. However, when F>150kmol/h, it is suggested to employ them. These strategies can guarantee the general HIDiC to be more flexible and energy efficient than the ideal HIDiC. Second, the influences of product specifications are examined. Figure 5 shows comparisons of the profits: J1, J2, and J3 with the top product specification as a varying parameter, when the bottom product specification has been kept as Xn=l-yl. The higher the specifications for top and bottom products become, the less the advantages of the ideal HIDiC will be. Beyond certain higher product specifications (here, 0.998), the ideal HIDiC will be more energy intensive than its conventional counterpart. This is because higher product specifications have to be achieved by higher pressure difference and this inevitably introduces higher electricity cost. For the general HIDiC, this problem has been alleviated substantially. As can be seen, the general HIDiC is, in a wider region, more energy efficient than the conventional distillation column, although we arbitrarily fixed, the pr-ps, at 1 atm in the simulation. It is the extemal reflux and reboil flows that provide degrees of freedom for avoiding the unnecessary higher electricity cost. Through process optimization the optimal region of the better HIDiC could be even larger than that shown on the figure. Regarding the influences of feed composition, although not shown here, the ideal HIDiC appears to be always more energy efficient than its conventional counterparts, and so does the better HIDiC.
~
4O
4O
2O
20
0
-20
-40 0
-20
100
200
300
F [kmol/h]
Fig.4. Comparisons ofprofits J1, J2 and J3
-40
0.99
0.9925
0.995 Yl [ - ]
0.9975
Fig.5. Comparisons ofprofits Jl, J2 and J3
666 4. CONCLUSION A better process configuration is synthesized through parametric studies. The process configuration is demonstrated to have higher operation flexibilities and higher energy efficiency than its original configuration. Simulation results justify the conclusion. 5. A C K N O W L E D G M E N T This work is supported by New-Energy and Industry Technology Development Organization (NEDO) through Energy Conservation Center of Japan and hereby is acknowledged.
6. N O M E N C L A T U R E A : area [m 2] C = cost or price [$/kmol] E = electric power of compressor [kW] F = feed flow rate [kmol/s] J = operating cost [S/s] L = liquid flow rate [kmol/s] n = number of total stages [-] Pr-Ps = pressure difference between rectifying and stripping sections [kPa] [-] q = thermal condition of feed [kJ/s] Q = heat duty [-] R = reflux ratio S = reboil ratio [-] [kJ/m 2K] U = overall heat transfer coefficient [kmol/s] V = vapor flow rate [-] x = mole fraction of liquid [-] y = mole fraction of vapor [-] z~ = feed composition <Subscripts> c = condenser f = feed b = bottom d = distillate p = compressor rs = from rectifying to stripping sections sr = from stripping to rectifying sections REFERENCES [ 1] T. Takamatsu, M. Nakaiwa, K. Huang, T. Akiya, and K. Aso, Comput. Chem. Eng., $21 (1997) $243. [2] M, Nakaiwa, K. Huang, M. Owa, T. Akiya, T. Nakane, and T. Takamatsu, Comput. Chem. Eng., $22 (1998) $389. [3] M. Nakaiwa, K. Huang, A. Endo, K. Naito, M. Owa, T. Akiya, T. Nakane, and T. Takamatsu, Comput. Chem. Eng., $23 (1999) $851. [4] P.C. Wankat and D.P. Kessler, Ind. Eng. Chem. Res., 32 (1993) 3061.
European Symposium on Computer Aided Process Engineering S. P i e r u c c i (Editor) 9 2 0 0 0 Elsevier Science B.V. All rights reserved.
- 10
667
Computer-aided screening of adsorbents and porous catalyst carriers F. St6p~nek~ M. M a r e k ,a M. K u b i ~ e k b a n d P. M. A d l e r c
aDepartment of Chemical Engineering, bDepartment of Mathematics, Prague Institute of Chemical Technology, Technickd 5, 166 28 Praha 6, Czech Republic cInstitut de Physique du Globe de Paris, ~ place Jussieu, 75252 Paris CEDEX 05, France
Abstract - A software package for computer reconstruction of mesoporous media based on image analysis, and their characterization by geometrically well-defined quantities such as chord-length distribution, the self-correlation function, pore-size distribution, porosity and tortuosity, is presented. The calculation of a key transport property - the effective diffusion coefficient - is demonstrated on the example of Gaussian-correlated porous media. 1 Background and O b j e c t i v e s Comprehensive software tools for computer-aided design of fixed-bed catalytic or sorption processes should enable simulations at up to five different length-scales: the molecular scale, the pore scale, the particle scale, the apparatus scale, and the process scale. Results obtained from simulations at smaller length-scales serve as input for "coarse-grained" models at larger length-scales. Software tools for simulations at the smallest (molecular) length-scale, as well as packages for process flow-sheeting and computer-aided design of individual unit operations, are nowadays commercially available. However, with a few exceptions (e.g., Adrover and Giona, 1997, or Rieckmann and Keil, 1999) neither a well-established methodology for pore-level (also called "meso-scale") modeling of processes in adsorbents and porous catalyst supports, nor software implementing it, are currently available, despite the obvious advantages such computational tools might bring during the catalyst screening part of a process development cycle (Mann, 1998). The efficiency with which the surface chemistry of a catalyst translates into the performance of a whole pellet depends on the effective transport properties (effective diffusivity, permeability, and thermal conductivity) of the porous particle. The effective transport properties can typically be expressed as a product of two terms, one accounting for the species (in the case of diffusion) or material (in the case of heat conduction) properties,
668
Figure 1: The process of computer reconstruction of a porous m e d i u m - (a) pore space imaging (here ESEM photo of a carbon extrudate); (b) image analysis and binarization; (c) calculation of the correlation function; (d) computer realization of a 3D sample. and the other for the morphological properties of the porous medium. As an example can serve the well-known relation for the effective diffusion coefficient Def = D,~(e/r), where D,~ is the Fickian diffusivity in a homogeneous medium, e is (open) porosity and r tortuosity. In the present contribution, we describe software for the calculation of effective diffusivity in general binary-encoded porous media by two m e t h o d s - a transient method of moments, and a steady-state diffusion cell method.
2 Methodology and Example The sequence of steps employed during computer-aided analysis of a porous medium from a physical sample to its computer reconstruction - is shown in Figure 1. Images of the cross-section of an original catalyst/adsorbent pellet are first obtained by a suitable technique such as the ESEM - Environmental Scanning Electron Microscopy (Philips, 1998). Assuming an isotropic medium and no close porosity, the statistical properties of the porous medium evaluated from the 2D image are representative of the whole 3D porous medium, provided that the image covers a sufficiently large (i.e., statistically significant) portion of the medium. The image is binarized using standard software capable of basic image analysis operations (e.g., MATLAB). The following formalism is employed.
669 Let p(x) : P -+ {0, 1} be the so-called phase function defined on a compact set P C R a as p(x) = 1 if x belongs to pore space and p(x) = 0 otherwise. We will further assume that the pore space forms a single percolation cluster and that p(x) does not contain any "singular points", i.e., that Vx and V7 > 0 the set {y E O r ( x ), p(x) = p(y)} has a nonzero measure. Then p(x) can be regarded as a spatially-correlated random function and characterized by means of a correlation function a0+ -~ (0,1/, defined as @(u) (p(x + u) - e). (p(x) - e)/(p(x) - ~)2, where c = p(x) is porosity (only nontrivial porous media for which 0 < e < 1 are considered), u = 1lull (for isotropic media, Gp does not depend on the orientation of the vector u), and the overbar sign denotes an average over the definition set P. The zero-th moment of Gp, the so-called correlation length La = f o Gp(u)du, is a geometrically well-defined quantity measuring the "mean pore diameter". The procedure of computer reconstruction of a porous medium having the same correlation function as the original is described in detail in the work of Adler, Jacquin, and Quiblier (1990). The generated sample of a porous medium - in the form of a three-dimensional matrix of zeros and ones, representing a spatial discretization of the phase function p(x) - then defines boundaries of the pore space and serves as input for the calculation of effective transport properties. Apart from porosity e and the correlation function Gp, calculation of the following quantities characterizing the porous medium (i.e., the phase function p defined on P) are implemented in our software: (1) chord-length distribution, Ep; (2) spherical cavity radius distribution, Fp; and (3) tortuosity, 7-. Exact geometrical definitions of these quantities are now provided. Let Sp C P be a set of surface points of the phase function, i.e., Sp - {x [p(x) = 0 A V7 > 0 ~ y C O r ( x ) [ p ( y ) - 1} and let e p ( x , a ) " S~ --+ R + be a measure of the distance from a surface point x to the nearest surface point in the direction of a vector a (or - a , whichever is oriented from the solid to the pore space). One can then define the mean chord length LE = %(x, a), where the overbar denotes an average over the definition set Sp of ep, and for isotropic media, LE does not depend on the orientation of the vector a. The distribution Ep(a), where a = []al], is defined as the ratio of the measure of a set of points x E Sp such that ep(x, a) < a to the measure of Sp (we assume that Sp is not fractal). For the sphere-radii distribution, let us define MT to be a set of all points x E P such that p(x) = 1 and x belongs to at least one sphere of radius r which lies completely in the pore space. Then Fp(r) is defined as a ratio of the measure of Mr to the measure of a subset of P representing the pore space (i.e., p(x) = 1). The first central moment o f - d F p ( r ) / d r , which we denote rp, can serve as a geometrically well-defined quantity for the term "mean pore radius". Finally, let us define tortuosity 7 = l i I n n ~ o o _1 E i n = l l(ui, Vi)/llUi -- Viii where (ui,vi) is a pair of randomly chosen points n from the pore space (i.e., p(ui) = p(vi) = 1), and l(ui, vi) is the length of the shortest path between the two points through the pore space. Since we assume that all points of the pore space belong to a single percolation cluster, there always exists a path between any two of them. While the algorithmic implementation of the calculation of e, Gp, Ep and Fp directly from their definitions is relatively straightforward, the method used for the calculation of "1-
670
Figure 2: Illustration of the "burning savannah" method for finding the shortest distance between two points through the pore space (2D medium, periodic boundary conditions). requires a brief comment. For finding the shortest path between two points U and V through the pore space, we have implemented the so-called "burning savannah" method, illustrated in Figure 2. A signal spreading with a known velocity is released from point U and the time that it takes for the signal to reach V is recorded. The "signal" is a reaction-diffusion front which develops as a solution of Oc/Ot = - D . V2c + r(c), where r(c) = k c ( 1 - c) is a reaction term. The simplest reaction having one stable (cs = 1) and one unstable (cu = 0) steady state has been chosen; the initial condition is c(x, to) = cu for all points except U, which is initiated by cs. This method for finding the shortest path through the pore space proved to be computationally more efficient than alternative search approaches. Two ways for the calculation of the effective diffusivity are implemented in our software, mimicking two most common experimental methods used for measuring Def: steadystate diffusion measurements in the Wicke-Kallenbach cell, and transient diffusion experiments, which can be realised either chromatographically (Pazdern/k and Schneider, 1982) or also in the Wicke-Kallenbach cell (pulse-response method). The software for the modeling of steady-state diffusion is a direct analogue of the Wicke-Kallenbach cell (Figure 3a). A concentration gradient Ac is imposed over a slab of a binary-encoded porous medium in one direction (say, x-axis), and periodic boundary conditions are applied in the other two directions (y- and z-axis). After an initial transient period, a certain mass flux Jef is approached, from which the ratio of the effective diffusivity to the Fickian diffusivity, Def/Dm, can be calculated using the relationship Jef = A D e f ( - A c / A x ) , where A = n y n z d 2, A x = nxd, d is the side of an elementary discretization cube and nx, ny, and nz are the dimensions of the slab. The transient diffusivity is calculated by the method of moments (Sall~s et al., 1993), represented schematically in
671
(a)
,...--, 0.15[
(b)
I 0.1f u 0.05
Cl
q)
0
0"
c:,
0.2
0.4
x/L [-I
0.6
0.8
1
Ax Figure 3: (a) Schematic representation of a simulated Wicke-Kallenbach cell for the calculation of steady-state diffusion. (b) Schematic representation of the method of moments, used for the calculation of transient diffusion (the peaks on the graph are y-z averaged concentration profiles, flattening with increasing time). Figure 3b. The initial conditions are c(x, t0) = 0 mol/m a (i.e., an empty medium) for all x from the pore space, with the exception of a plane perpendicular to the x-axis, positioned at x = x0, for which a "Dirac"-pulse non-zero concentration Co is set. Let ~(x, t) be a concentration profile, averaged over the y-z plane at each axial position x. Assuming the moments Mo(t) - foL~(x,t) dz, ]~/l(t) ( 1 / M o ( t ) ) f o L ~ ( z , t ) x d z , and M2(t) - (1/Mo(t)) foL ~ ( z , t ) ( x - Ml(t)) 2 dx, the transient effective diffusivity D'~f is de n d D5 - limt_,~(1/2)dM2(t)/dt. In practical terms, the asymptotic value of d M 2 ( t ) / d t is reached already after a relatively short transition period. Obviously, the slab of the porous medium has to be sufficiently large in the x-direction for the spreading pulse not to be affected by boundary effects. As in the case of steady-state effective diffusivity, the ratio D'eI/Dm is calculated. -
-
4 Conclusions
An example of a calculated dependence of the ratio Dlef/Dm (i.e., the transient diffusivity) on porosity, parametrized by the pore-space correlation length, is shown in Figure 3 for the so-called Gaussian-correlated medium (its correlation function has the form exp[-(u/Lc)2]). As expected, the effective diffusivity is an increasing function of e. It follows from the comparison between simulations of dynamic and steady-state diffusion that when tortuosity is evaluated from dynamic D~ef using the formula Dlei - Dm(c/~-), it tends to be overestimated when compared with its geometrical definition as introduced above. The reason is that in the transient regime, the diffusion front samples dead-end pores as well as the conductive ones, while at the steady state, the diffusion flux is located in the conducting pores only. The role of computer-aided catalyst design and screening can be expected to gain on significance as the requirements on ever shorter process development lead times, imposed by competitive pressures, intensify. This trend will be further promoted by the increas-
672 0.04 0.035 r~ E
~,~ 0.03 E3
....................
0.025 0.02 0.01~
.3
0.4
porosity,
0.5
0.6
Figure 4: Calculated dependence of the transient effective diffusivity on porosity in a Gaussian-correlated porous medium with correlation length Lc = 2, 4, 6 and 8 voxels (from bottom to top line). ing power of computer hardware and the availability of sophisticated electron microscopy techniques. Software for the calculation of the effective diffusivity, described in this contribution, together with a recently developed module for the calculation of equilibrium distribution of vapor and liquid phases in a porous medium (St~pgnek, Adler, and Marek, 1999) helps gaining insight into pore-scale processes occurring in industrially important systems, such as gas-liquid-solid catalytic reactions (St~p~nek et al., 2000). References Adler, P. M., C. G. Jacquin, and J. A. Quiblier, Int. J. Multiphase Flow, 16, 691 (1990) Adrover, A., and M. Giona, Ind. Eng. Chem. Res., 36, 4993 (1997) Mann, R., "Computer-aided characterization and design of catalyst pore structure," pp. 617-643, in Structured Catalysts and Reactors, (A. Cybulski and J. A. Moulijn, eds.), Marcel Dekker, Inc., New York (1998) Pazdernik, O., and P. Schneider, Appl. Catal., 4, 321 (1982). Philips Electron Optics B.V., "Seeing things you've never seen before" CD-ROM, ISBN 9498-701-21712 (1998) Rieckmann, C., and F. J. Keil, Chem. Eng. Sci., 54, 3485 (1999) Sall~s, J., J.-F. Thovert, R. Delannay, L. Prevors, J. L. Auriault, and P. M. Adler, Phys. Fluids A, 5, 2348 (1993) St~pgnek, F., P. M. Adler, and M. Marek M., AIChE J., 45, 1901 (1999) St~p~nek, F., M. Marek, J. Hanika, and P. M. Adler, "Meso-scale modeling in multiphase catalysis," accepted to 3 rd Int. Symposium on Catalysis in Multiphase Reactors, Naples, Italy (2000) Acknowledgements: Project VS 96073 (Czech Ministry of Education), Grant 104/99/1408 (Czech Grant Agency), Bourse du Gouvernement Fran(~ais No. 11874.
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V,All rightsreserved.
673
A Hierarchical F r a m e w o r k for Modelling B i o p h a r m a c e u t i c a l M a n u f a c t u r e to A d d r e s s Process and Business Needs Suzanne Farid a, John Washbrook b, John Birch c, Nigel Titchener-Hooker ~ aThe Advanced Centre for Biochemical Engineering, Department of Biochemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK.* [email protected], [email protected] bDepartment of Computer Science, University College London, Gower Street, London WC1E 6BT, UK. [email protected] CLonza Biologics plc, 228 Bath Road, Slough, Berkshire, SL 1 4DY, UK. [email protected]
Abstract This paper describes the development of a hierarchical framework for modelling biopharmaceutical manufacture. Emphasis is placed on how a closer integration of bioprocess and business process modelling can be achieved by capturing common information in an object-oriented environment. The steps to use the developed decision-support software tool for addressing the impact of manufacturing options on strategic technical and business indicators, at different levels of detail, are identified. 1. INTRODUCTION The biopharmaceutical industry faces mounting pressures to achieve cost reduction whilst increasing speed to market. Global competition is driving the need to enhance manufacturing operations by achieving accelerated process development, maximized process yields, improved resource utilization, and reduced cost of goods. To achieve these objectives companies might typically use a number of separate software tools and consequently, administrative and manufacturing-oriented information systems frequently remain isolated (Schlenoff et al., 1996; Mannarino et al., 1997; Puigjaner and Espuna, 1998). A new kind of computer-aided design tool capable of reconciling business and process needs is required. The development of such a tool and its application to the production of biopharmaceutical products is the subject of this paper.
2. DOMAIN DESCRIPTION Manufacturing of biopharmaceutical products is primarily by batch processes. Each batch is produced by a series of operations that proceed from fermentation through to product recovery and finally purification, in a train of vessels. Additional manufacturing operations, involved indirectly in the production of a batch, include the following: the preparation of intermediate materials such as media and buffers; and the preparation of equipment used to
* Financial support from the Biotechnology and Biological Sciences Research Council (BBSRC) and Lonza Biologics is gratefully acknowledged.
674 produce a batch by, for example, cleaning-in-place (CIP) and sterilising-in-place (SIP) operations. Successful operation of a production facility requires both a technical understanding of the process and a command of the logistics of operations affecting the business. A tool capable of capturing and integrating these process and business issues is needed to accomplish this. 3. THE M O D E L L I N G A P P R O A C H
The framework introduces a hierarchical approach to represent the key activities in a manufacturing process through a series of levels. Such approaches have often been applied to systems in other sectors (Manivannan, 1998; Lakshmanan and Stephanopoulos, 1998; Subrahmanyan et al., 1996; Puigjaner and Espuna, 1998; Johnsson and Arzen, 1998, Davies et al., 2000). The hierarchical framework presented in this paper focuses more specifically on the manufacturing operations. The procedure to manufacture a product can be represented at different abstraction levels according to the desired goals of the user. Modelling high-level activities helps provide an overview of the process and a summary of the key operational and financial parameters. With more information, each high-level activity can be broken down into sub-tasks which generates more accurate estimates of the key parameters. This hierarchical approach to modelling enables selected activities to be examined in great detail. This is a computationally efficient approach and also reflects the fact that process data is often sparse. An example of the hierarchical levels used to describe product manufacture is illustrated in Figure 1. The levels correspond to the procedural control model in $88 guidelines where a procedure can be subdivided into unit procedures, operations and phases. Manufacturing campaign Intermediate material preparation recipes
Product manufacture recipe
Fer,~
Centriugatioo I [ Burpreparation II Charge ,11
I ManufacturingOperations
I I Mediapreparation II
etc.
etc. React II
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S,P etc.
I1
Manufacturing Procedures
Equipment preparation recipes
Mix
]1 Empty I
I ManufacturingPhases
Figure 1: The hierarchical representation of biopharmaceutical manufacture To satisfy both process and business applications each level specifies the manufacturing process in terms of the manufacturing tasks, the resources within the plant, and the process
675 streams that flow between material-handling tasks. Outside these core knowledge requirements exist characteristics that are more specific to each individual application. For process applications, examples include mass balance data and associated mathematical procedures. Business applications require cost dats. and knowledge of resource availability and utilisation. The developed framework permits the user to investigate different production strategies in terms of the process performance, resource demands and bottlenecks, and the cost of goods. 4. I M P L E M E N T A T I O N All work was implemented in ReThink (Gensym Corporation), which runs in the G2 objectoriented programming environment. ReThink was customised to reflect the processes in biopharmaceutical manufacture. The models built with this tool possess the benefits of objectoriented design, including the use of encapsulation, inheritance and polymorphism. The first stage of implementation involved mapping the declarative and procedural knowledge identified for each level into standard object-oriented design techniques. The three main constructs identified to specify a manufacturing process were the tasks, the resources, and the process streams. Each type of construct required descriptions at multiple levels of detail and was represented as objects that belong to a class defined by a set of attributes and methods describing its functions. These constructs have been classified into class hierarchies to allow inheritance of common properties and behaviour. Encapsulating the common features and behaviours of these objects into generic classes facilitates reuse of code and the extension of knowledge representation.
4.1 Manufacturing task classes The tasks describing the manufacturing operations are the core of the hierarchical framework. The procedural hierarchy creates sub-systems through the use of the levels (Figure 1). The biopharmaceutical manufacturing tasks were classified into the product-handling tasks, eg. fermentation, and ancillary tasks, eg. intermediate material and equipment preparation steps. This was necessary to prevent tasks inheriting inappropriate attributes and methods. The tasks' properties and behaviour were further influenced by the recipe structure selected. Separate recipes, and hence execution threads, were used for the sequence of product-oriented tasks and each of the ancillary operations (Figure 1), to facilitate scheduling. Similar separation of product recipes from recipes performing ancillary tasks can be found in Bastiaan (1998) and Crowl (1997). ,
FERMENTATION,a r .................................
definition t M..~NUFACTURIN,3-OF'ER.-'.TIONS. a EQU,PMEI~IT.PREP.~,.RA TION,, el....
J t
t
CENTRIFUGATION, s cla~s.d,~initbn 1
.__JF'F;,:':,DI.ICT.bI,~.NU F.,'.CTURE,s cls~.~-
blEf,.,~IEIR,~,.t'IE-FILTFIATION, e.clo,~. ]
~_.~ INTERblEDt'~tTE't"I"~TERIAL" F'FIEF',~R.~.TION, a ,::la~.~-detinitionJ
CHRObI.~T(:,GR.~PH'a ~cta~-defini~on ',
CLEANING,a r .... delhi,on I
P,~.CKED-BED-.~.DSORF'T1ON.
I C,, ,,,,0':a",,iOo~
Figure 2: The manufacturing task class hierarchy The task class hierarchy is shown in Figure 2 where the definitions of the tasks have been extended from ReThink's system-defined class of a task block. The task blocks represent
676 discrete events and have cost and duration attributes as well as a method defining default behaviour. These are all inherited by the custom tasks. The product manufacture and intermediate material preparation operations all share some characteristics and behaviours as they all make material. For this reason, the superior class, material preparation, was created to encapsulate the common information in a single class. This minimizes the use of redundant code, making the class hierarchy easier to extend and maintain. The subclasses have more specific attributes to suit their function. 4.2 Manufacturing resource classes The resources include the staff, equipment, materials and utilities required to execute the tasks and may also be generated as a result of a task execution. The resource class hierarchy, depicted in Figure 3, categorises the resources. Since ReThink already has definitions of the resources, these were customised to reflect the nature of the current domain. As a result, all the customised resources have inherited the attributes and methods of ReThink's resource class. The attributes include "id" and "maximum-utilization".
~t EQuiPbIENT,ar
.. iii~~ USP.EQUIPMENT,a clase-d~ini'0on
FERMENTEaacl.... definitioqI ROLLER-BOTTLE, a ob.s~-d~ini~on}
_~ DSP.EQUIPMENT,s r
CENTRIFUGE,e ele~-defini~on MEMBRANE-FILIRATION-RIG, ar I definition
I
E
~ STEAM,e r UTILITIES,~ r
I
WFI,,~r f
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CHROMATOGRAPHY-COLUMN, s cla~de,nition I
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1-'~
CHROMATOGRAPHY-MATRIX, a cla~e-I de~nition I
--[ HLIMAN-RESOUFCE,, el.... de'ini,on1 .... { OF'ERATOR,,el.... definition I
Figure 3: The manufacturing resource class hierarchy Equipment resources were given additional attributes such as "equipment-status" ("clean", "dirty" or "sterile"). Equipment resources were assigned the attribute "purchase-cost" to permit calculation of the capital investment and depreciation used to compute the cost of goods. Material resources were classified into chemicals-and-biochemicals eg. glucose, and equipment-related-material, eg. membrane-filter. The former was assigned a physical state attribute as well as composition tables. Each subclass of equipment-related-material was assigned attributes to indicate its size, eg. "area" and "number-of-units" for membrane-filters. 4.3 Process Streams The process streams refer to the flows of material out of product manufacture operations and intermediate material preparation operations. The process stream object was given attributes
677 to define its composition in terms of the component masses, volumes, and concentrations and their respective totals. 4.4 Defining the behaviour of the main objects To customise the behaviour of the material preparation operations, a method was configured to perform certain actions. Operations call methods to create the outlet process streams and determine their composition. In the present model the simple mass balance calculations for each of the unit operations have been coded as methods for the respective classes. The modular structure of the application means that more complex mass balance models can be easily incorporated later. Equipment preparation operations call equipment methods, which adjust the status of the equipment resources attached to the task. The resources were configured to behave as renewable and non-renewable resources as appropriate. 4.5 The cost objects and their behaviour The calculation procedures to compute the capital investment and the cost of goods can be used after running the model of the manufacturing process. Fixed and variable costs are assigned to the resources allocated to the manufacturing activities. The cost of each manufacturing activity is based on the costs incurred through the use of the allocated resources. The cost methods, associated with cost table objects, compute the variable costs based on the utilisation of the material, operator and utilities resources. The fixed facilities overhead costs are derived from the capital investment. 5. USING THE PROTOTYPE APPLICATION The application has the basic building blocks to enable a model of a particular manufacturing process to be assembled at increasing levels of detail. The following stages have been identified for a user to simulate operation of a plant: i. Initialize the chemicals-and-biochemicals components by cloning their representative objects and placing them on the initialize-components workspace. ii. Select the resources that are present in the company and put them in the resource pool. iii. Specify the specific attributes of each resource in the resource pool eg. "maximum-utilization" and "cost-attributes". iv. Set up the task sequences in the product recipe (Figure 4) and ancillary recipes. The user, through the selection of the number of subtasks and the sophistication of the models chosen to perform the calculations, chooses the level of detail required. v. Allocate the required resources to each task and specify their "utilization". The level of detail adopted to model a task determines the degree of complexity in which the associated resources are represented vi. Input the parameters in the mass balance tables of the tasks. Figure 4: An example of using vii. Run the simulation and view where resource the prototype application to bottlenecks occur as well as the output mass balance assemble a model and cost of goods results.
678 After a particular case is set up the impact of different production strategies on the process performance, resource demands and bottlenecks, and the resultant cost of goods can be evaluated. The hierarchical representation of a manufacturing process permits users to prototype a process at the required level of detail and to perform a series of "what-if scenarios" rapidly. Consequently, users can perform strategic, tactical and operational analysis according to their goals. 6. CONCLUSIONS The development of a prototype application has been presented, which provides a framework for modelling the operation of a biopharmaceutical manufacturing plant at various levels of detail. Its use to assemble models and evaluate the effects of different production strategies has been indicated. Future work will concentrate on two main areas. Firstly the knowledge within the application for computing mass balances and economic calculations will be enhanced to allow more rigorous modelling of these features. Secondly, case studies that analyse different manufacturing routes with respect to their relative operational and financial benefits will be carried out. They will serve to illustrate how the prototype application can be used to achieve more cost-effective planning of production. REFERENCES
Bastiaan, H.K. (1998) Process model and recipe structure, the conceptual design for a flexible batch plant. ISA Transactions, 36, 4, 249-255. Crowl, T.E. (1997) $88.01 concepts streamline control software application for biotech plant. In ISA Tech/Expo Technology update conjerenceproceedings', 1,2,131-141. Davies, E., Karri, S., Dunnill, P., Washbrook, J. and Titchener-Hooker, N. (2000) Biopharmaceutical Process Development: Part 3 A Framework for Manufacturing Decision Making in Biopharmaceutical Project Portfolios. In preparation for
BioPharm. Gensym Corporation (1999) V.5.1. manual. Johnsson, C. and Arzen, K.-E. (1998) Grafchart for recipe-based batch control. Comp. Chem. Engng. 22, 12, 1811-1828. Lakshmanan. R. and Stephanopoulos, G. (1998) Synthesis of operating procedures for complete chemical p l a n t s - I. Hierarchical structures modeling for non-linear planning. Comp. Chem. Engng., 12, 9, 985-1002. Manivannan, M. S. (1998) Simulation of Logistics and Transportation Systems, In Handbook of Simulation, J. Banks, ed., John Wiley & Sons. Mannarino, G. S., Henning, G. P. and Leone, H. P. (1997) Process Industry information Systems: Modeling Support Tools. Comp. Chem. Engng., 21, $667-672. Puigjaner, L. and Espuna, A. (1998) Prospects for integrated management and control of total sites in the batch manufacturing industry. Comp. Chem. Engng. 22, 1-2, 87-107. Schlenoff, C., Knutilla, A., Ray, S., (1996) Unified Process Specification Language: Requirements for Modeling Process, NISTIR 5910, National Institute of Standards and Technology, Gaithersburg, MD. Subrahmanyan, S., Pekny, J.F. and Reklaitis, G.V. (1996) Decomposition approaches to batch plant design and planning. Ind. Eng. Chem. Res. 35, 1866-1876.
European Symposium on Computer Aided Process Engineering - l0 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
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Study of the insertion of partial oxidation gas turbine to satisfy high temperature requirements of industrial processes using energy integration techniques. F. Marechal a, B. Kalitventzeffa, b aLASSC, University of Li6ge, Sart-Tilman B6a, B-4000 Li+ge- E-mail: [email protected] bOxipar d'Ex, 61, avenue du Parc, B 1330 La Hulpe - E-mail:[email protected] 1. INTRODUCTION In the chemical industry a wide range of processes require energy at high temperature to perform endothermic reactions. In such systems, the energy of reaction is supplied at high temperature in the radiation zone of a furnace. This is the case in steam reforming, cracking or pyrolysis reactions. In these systems (figure 1), the flow-rate of fuel is determined by the high temperature requirement above a given temperature (usually higher than 500 to 1000~ that defines a utility pinch point between the fumes and the process. Below this temperature, the energy remaining in fumes (between the high temperature pinch point and the stack temperature) and the energy for cooling down the reaction products are usually not all useful for the process needs. Combined heat and power is therefore used to transform the energy excess into mechanical power in a steam condensing turbine. This is usually a benefit since such processes require also mechanical power for compression. Rational use of energy will be obtained 91) by reducing the energy requirement above the pinch point; 2) by optimising the combined production of mechanical power below the pinch point; 3) by reducing the mechanical power demand; 4) by optimising the transformation of fuel into useful energy for the process. Each modification did affect the others and therefore there was a need to have a tool able to compute this influence in order to quantify the real benefit of the modifications. Of course, simulation models are useful but these suffer from the disadvantage of being based on fixed process structure. To allow heat exchanger reallocations to attain the energy savings, we used a method which combines simulation models and energy integration techniques to model the heat exchanger network and the steam network using optimisation. The method: Effect Modelling and Optimisation (EMO) has been defined in Mar6chal and Kalitventzeff (1997 and 1998). Our goal is to present the insertion methodology and to give some inovative results concerning the integration of energy transformation technologies in a given process. High temperature cogeneration is known to be a domain in which efficient technology is lacking, as it has been reported by Worrel et al. (1997). Partial oxidation gas turbine (Oxipar d ex, 1998) is an emerging technology which can be applied for this purpose 9oxidation with fuel excess upstream from the expander produces a fuel able to deliver heat at high temperature. To illustrate the application, we consider a classical ammonia production plant. The process is made of the following steps 9reactant preheating, steam reforming at 35 bar, 2~yreforming, high and low temperature shifts, CO2 removal using a hot potash absorption system, methanation, compression to 165 bar, synthesis loop and purge recovery. The overall energy requirements of the process is represented on figure 1 by the grand composite curve of the process. The process pinch point is defined by the high temperature requirement of the steam reforming (endothermic reaction) that occur above 765~ In this case, the fumes resulting from the combustion leave the reformer at 850~ Below the pinch point, the process features an excess of energy that will be transformed into useful energy by combined heat and power production in a steam network. The way the high temperature requirement may be reduced is out of the scope of this study. Even at fixed MER (Minimum Energy Requirement), as both high temperature energy and mechanical power are required, the performances
680
of such processes are determined by the efficiency of the combined heat and power production from natural gas used as fuel. Here, we will compare the integration of three of them: air preheating, gas turbine and partial oxidation gas turbine (OXIPAR system).
Figure 1 9Schematic description of the system 2. DESCRIPTION OF THE TECHNOLOGIES AND THEIR MODELS The process requirements are defined knowing the temperature and the heat loads of its hot and cold streams taking into account the chosen value of the DTmin contribution of each stream. When utility streams are considered these are added as hot and cold streams with unknown flow-rates. The optimal utility flowrates are computed by solving a Mixed Integer Linear Programming (MILP) problem modelling the ideal heat exchanger network by the representation of the heat cascade: nw
Minimise ~ ( C l w y w Rk,yw,fw w=l
+C2wfw) nw
(1) 13
subject to heat cascade balances: ~ fw qwk + ~ Qik +Rk§ R, = 0 V k=l .....n, w=l i=1 nw nw Export of electricity' ~ f. ww + weli- wele+ W=0' Import of electricity" ~ fw Ww+ weh + W = 0 w=l w=l fminw yw = f. = frnaxw yw ;yw ~ {0,1} '~' w=l .....nw R1 = 0, Rnk§ = 0" Rk = 0 V k=l .....nk§
(2)
(3) (2.2) (2.3)
n is the number of process streams; Rk: the energy cascaded from the temperature interval k to the lower temperature; Q~, the heat load of the process stream i in the temperature interval k. Q~, is > 0 for hot streams; nwthe number of utility streams; q~k the heat load of the utility w in the temperature interval k; f, the multiplication factor of the reference flowrate of the utility w; fmin(max)~ the minimum (maximum) value accepted for fw; yw the integer variable associated with the use or not of the utility w; Clw (C2w) the fixed (proportional) cost of using the utility w; nk the number of temperature intervals; Wwis the mechanical power produced by utility w: ww < 0 for a consumption; W is the mechanical power required by the process; weli the electricity imported from the grid; wele the electricity exported. 2.1. Air preheating Air preheating aims at recovering the heat of the fumes at low temperature to raise the temperature of combustion and therefore increase the heat available at high temperatm'e. This leads to a reduction of the fuel flow-rate. The energy saving is of the order of 12% depending on the temperature of
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preheating and the energy requirement of the process at low temperature. To model the combustion, we use the EMO concept as it has been presented in Marechal and Kalitventzeff (1998) and as it is summarised below. The combustion generates two hot streams, the first at constant temperature Tr (assumed to be the radiation temperature) will have a heat load denoted Qr, the second goes from Tr to Tstack (stack temperature) with a heat load Qc. To model the combustion, we distinguish two effects that constitute the LHV (Lower Heating Value): the fuel and the air are heated up from 25~ to the adiabatic temperature, then the combustion takes place and the resulting flue gases are cooled down from Tad (adiabatic temperature) to Tstack. Usually, we will use several fuels in the combustion: e.g. natural gas but also purge gases in the case of an ammonia plant. In the model, the heat balance above Tr defines a constraint that computes Qr, the heat load of the hot stream at Tr: nf
ff Hrf + fao cpa(Ta- Tr) + Qp- Qr = 0 and Qr = 0 with
TadO~- Tr O2~ Hrf = LHVf * ~ - ~ cpa (TO- Tr)
(4)
f=l nEf is the heat of combustion of one unit of fuel f; ff the flow of fuel f; fao the flowrate of ambient air; cpa the specific heat of air; Ta the ambient temperature; Tr the radiation temperature; Qp the heat of air preheating. O2f is the oxygen required by the complete combustion of one unit of fuel f (in kg O2/kg fuel); O2a the mass fraction of 02 in the air; LHVf the lower heating value of fuel f; TO the reference temperature; Tad~ the standard adiabatic temperature of combustion of fuel f.
The heat balance between Tr and Tstack defines Qc. We dissociate the contribution of the fuel and the contribution of air by considering the fumes as being made of a first contribution of burned fuel and a second of air: nf f=l
Tr- Tstack O2f ff Hcf + fao cpa (Tr- Tstack)- Qc = 0 and Qc = 0 with Hcf = LHVf* -l-adOf: TO f- ~ cpa (Tr-Tstack)
(5)
This approach allows us to decouple the contribution of fuel and the contribution of air in the combustion. Each fuel f will introduce its own contribution Hrf to Qr and Hcf to Qc. Of course, we have to introduce a constraint that will secure that the air flow-rate is sufficient to burn the fuels: nf ~, (1+el) O2f f,- O2a fa = 0 f=l
(6)
with efthe (minimum) oxygen excess required to insure a complete combustion. The heat of preheating Qp is computed by (7) it allows to compute the preheating temperature by representing the air preheating by a list of cold streams from Ti to Ti+l = Ti + AT. fli
QP- 7_., faicpa (Ti+l- Ti) = 0 i=1
and faj = fai+l V i=O..... ni
fa i is the flowrate of air preheated from Ti to
Ti +
(7)
AT; nxthe number of discretising intervals.
2.2. Gas turbine A gas turbine (figure 1,a) is used to produce preheated combustive agent to be consumed by a post combustion. The mechanical power produced by the gas turbine reduces the cost of the energy but also the heat available at high temperature. This leads to an increase of the total fuel flow-rate. In order to maximise the heat available at high temperature and therefore reduce the fuel flow-rate, complete combustion is required. This leads to an electrical efficiency of about 8% if we consider the LHV of the natural gas put in the system. Gas turbine and air preheating technologies are excluding each other: both play the role of producing a hot combustive agent for the combustion. To model the gas turbine integration, the outlet of the turbine is a hot stream with an unknown flow-rate fGT that will be cooled from TOT to Tstack. The heat load, i.e. QGT = fGr cp~ (TOT - Tstack), is computed using a simulation model where the efficiencies and the compression ratio are determined according to existing engines available on the market : Kalitventzeff and Marechal (1998). The gas turbine introduces also a contribution to the mechanical power constraints (3) :ftT * WtT. At the outlet of the turbine, the hot gases contain an excess of oxygen that may be consumed in the combustion. The flow
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sent to the post combustion (fPc) acts as the combustive agent of the combustion. It reduces the amount of heat available above Tr (Qr) from the heat load corresponding to heat the hot gases from TOT to Tr, i.e. QPc = fr~ cpf~ (Tr - TOT) . Of course after post combustion this heat becomes available as a hot stream from Tr to TOT and is mixed together with the heat from the gas turbine. A constraint indicates that the flow going to post combustion is limited to the flow delivered by the gas turbine 9f c T - fPc = 0. The gas sent to the post combustion also introduces a contribution to the oxygen balance of the combustion (6)" fPc * O2vc where O2vc is the amount of oxygen in the exhaust of the gas turbine per unit of flow. The contributions of the gas turbine used in the example are given on table 1. This representation allows to generate solutions where only the part of the gas turbine gases necessary to satisfy the high temperature requirement will be sent to the post combustion, the remaining gases being used to satisfy lower temperature requirements. The formulation allows also to introduce different turbines (with different performances) in the system and to select the optimal one(s) using the optimisation tool and the integer variables. It allows also to burn a mixture of fuels in the post-combustion. 2.3. Partial oxidation gas turbine The partial oxidation gas turbine is a technology developed by a Belgian company 9Oxipar d Ex. The principle is to transform by cogeneration part of the energy of the natural gas used in the combustion. This is obtained by compressing about I/3 of the combustion air and mixing it to the natural gas. A certain amount of steam (fs) is also added to avoid coke deposits on the catalyst. A high pressure catalytic partial oxidation of the mixture acts as the burner of a gas turbine to produce a high temperature and low molecular mass fuel gas comprising mainly H2 and CO. The resulting gases are expanded through a turbine to produce mechanical power before being bumed in the combustion. The mechanical power produced (Wox) is between 6 and 9 % of the LHV of the fuel, depending on 0 2 to C ratio, steam to C ratio, and on T and P. This technology produces a fuel. It may therefore be used in synergy with the air preheating or the gas turbine technologies that will produce the oxygen required for the combustion (about 72% of the one required by the normal combustion). From the results of the simulation for one unit flow-rate of natural gas, we compute Qa=fa cpa (Tr-T,) the heat required to raise the combustion air above Tr; Qs - fs cps (Tr-Ts) the heat required to raise the steam entering the partial oxidation gas turbine at Ts above Tr and Wox the mechanical power produced by T o the gas turbine. The heat available above Tr is a constant term " r~'~" t n l ~9Tad ad'ofTr -Wox - Qa - Qs f- TO computed for one unit flow-rate of natural gas entering the partial oxidation reactor and to be added in equation (4). The heat available from Tr to Tstack is identical to the one of the fuel increased by the contribution of the steam added in the system Qs c = fs cps(Tr-Tstack). The steam production introduces a cold stream defined in three sectrions: preheating from the ambient temperature to the saturation, vaporising and finally superheating the steam to reach T~. The values used in the example are given on table 1. The values are computed for Tr = 1200 ~ and a specified remaining 02 percentage in the fumes. The natural gas has a lower heating value of 48380 kJ&g. Table 1 9Contribution of the technologies to the different equations of the model ......... Eq, Combustion Air preheating OXIPAR Gas turbine Reference flow . . . . . . 1 kg/s of fuel !kg/s of air l kg/s of fuel 1 kg!s of fuel Cost 1 Euro/kg 0.168 0.168 O,168 Heat above Tr 4 kJ/kg 44482 -t062 35894 -44650 Heat below Tr 5 kJ/kg 3571,2 950.44 7869.7 67530 Mechanical power production 3 kJ/kg 0 0 3353 15048 Oxygen consumption 6 kg 02/kg -4,6673 0,2333 -3.394 12.789 Steam 2 kJ/kg 0 2.4. Steam network Below the pinch point, heat from the process and the one resulting from the combustion is an excess of energy whose exergy will be valorised by combined heat and power production. Usually, this energy is used to produce high pressure steam to be expanded into back pressure and condensing turbines. The pressure and temperature levels as well as the flow-rates of steam production and draw-
683
off have to be determined in order to maximise the mechanical power production. Below the pinch point, the different technologies will produce different flow-rates of fumes but also different requirements: steam production or air preheating. The back pressure turbines will be used to satisfy the energy requirements of the process but also the steam requirement of the partial oxidation system or the air preheating. In order to compute the optimal matching of the steam network to the integration of the different technologies, we used the steam network model described in Marechal and Kalitventzeff (1999). The complete model is made by summing up different models 9the ideal heat exchanger network, the ideal steam network, the transformation of energy sources into useful energy and mechanical power balance of the process. It has been used to compare optimal integration of the different technologies: normal combustion, air preheating, partial oxidation gas turbine, gas turbine with more or less post combustion, but also the possible combined usage of a partial oxidation gas turbine and a gas turbine. The model allows also to use different kinds of fuel in the combustion. On table 2, we present the results obtained for the integration of the systems to produce 100 GJ at a temperature above 850~ in the fumes and when accepting a maximum preheating temperature of 280~ Marginal efficiencies are computed with respect to the air preheating solution. The total losses include the stack losses and the condensation in the steam system. The mechanical power of the steam system has a calculated efficiency of 25% (expansion from 120 b superheated to 0.12 bar, with a condensing turbine having 70% of isentropic efficiency). The gas turbine and the partial oxidation gas turbine system (OXIPAR) have been computed with the same efficiencies for the compressors and turbines, the same pressure and the same inlet temperature in the expander. For the OXIPAR turbine, we considered no steam injection. Table 2." energy efficienc!es of the different systems Natural gas Gas turbine production Heat above 850~ Useful heat below 850~ Air preheating Total Losses Possible production by steam system Total mechanical power Marginal efficiencies
GJ GJ GJ GJ GJ GJ GJ GJ %
Combustion 181.6 0 100 70.9 0 63.9 17.7 17.7 -
Air pieheating 150.4 0 100 41,5 17.2 40 10.4 10.4 -
OXIPAR 177 12.2 100 54.4 14.72 51.2 13.6 25.8 57.9 %
Gas turbine 216.3 19.1 100 84.5 0 76.1 21.1 40.3 45.3%
3. APPLICATION OF THE MODEL The model is applied to compare air preheating, partial oxidation gas turbine and gas turbine to supply energy to an ammonia plant. We compare the technologies assuming that all the efforts concerning the energy integration of the plant have been made and that the Minimum Energy Requirements of the system is obtained with this configuration. This approach allows computing the size of the CHP systems to satisfy the MER. The data correspond to a real plant producing about 1000 t/d of ammonia. The results are presented in kJ per unit of product and in MU (monetary Units). The cost of electricity is 4 times the one of natural gas (3 if we export). In table 3, column 2 represents the best solution using air preheating as it is used in the present situation. The process globally imports electricity from the grid. An alternative would be to produce it using the condensing turbines, these results are presented on column 3. Column 4 presents the integration of the partial oxidation system and column 5 the results of the gas turbine system. In the table, we compare the energy efficiencies of the systems, the operating cost savings and the investments. With the partial oxidation system, the process becomes nearly self-sufficient. The marginal efficiency of the steam injection (33%) being higher than the one of the steam network (28%), it is worth to adjust the steam flow-rate to reach the self-sufficiency. In the solution presented the steam to carbon ratio is 0.5. Compared to the other solutions, partial oxidation induces smaller perturbations on the steam network. This is due to the fact that the air preheating is preserved and that the excess of energy in the fumes is lower. This will be an advantage if the steam turbine is the bottleneck of the process. The marginal efficiency of the partial oxidation turbine (49%) is higher than the one of the gas turbine (43%). One should also note that the integration of the partial oxidation gas turbine leads to a positive
684
net result including the investments. This is explained by the benefit on the operating cost but also by the smaller investment (32%) related to the smaller size of the cogeneration machine (both in terms of size and cost per MW). It should be mentioned that the purge gases were only available for combustion (these account for 12% of the energy) by consequence these have not been used in the partial oxidation system but effectively used as fuel in this solution. Table 3 9Comparing the different solutions
. . . . . . . . . . . . . Reference Natural gas (kJ/unit of product) 100 Net electrical production (kJ/unit of product) -11 Steam network (kJ/unit of product) 25.2 Gas turbine system (kJ/unit of product) 0 Air preheating (kJ/unit of product) ..................12.4 Marginal efficiency of the system na Efficiencies of gas turb!ne system ......... na Operating cost including maintenance (MU/year) 101.1 Benefits on operating costs (MU/year) 1.1 Investments gas turbines (MU) 0 Rate of return na
Self sufficient 0XIPAR 142 122 0 -0.1 37.4 29.7 0 9.7 16.8 11.1 25.91% 49.33% na 7.91% 100 89.7 0 10.3 0 43.5 na 4.2 .
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Gas turbine .......... 136 4.6 31.7 12.2 0 43.29% 8.98% 90 10 65 6.5
4. CONCLUSIONS MILP modelling techniques and concepts have been used to compare the integration of high temperature energy saving techniques in industrial processes. The method proposed allowed to represent the influence of the available fuels, the synergies between the technologies, all the processes being considered as perfectly integrated. This allows to set targets on the possible benefits of the different technologies and to identify the impact of this integration on the rest of the process without having to consider heavy calculations for simulating and modifying the heat exchanger network and the steam network. The main interest of this approach is the representation of the ideal heat exchanger network, the adaptation of the steam network and the integration of the combustion by a set of linear constraints included in an optimisation problem. The results show that CHP located above the pinch is not 100% efficient and influences the transformation of energy below the pinch, what is in opposition with several statements found in the literature. This method has been used as a preliminary step to compare technologies in the optimal integrated situation. The model proposed to represent the combustion and the cogeneration using partial oxidation turbine or conventional gas turbine is generic and may be applied to different processes without modification. It has been applied to study the integration of such technologies to an ammonia plant and allowed to illustrate the competitive advantage of the partial oxidation gas turbine in the case of high temperature cogeneration. If the performances are very similar, the calculation shows that the investments will be smaller and that the existing process will be less modified by the insertion of the new technology then by the conventional gas turbine solution. REFERENCES Worrel E., Bode J.-W., de Beer Jeroen Analysing the research and technology development strategies - the 'ATLAS project. Report 9700, Department of Science, Technology & Society, Utrecht University, ISBN 90-73958-25-3 (1997) Martchal F., Kalitventzeff B. Targeting the optimal integration of steam networks. Computers and Chemical Engineering, Vol. 23 Suppl., pp. S133-S136, (1999) Martchal F., Kalitventzeff B. Effect modelling and optimisation, a new methodology for combined energy and environment synthesis of industrial processes. Applied thermal engineering, Vol. 17, n~ - 10, pp. 981-992, (1997) Marechal F., Kalitventzeff B. Process integration : Selection of the optimal utility system Computers and Chemical Engineering, Vo122 Suppl., pp. S 149-S 156, (1998). Oxipar sprl, Patent " "Syst~me 6nerg&ique thermique b. turbine & gaz avec oxidation partielle catalytique de combustible", n ~ de publication 1009707A6 (1998).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
685
Abstract design in the development of pharmaceutical processes Mona Sharif, Nouri J. Samsatli and Nilay Shah* Centre for Process Systems Engineering, Imperial College, London, SW7 2BY. The specialty chemical manufacturing industry is under increasing pressure to reduce costs and increase its speed of response to customer requirements. The response, currently, has been to stay with old manufacturing technology and to move to lower cost economies. This paper describes one aspect that can be applied to change the design and operation radically in this sector in order to achieve significantly improved business performance. 1. INTRODUCTION The main concern with current industrial practice is that processes are generally developed within the restrictions of standard plant items, and that standard plant items are designed to match closely the behaviour of laboratory equipment. In order to develop a significantly more efficient process, it is necessary to examine how it behaves on a physicochemical basis alone. To this end, we can define a process only in terms of fundamental properties and the "conditions" that are required to allow the process to perform at its best. Depending on the level of abstraction, the conditions might be properties such as pressure, rates of heat transfer, rates of mass transfer, rates of mixing, etc. from which the behaviour (and hence performance) of the process can be calculated. We can then relate the required conditions for the best process to the capabilities of the equipment in which the process may be carried out. This will highlight any inadequacies in standard equipment and hopefully stimulate new equipment designs that have significantly greater ability to provide the fight conditions for the process. At worst, if no novel equipment design is available, we will have a precise understanding of the effects on the process of using a standard plant and the ability to make the best use of existing plant without the problems associated with scale-up. Similarly, processes are defined in terms of "unit operations" too early in development. This can stifle innovative thinking and lead to inefficient processes. If we use the above philosophy to define a process that is independent of equipment and any pre-conceived unit operations, we have what we term an "abstract" process, where individual component flows can be manipulated to generate the ideal conditions for the process. From this abstract process, an idea of what types of operations are best for the process can be obtained; as well as indicating which standard unit operations are suited to the process, more importantly it should indicate where novel techniques can be of benefit. Our approach is a hierarchical series of procedures. It begins with a completely abstract description of the process (one with no notion of unit operations or equipment limitations), passes through various intermediate representations, and finally arrives at a "concrete" process and a plant design. This provides a mechanism that takes direct account of the effects of any
* Author to whom correspondence should be addressed (email: n . s h a h @ i c , a c . uk). This research was partially supported by EPSRC grant ref. GR/L63020.
686 decisions on the performance of the process. The hierarchical procedure is summarised in the following steps: 9 "abstract" process design; 9 "conceptual" process design; 9 "concrete" process design and mapping to equipment (including consideration of novel operations and equipment); 9 plant design. Associated with the above hierarchy are a number of key issues, the first of which is the handling of data and its link to experimentation: this gives rise to an associated data hierarchy. A key feature of the approach is the way in which experiments are driven by the process design activity in order to generate the necessary data. Currently, experiments are carried out with the goal of obtaining the "best" reproducible process in the laboratory. Any data that are available for modelling are almost purely as a consequence of this, and quite often the data are insufficient to build a reliable model of the process (because the right properties were not measured or the data are valid only for a small region of operation, or more than likely both). In our approach, we use the model to predict the best process (since this can be done without equipment limitations) and discriminate quickly between alternative routes; use of experimentation is mainly to support and validate the modelling activity. The overall aim is to develop processes quickly while still exploring as many sensible options as possible. 2. OVERVIEW OF PREVIOUS RELATED WORK
Hierarchical method (e.g. Douglas, 1988) This approach is based on the use of models of increasing complexity and rules which evolve a series of candidate designs with evaluations at each stage to prune the decision space. It has been widely applied to continuous processes but is less immediately applicable to batch processes.
Superstructure optimisation (e.g. Grossmann and Sargent, 1978) This technique has again been widely applied to continuous processes, where relatively detailed steady-state models are used for a superset of interconnected equipment items and the required parts of the process and their operational details are determined through the solution of an optimisation problem (usually mixed integer non-linear).
Global~local variable decomposition (Tricoire and Malone, 1992) This technique has been developed specifically for entire batch processes, whereby the decision space is divided into "global" variables important to the overall process (e.g. conversions, recoveries) and "local" variables which are only essential for individual steps. The local variables define the operating policy of a step and therefore determine the values of the global variables.
Detailed~simple model iteration (e.g Salomone et al., 1994) This approach is based on an iterative procedure which develops overall processes using simple models (based on parametric fits through experimentation or through detailed modelling) and then uses detailed models for simulation and scheduling to assess processes so derived more accurately.
Hierarchical material-balance oriented approach (Linninger, 1996) This group has developed a toolkit that explores and supports different levels of the development hierarchy. The emphasis is on the use of knowledge bases and material balancing to choose and assess options. The process was particularly applied to the problem of developing low environmental impact processes, including route synthesis.
687
Functional operator method (Smith and Pantelides, 1995; Sharif 1999) This approach is applicable to batch, continuous and batch/continuous processes. The idea is that a number of functional operators (e.g. reaction, separation) are defined through detailed models. A series of mixing and splitting operations allows all operators to be interconnected in any possible way. Given a potential starting set of raw materials and some final product specifications, the technique uses optimisation to select the required operators, their operating characteristics and their interconnections. Smith and Pantelides (1995) also demonstrated that, provided the models fulfil a certain condition, the number of discrete altematives that need to be considered is relatively small. Single-level detailed model approach (Charalambides. 1996) This approach is used to develop the best operating policy and resourcing for a process with a fixed structure. It may therefore be used to assess different process in the best possible light. It used detailed dynamic models to trade-off intensities of processing at each step as well as to dete.wnine their operating policies. 3. NEW APPROACH FOR PROCESS DEVELOPMENT As mentioned earlier, we consider a hierarchical approach (Figure I) which I .... ! DATA PROCESS allows the evaluation of a re, tuber of ,I [ alternative routes at different levels of detail. [ "first set" [ "first set" This facilitates ~vo thin~s in .Da_~icular: (i) [. fi-mdamenta! A ~ abstract the screening of altemati-ves Without having 1 derived % ~& conceptual ta ~enernte eamnlete de.~ions (a fent, wo in . conditions ~ . concrete common with the approach of Douglas [plant/process [ [ ,~:.~ ,u.. parameters" ('9S8)): ..... ~--,~ v--.., ~-',~ -~_b'..'-litv .. . . .to . . . . . -~!:-r .. processes in a fashion that transcends tradition_a! '! :! ='-_1":i1~.+~! " * '~.I~V .-Li.~. . . . <.!Li.U~ w"u':"h*s --'_t I ~ i - ! i ~ _ ' . ~ s selection ot ~ 1 ~ experimentation fundamental limitations associated with the ^u..: ~ ..c .... ,.. process L-Lit:-L~.
t~i
l':--:'-~[-rC,
I
The top level activity attempts to design ,,-,,-~-.:.~.-. ............ a-~+ t ~ V C ~ 4 abstract ', ,.... c v c ~ ., ...u:~t. w-~_,'it:-:,.~ ;_ ~ concerned with the spatial and temporal A:.~+MK,:+: . . . . .
c
1. . . . . . . . .
:.....
t "
~' new
d-:-t:-;sct.~-
d
conditions. At the second level, the abstract
Figure 1 Schematic of dual-hierarchy methodology.
process operations that might realise it (e.g. heat and mass transfer). This level still operates at a de~ree_ ,,_,~'eabstraction and is n~t..,c o n c e d e d with specific, ~q,_<,.)..,.-:-~.,:.,~" "';. . . . . . ) and therefore bears some similarity with the mass/heat exchange module approach of Ismail et al. (1997). The conceew_a! !.... ' is cu~ent!v b,~in~ evolved bv ~t ),-~- re,--.earch croups ,..a will ,'-o* be a~s---; -u--~further here. The concrete process design activity seeks to implement a given process in a plant =, proce.~s p e r f o m l a n c e '~'--iDaref,~.-~r'h ~--~'.'~ ( r ~-" o ~har{f !QO(}~ A.~.~.r c o l l s e o u e n e e s o11 §,_~,:,_. , . A l l be identified here. T!:'-... . . . . *,'~-+~<:,~.:.~w';th,. . :._ "~--.~,~,~+,-~;+:~._-,,_,:,.~c,,.~,~role of experimentation., where experiments are carried out only to "optimise" the process at the laboratory scale: the experiments are aimed to co:-v.~eree._, on a f.-~n-~ib~e_. .... __.:.. :nrocess-.. ,~:r..o,-~a-~. .... ' .... .... our approach 1~" ~a,,,m-.' ........... -:-~..._~
The next section of the paper will describe the abstract level design.
688
4.
ABSTRACT PROCESS DESIGN
-.b~wac~ nroccsseq are ~ene,-a*e~ u~in,., .~ scr!cs o f interconnected nrocess cc ~" as illustrated in Figure 2 and Figure 3.
FiNu-e 3. Multi-cell abstract design
Figure 2. Single-cell abstract design.
E a c h cell is a s s u m e d to be well m i x e d and to operate in a d y n a m i c fashion, with timevap.:ina . . flows, . . . c o n m o n e m ho1~_,-ue.~' ,,.-a and conditions. The onlv. vhvsico-chemical . . . . w:'ocess that is m o d e l l e d explicitly in a cell is chemical reaction. There are two types o f flows (which are _.._-, 9 ...... ,_........... . . .. . . .buiiu ........ ~. . . . .flows . . . . . . he.>a~een .~. n~r.~ h;~i,-o,-,; . . . . r~_. .T~ ':'~ lines ;nd-.:~.'~ ..... eux~i.:_,uh,.-~i-~ ~. . . . . . . . . . -* reseiwoirs " ...... "".~ ,.*_hecell; there is a t i m e - v a r y i n g flow for each c o m p o n e n t from or to a cell. These represent either Da-:~.c,e, or --ve,_,,--e,~.~, ~:_ ............. ~,~u~.,.~s ...... ._,-~,.,, to r e m o v e sina!e species with perfect resolution. The dotted flows are b e t w e e n cells and are a s s u m e d to be at the -,_~ ~:x~ source ~_.~,. It can be s h o w n that in the absence o f explicit treatment o f multiple phases, a single cell is all that is ,'.~,~,---;,'edtn describe "-v ":''--~-~. . . . --~-~*i. . . . . . narat;~,'.-- processes .::.._-*a very abstract '~.~vo,. There may, h o w e v e r , be computational benefits to a multi-cell structure. ~ e ce!l c o a c e p t is utiiised as fbilows' i. a d y n a m i c m o d e l is d e v e l o p e d for each cell and reservoir, taking account o f reaction stoichiomettT and kinetics; ii. a d y n a m i c optimisation is p e r f o r m e d w h e r e b y the cell conditions (e.g. T, P) and flows are varied d v n a m i c a i i v to optimise a fimction based on the rate o f v a l u e - a d d e d associated with the contents o f each reservoir; . . res~Its . . o f u-_,e abstract desi.>,_,~ ;,; iii. w..:e "L"" m a y be used *:-, "-" screen poor solutions (bearina .. . . . . . m i n d that at this level there is large uncertainty in the design); iv~ ---+~:._._~results:;_ . are . . "-q~d . . to. attempt . . tc.~.identiPv .tms~ib!e . . . . operations_ to achieve the i m N i e d conditions. T w o example~ are used below to illustrate this concept. "_r-'* * ~ ~ ~" . x - = a a &
-
-
Industrial case study 1 H~-e w e ~c-,:---~sider:m indus~ciai case .~w&-.-witL-: the reactio~_ scl-_,eme A + B ~ c,- A + CD, w h e r e C is the desired product and D is a by-product. A c o m p a r i s o n o f the existing, best corlvemionai and abstract np ~ ,:~ra.tng'-" . eoi';c'"ies can be seen in ~igure,. 4. it can be seen that the C:D selectivity is greatly e n h a n c e d in both the best conventional and abstract processes. The l_.ioaa_up ...... "t~ tt'_,e' " for the com~one~t~_. .... in the abstract .c...rocess. is shown in Figure 5. N o t e that the a m o u n t o f b y - p r o d u c t D generated by the abstract process is significantly less than the
689
amounts produced by the best conventional and the base case. This is achieved by selectively removine m'oduct '-" as soon as it is o~ne,'~+od in the ceil. The State-Task Network (STN) for the abstract process can be seen in Figure 6. (The ,--:-,,-'-'~ ,'-~p,'+.~-~:---+..... +....;~+ states and the rectangles processh:,g tasks).
Figure 4. Compari~n ~ffthe exk~ing, be~ convemional and abs~xactprocesses, The:best .conventional :process generates a rate o f value-added that is 7 times greater -than the base case process. Nevertheless, the abstract process has a value-added rate o f 33; roughly v~ice, that o f t h e bestconve~.~+tio~m!,pmcess,
Figure 6. STN for the abstract proce~.
Figure 5. Hold-up in the reservoir.
Industrial case stud), 2 This example problem is s:imilar t o the r . . . . . ~:-, r a~,;ila~ion i l prc, cess used bv Charaiambides, which in turn is based on that o f Barrera (1.9,.)0). Here, reactants R I, R2, R3 ..... ~ R4 arc tr:-ms.:q:,,q~cd into p.n;ducts A_ D and E via the- ti-ilowin~ reaction scheme: Ri + R2 --~ il; R1 + II --~ A; I1 -+ C; C + R3 --+ E; ii + C ~ I2; Cat -+ Cat*; 2E + R4 --~ D + 2R3 ~ c rcactior-.:s . . . . . . . t:+kc ---;+...... .,aur " a so!vent " fl~cpresence o f actlve" cata.ys,* I + part~c!cs'' (:Cat). T M concentration profile o f the products in the celt is show~ in Figure 7. The product '.-r.._--o, . 41-ups . . -_,nthe reservoir can be s e e n Fi~_~.+_vv8. ~"=_, ,,r abstract process has a rate . . .of . . . v.a!+--:e. added which is t2 times greater than that o f the convemional process reported by, Charaiambides. Figure 8 shows finat the rusk ca,~,=;+~,,-~for v,vo main steps. Product A is generated during the first mep, with very small quantities o f E and D, whereas most o f product --7-_;_is ~.ee-"---eratedin the second .sten.. .+Th.is . ". .anaicwo-.:s_ . *,; . the . conventionai . . . vtructu_re,, used_ by Charalambides, which consisted o f three reaction tasks with product A being produced in the +ir~+ +:+~k a-!d product l) + the tb.{..--,i The STN +2:.,rthe abstract pr_:,ccss ca-i bc scc-n in Figure 9. ~i
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_i-'igure 81 t b k M ) p i a ~;[ter ~ e r v ( u f
..... Cxa-:--:-pJc8 considered that the >:due-added rate o f ~i:e :,-,-:a~_--: ~roceas at least e,vice (h-.-: g::e case o f e x a m p l e 3, it is 12 times) ti-e rate o f the convei-:tiona: ~,:'nc.~a~ T:,.. nI~t,-n,-.+ .nrocea~ is able to u.nlock the Et[[ p,-:.:...,_er:::a.:~ . . . . . .-:,; the chemistry. The r.,':.o)iva, ion i2-.,rcor:.sidering the types ,:>f ,:>pcratic,~s to, ach{cvc suci- prc:ccsscs is c icar a ~ d thus a -~cw chaiic>gc is posed. There " no -J~,-,~:ave c,f techniq--,Jes to ri~:e to ~-:"--:~'-~--g-,~ {e-'..e. ~i~emhr;m:.--: reactors. che:--:icai conversio:::Anactivation etc.); vie critical step " idemi#,,inp th,-e oppommi~< a.,:--d lqgure 9. STN for the abstract process motivation for targeted process development. RI.:-..c . . . . . -l"; Chemical Processes. Comput. Chem. Engng, 7, 439. ismaii, < t? E . N . Pi.q.{ik-,.-,6-,:;.,..:.h-:,~and K. p Papaiexandri ,': 997). ~:o~:,r~:J;,-,~, ,,-)f not:ideal mix:ures :
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based on mass/heat exchange principles. The entrainer selection and sequencing problem. Cornput. Cheer.,7. 1..7,:,_,::,_:2i. $211 Linninger, A. A., A. A. Shahnin and G. E. Stephanopoutos (t_9961). Knowledge-based validations and .. waste )nam~geF,---ent of batci: :u:::::..,, . . . . . . n. .a~e.--:,., . . . . . . .ca . ~ process designs. C..::,mpuL Chem. Engng, 20.. . .r .i.431 Saiomone, H. E., j. M. Montagna and O. A. iribarren (i994). Dynamic simulations in the deign o f ,;~,c: r,roce:<e: C,..?m2e~.t.Chdl}i. E-.,'i~nP, i6_ 17"~ Sharif, M. (i 999). Design ojbuegrawd Bar Processes, PhD thesis, University of London. Smith, E, M, B, ,~.na (!995), D,--:s ~;~,,. . .of . . C, . .C,. Pante!ides . . . . . . .reaction .. separation networks :,sing detailed iiio(:-~[s. (.7otn[3ui. C7}e'n',. Engng, 19, SS3. Tricoire, bl, and M, F, Malone (t992). Design a f multipraduct batch processes for polymer P.'-Ki ]u,.J. 9.i .....~~ : ~.... i:,',2:.'oceedi
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
691
B a t c h D i s t i l l a t i o n of A z e o t r o p i c M i x t u r e s in a C o l u m n w i t h a M i d d l e Vessel M. Warter and J. Stichlmair Lehrstuhl ffir Fluidverfahrenstechnik, TU Mfinchen, Boltzmannstr. 15, D-85748 Garching, Germany, email: [email protected] Separation of azeotropic mixtures is an important task in the fine chemicals industry. We show for a middle vessel column the general behaviour in systems with azeotropes and give an overview over processes separating binary azeotropic mixtures using an entrainer. All aspects are explained by results of computer-based simulations with concrete systems. 1. INTRODUCTION Batch distillation is a very efficient unit operation for the separation of multicomponent mixtures into pure components. However, if the mixture exhibits azeotropes distillation becomes very difficult since liquid and vapor have the same concentration at the azeotrope and, therefore, no driving force for the separation exists at this point. There are several possibilities for separating azeotropic mixtures by batch distillation using a regular or inverted batch column (see Fig. 1) [1 ]. Only few publications deal with the usage of a middle vessel column, although it may offer many advantages like lower temperatures in the feed vessel during the process. (3)
(2)
(1) r
C
....
,
\
I
~=--
(M
r . . . . .
Fig. 1: Regular (1) and inverted (2) batch column, middle vessel columns (3) of modifications A, B and C. The middle vessel column is a combination of a regular and an inverted batch column [2]. Therefore, it is possible to obtain the light and heavy fractions simultaneously from the top and bottom of the column, while an intermediate boiling fraction may be recovered at the end of a process from the middle vessel. There exist several modifications of this column type, different with respect to the streams connecting the middle vessel and the two column sections [3]. The middle vessel column of modification A, in which both streams are fed into the middle vessel, was generally used in the simulations unless something different is indicated.
692 2. GENERAL BEHAVIOUR IN AZEOTROPIC MIXTURES 2.1 Systems with a Distillation Boundary
The behaviour of the middle vessel column in azeotropic systems with a distillation boundary is explained in Fig. 2 left. Three different locations of the feed have been studied (F1, F2 and [:3). Simulated was a column with a high number of stages, high reflux and reboil ratios and different vapor ratios q. The curves in each figure mark the concentration paths of the liquid in the middle vessel during each process. For comparison sakes the concentration paths of the regular and inverted batch column are shown, too. , , 9..... i i '
" ,,
.
...... ~ , ~ maximum --1 nazeotrope nverted i L! ~ I ~ 64.0 ~ column "
ethanol 78.3 ~
acetone 56.1 "C
middle vessel
column q=l " , ~ ,
/~ minimumazeotrope / / - ~ , ~ 78.0~ ,f~::~)~
~:/~
inverted ~
/
/
~
." X~'batrcehguto~rum n
.
~
....... x,
~q-1 azeotrope
80.1 ~ benzene
61.2 ~ chloroform
/
197.5 ~
~
ethylene glycol
-
-
'~< midde{ vessel
100 ~
water
Fig. 2." Systems with a distillation boundary (left) and with two intermediate boilers (right) concentrations paths of the liquid in the feed vessel jbr d!fferent feeds F and vapor ratios q. In a process with feed F1 the products are at the top of the column the low boiler of the upper distillation field acetone and the high boiler benzene at the bottom. For nearly equal vapor flows in the upper and lower column section (q=l) the concentration path moves to the intermediate boiler, the maximum azeotrope. With increasing vapor ratios (q t) the column adopts the character of a regular batch column and for high vapor ratios the high boiler benzene is at the end of the process in the feed vessel, similar to a regular batch column. For low" vapor ratios the column behaves like an inverted batch column (see q;). Similar results can be obtained for a feed F2 in the lower distillation field, with chloroform as low boiler. In Fig. 2 left the feed F3 lies in the concave zone of the distillation boundary. With an inverted batch column a crossing of the distillation boundary is possible because the product benzene drawn off with this type of column is the same on both sides of the boundary. In a middle vessel column with low vapor ratios the column adpots the character of an inverted batch column and, therefore, it is also possible to cross the distillation boundary (see qS). 2.2 Systems with two intermediate boilers
As shown before the concentration path of the liquid in the feed vessel of a middle vessel column with nearly equal flowrates in the upper and lower column section ( q - l ) moves to the intermediate boiler of the distillation field. In systems consisting of two opposite intermediate boilers (see Fig. 2 right) this results in two different directions in the area above and below the connection line between the high boiler and the low boiler of the system (see dashed line in Fig. 2 right and FI and F2) [4]. The reason for this is that the two products drawn off at the top and the bottom of the column, the minimum azeotrope and ethylene glycol, determine by vector addition the direction of the concentration path of the liquid in the feed vessel [5].
693
3. P R O C E S S E S F O R T H E S E P A R A T I O N OF A Z E O T R O P I C M I X T U R E S 3.1 Process in one distillation field
In the process in one distillation field a homogeneous entrainer is added to the binary azeotropic feed that enables the separation into pure components. The entrainer has to be selected in that way that both products, a and b, lie in the same distillation field. This process was shown by Bernot et al. for an operation with a regular or inverted batch column [6]. acetone
acetone (a)
56.1~
azeotrop /v ~1~
"'~
kk r
~
,
,
x
b-
":
/
~ y
\
~
internalcolumnprofiles
"
-e
b-
"
-e
x
b
binary minimum azeotrope
Min
a
e
98.4~ heptane (b)
Mm
80.1~
benzene (e)
binary maximum azeotrope
Fig. 3. Process in one distillation field. separation of aceton/heptane using benzene as entrainer (left) - criteria for the entrainer (right). Fig. 3 presents the process in a middle vessel column for the separation of aceton a and heptan b which form a minimum azeotrope. Benzene is chosen as entrainer e being the intermediate boiler. No distillation boundary between a and b exists since there is just one start and one end point of distillation lines. In Fig. 3 left the concentration path of the liquid in the feed vessel during the process and two internal concentration profiles of the column are shown. At the beginning of the first process cycle the feed F is fed together with an entrainer-rich fraction Mo, into the middle vessel (Ms). During operation the low boiler acetone and the high boiler heptane are simultaneously drawn off as top and bottom product of the column. Therefore, the residuum in the middle vessel depletes in both components and the entrainer e accumulates. The entrainer-rich fraction Mo, remains at the end of the process in the middle vessel. In the next process cycle it is just necessary to fill the new feed into the middle vessel. Because of this the entrainer need not to be handled externally in a process with a middle vessel column, as in the conventional process with a regular or inverted batch column [7]. Fig. 3 right summarizes the entrainers which allow a separation by a process in one distillation field. The process itself is very simple and elegant. However, application to practical problems is seldom possible since the entrainer e has to be in most cases the intermediate boiling constituent. Azeotropes are most likely formed by close boiling mixtures. Therefore, it might be difficult or even impossible to find an entrainer that boils between. One exception 1-nay be mixtures with a very large boiling point difference.
694 3.2 Process in two distillation fields
In this process class an entrainer is added to the feed a-b which constitutes a curved distillation boundary. The boundary begins or ends at the a-b azeotrope and, therefore, the constituents a and b lie in two different distillation fields. The entrainer selection for this process class is more flexible than that for the process in one distillation field see (Fig. 4 right). Duessel and Stichlmair presented the process with a regular or an inverted batch column [ 1].
~
(a) 56"1 ~ D1 p
/
... ~
~
~:~ "
'"it":ii. " k
acetone
a
O
acetone
1
e ~j~l,~" Mlc~f I
~
int.... I c~176
: Mlot/..:fi'ii::: ofthe liquid - - - - - - ~ ~ ' . ~ ~ i
(~ "--"
chloroform ~L. 2 ~
benzene
(e)
~
= 1.013bar M2u/
i~. . . .
e
~
~
T
xb---I> bb I T b binarymaxi. . . . . . .
trop
~
e b T>Tb
T>T~ o
'1"
,:.:;...... 2
benzene
80.1 ~
~
b
eb xb-~l> T
binary minimumazeotrop
61.2 ~ chloroform(b)
Fig. 4. Process in two distillation fields. separation of aceton/chloroform using benzene as entrainer (left) - criteria for the entrainer (right). The process with a middle vessel column is presented in Fig. 4 left for the separation of acetone and chloroform using benzene as entrainer. A curved distillation boundary starting at the high boiler benzene and ending at the maximum azeotrope devides the composition triangle in two distillation fields. At the beginning of the process the starting mixture F, the residuum M2o)and the entrainer-rich fraction B2 are fed into the middle vessel. This mixture 1141 lies in the upper distillation field in which the compound a has the lowest boiling temperature and is, therefore, the top product of the column. In this first process phase the middle vessel column is operated as a regular batch column with no bottom withdrawal. The concentration path of the liquid moves towards the concave zone of the boundary. In the second process phase the column is operated as a middle vessel column. The entrainer e has the highest boiling point and is the bottom product of the column. The vapor ratio q of the column is chosen in such a way that the vapor flow in the lower column section is higher than that in the upper one and, in turn, the column behaves similar to an inverted batch column (q
695 3.3 Extractive distillation
One example for a hybrid process is the combination of distillation and absorption called extractive distillation. In this case a high boiling entrainer is added at the top of the column which acts as absorbens. Many publications investigated the operation with a regular batch column [eg. 8,9] but only few with a middle vessel column [3, 10-12]. ethanol
ethanol ethanol (a)
'~ ~.
78.3 ~
t" " /)I ~ minimumI:i .... ~,ng y ~ azeotrop t::i ....... Z^/ p ' ]/gx"X / V/IE~k 78.0 ~ [:i internal column F :ii: profile ~ M -'~ . [ " ' - - ~
p= 1.013bar
197.5 ~
AV',\AA
j/e/xtra~t;ve~).~ ~
ethylene glycol (e)
r
~/~" L~}~'ct
\
M ~--=
~11
100 "C co
water (b)
rectifying ~ / ~ minimum....... Lm ~section t i o n / / / / "~z~ .... internal column azeotrop
p= 1.013bar /
/ ........X \
197.5 ~ ethylene glycol (e)
-',
ex2j:t~2n ~ k \
F
F .Mot
~
N~ "~
~\
~ /,---
~('''
water
100 oC water (b)
Moo
Fig. 5. Batchwise extractive distillation. separation of ethanol~water using ethylene glycol as entrainer - process with middle vessel column A (left) and B (righO. Fig. 5 shows the separation of the mixture ethanol/water which forms a minimum azeotrope. The entrainer is the hygroscopic high boiler ethylene gylcol. At the beginning of the process the starting mixture F is fed into the middle vessel. Normally, the minimum azeotrope between ethanol and water would be the top product of the column because it has the lowest boiling temperature. Here, the high boiling entrainer is fed near the top of the column and absorbs the water by countercurrent contacting. Hence, pure (i.e. water-free) ethanol is the top product of the column. In an operation with a regular batch column the high boiling entrainer would accumulate in the feed vessel. In an operation with a middle vessel column there exists an additional stripping section which makes it possible to recover the high boiling entrainer immediately at the bottom of the column. The second product, water, remains at the end of the process in the feed vessel (see Mo)Fig. 5 left). The middle vessel column has several advantages over a regular batch column. During the operation the temperature in the feed vessel is lower than in a regular batch column because the high boiling entrainer is steadily removed from the feed vessel during the process. Additionally, the amount of entrainer necessary for the separation is smaller since the entrainer can be recovered immediately from the bottom and reused at the top of the column. Furthermore, in some cases the energy and, in turn, the time demand of the process is reduced with this column type. Best results were obtained with a middle vessel column of modification B in which the liquid from the upper column section is directly fed to the lower column section (see Fig. 5 right). In this modification of the column nearly none high boiling entrainer accumulates in the middle vessel [3, 12].
696 4. S U M M A R Y
In this paper the general behaviour of a middle vessel column in azeotropic mixtures and three different classes of processes for the separation of binary azeotropic mixtures using a middle vessel column and an entrainer are explained: processes in one and two distillation fields, and hybrid processes, like the batchwise extractive distillation. In many cases the usage of a middle vessel column has advantages over the regular or inverted column: it leads to an easier handling of the liquid fractions involved in the process, it reduces the temperature in the feed vessel which is important for substances that tend to degrade at high temperatures or it is possible to reduce the energy demand of the process. 5. NOTATION a,b Feed components B bottom fraction D Distillate e Entrainer F Feed M intermediate fraction
q T V-I, V-2 1,2 --y a; co
vapor ratio Temperature Vessel process step decreasing boiling temperature start/end of a process step
REFERENCES
1. Dfissel, R. and Stichlmair, J., (1995), Separation of Azeotropic Mixtures by Batch Distillation Using an Entrainer, Computers chem. Engng., 19, Suppl., S 113-S 118 2. Robinson, C. S. and Gilliland, E. R., (1950), Elements of Fractional Distillation, 4 th ed. McGraw Hill, New York 3. Warter, M. and Stichlmair, J., (1999), Batchwise Extractive Distillation in a Column with a Middle Vessel, Computers chem. Engng., 23, Suppl., $915-$918 4. Hilmen, E. K., Kiva, V. N. and Skogestad, S., (1999), Analysis of Closed Multivessel Batch Distillation of Ternary Azeotropic Mixtures using Elementary VLE Cells, Computers chem. Engng., 23, Suppl., $347-$350 5. Safrit, B. T., Westerberg, A. W., Diwekar, U. and Wahnschafft, O. M., (1995), Extending Continuous Conventional and Extractive Distillation Feasibility Insights to Batch Distillation, Ind. Eng. Chem. Res., 34, 3257-3264 6. Bernot, Ch., Doherty, M. F. and Malone, M. F., (1991), Feasibility and Separation Sequencing in Multicomponent Batch Distillation, Chemical Engineering Science, 46, pp. 1311-1326 7. Stichlmair, J. and Fair, J. R., (1998), Distillation - Principles and Practices, J. Wiley, New York 8. Lelkes, Z., Lang, P., Benadda, B. and Moszkowicz, P., (1998), Feasibility of Extractive Distillation in a Batch Rectifier, AIChE Journal, 44,810-822 9. Warter, M., Dfissel, R. and Stichlmair, J., (1997), Separation of Azeotropic Mixtures by Batchwise Extractive Distillation, Inst. of chem. Eng., Symp. Ser. No. 142 Vol. 2, 705-714 10. Safrit, B. T. and Westerberg, A. W., (1997), Improved Operational Policiesjbr Batch Extractive Distillation Columns, Ind. Eng. Chem. Res., 36, 436-443 11. Hilmen, E. K., Skogestad, S., Doherty, M. F. and Malone, M. F., (1997), Integrated Design, Operation and Control of Batch Extractive Distillation with a Middle Vessel, presented at the AIChE Annual Meeting 1997, Los Angeles, paper No. 201h 12. Warter, M. and Stichlmair, J., (1999), Batchwise Extractive Distillation in a Novel Modification of a Middle Vessel Column, ECCE 2, 5.-7.10.99 in Montpellier, France
European Symposiumon ComputerAided Process Engineering- l0 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
Development and design of a forced unsteady-state numerical simulation
697
reactor through
Marco Cittadini, Marco Vanni, Antonello A. Barresi and Giancarlo Baldi*
Dip. di Scienza dei Materiali e Ingegneria Chimica. Politecnico di Torino Corso Duca degli Abmzzi 24 - 10129 TORINO (Italy) e-mail: cittad(&athena.polito, it, vanni(&athena.polito, it, barresi(&athena.polito.it, gbaldi(&athena.polito.it A detailed mathematical model has been used for an extensive investigation on the behaviour of the reverse-flow reactor for the combustion of lean mixtures of pollutants. The final aim was the design of a bench-scale reactor with performances very similar to those of a large-scale reactor. For this reason, the model had to include the heat losses at the wall, the energy balance for the tube and the possibility to simulate ancillary and measuring devices. In these conditions, no simplifications were allowed and the employment of the numerical simulation and of the most powerful computing machines has proved to be essential for the solution of the model. 1, I N T R O D U C T I O N The reverse-flow catalytic combustor is a fixed-bed reactor, in which the direction of the flow is periodically changed, in order to keep the heat of reaction inside the bed, and, as a consequence, to allow autothermal operation even for cold and very lean mixtures. After the transient period following the start-up, a pseudo-steady-state is attained, in which the hot front shifts periodically in the central part of the reactor, while the external parts of the bed remain cold, allowing a very low heat loss in the outlet flow. For this reason, in some cases the external parts of the bed are composed by inert material. Extensive investigations about the reverse-flow reactors have been performed in the past twenty years (a complete review can be found in Matros, 1996), including both numerical simulations and experimental works. There is, nevertheless, a wide difference between the performance predicted by the numerical simulations and the results supplied by a bench-scale reactor. The reason is simple: numerical simulations are usually aimed to investigate the behaviour of the large-scale industrial reactor, in which heat losses at the wall and thermal conduction in the tube are negligible and thus not considered in the model; in the bench-scale reactors, on the contrary, these wall effects are relevant and worsen the performance considerably. On one side, the inclusion of the wall effects in the model makes its solution much more demanding; on the other side, it is very difficult to build bench-scale reactors in which these effects are negligible. As concerns the former aspect, modern computers allow fast convergence to the pseudo-steady-state also for complex models. The solution of the latter problem is much more complex. Attempts have been carried out in order to reduce the wall
This work has been financially supported by the European Community (Contract ENV4-CT97-0599).
698 effects: van de Beld et al. (1994) endowed their reactor with a jacket in which vacuum was attained, Zafle and Turek (1997) and Nieken et al. (1994) compensated the heat losses by uniform electrical heating on the surface of the reactor; moreover, to further reduce these effects, the size of the bench-scale reactors is usually kept quite large, with a diameter ranging from 50 mm to 160 mm, leading to very expensive operation. In spite of all these efforts, really satisfying results have not been obtained: the measured profiles are very different from the theoretical ones, calculated by numerical simulation for an adiabatic reactor, and the performances are much poorer. The aim of this work is to use a complete mathematical model for the design of a bench scale reactor that is really representative of a large-scale quasi-adiabatic plant. 2. THE M O D E L
The adopted model (previously presented in Cittadini et al., 1999) is monodimensional and is described by the following equations. Energy balance for the gas phase: &co
8s
Po,0 &co
Po
c~z
k~#s
c')2-co
hal
4Lh~v,i
c;,oLuoPo c?z2 ~ c p,(; Uop (~
D R c p,(~ Uop (,;
Mass balance for the gas phase:
kcaL ' (go-L) uo
(2)
haLe c>cx = Zx~ c?2"cx+ k(flL(-AH)Pce'Yc~" (Yc~- Y ~ ) (1-~)u0ps (z s - -cc;) &s' c;,xLuoP x C~Z2 cp, s(1 - S)Uops,M7 o " c;, s
(3)
8Yc '= 8s
p < 0 8 F c De~S 8 2Ec:; '+
Po
8z
Lu o 8z 2
Energy balance for the solid phase:
Energy balance for the tube:
c'?rt.~
- - = OS
Zp Cp,pLVadPP
~2"cp+ &2
4L Cp,pVadPp(d2e_dR,i)
2
[dRihwi(z o .
.
-.
z~) .
--
d~eht,vc(T,r, -- -co)
(4)
Mass balance for the solid phase:
koa(go - g ~ ) = r(Yx,'cx)
(5)
For the catalytic part of the reactor, a first order rate equation is considered:
(6)
r-qokoo e x p ( ~ T o , c s )yc:;,oF~
while for the inert part: r = 0 In the equations, s = tuo/L~ and z = x/L are the dimensionless time and axial coordinate, respectively, and ~ = TIT o and go = Yo/Yo,o
are dimensionless temperature and molar
fraction. Conventional Danckwert's boundary conditions are assumed.
699 Mass accumulation on the solid surface and pressure loss inside the reactor are neglected. The physical properties and kinetic parameters adopted for the simulation refer to lean airmethane mixtures, while the design parameters are always specified in the figure captions. The solution of the system has been carried out taking into account the temperature dependence of all the thermodynamic and transport properties. In comparison with the models that are usually employed for the numerical investigation of the reverse-flow reactor, the main difference is the addition of equation (4), that allows to take into consideration the wall effects. The same equation has been already used by Nieken e t aL (1994). The model includes also the heat-exchange term with the outside, in equation (1), and the effectiveness factor for the catalyst (qo), which depends on "rs, in equation (6). Therefore, the model can be considered accurate also for a bench-scale reactor. The PDE system is reduced to a set of ODEs, by discretizing the spatial derivatives on a 100-point grid, which is usually sufficient to ensure a grid-independent solution. In a few cases only it has been necessary to increase the number of grid points up to 160, in order to avoid the onset of spikes, due to spurious terms in the numerical solution. The ODE system is solved through the fortran routine "LSODE" included in "ODEPACK" (Hindmarsh, 1983). After a transient period, the solution of the system evolves towards a pseudo-steady-state: the behaviour of the reactor (temperature and concentration profiles) is the same within every cycle (every two switches). This evolution requires a large number of cycles, therefore the solution of such a complete model needs a large amount of computing resources: a Sun Enterprise 450 (250 MHz) has been adopted for all the simulations. 3. RESULTS AND DISCUSSION 3.1. Transient effects during start-up A first set of simulations concerned industrial-scale reactors, in which wall effects are reduced A particular effect, not reported by previous investigations, deals with the conversion during the period of start-up in a non-adiabatic reactor, with a wall 1 heat transfer coefficient ~ ............ corresponding to standard industrial insulation. It has been 0.99 found that even when conversion .g tyv477~C L27~ Pre-heating temperature ~, is very high (> 97 %) at the 0.98 pseudo-steady-state, during the transient period that follows the 0 beginning of operation, it may 0.97 / drop to low values. Depending on the conditions, this effect, that 0.96 [ . . . . . . . . . . . . . . . . . . . does not appear in an adiabatic reactor, can last for some hours.
It can be explained considering the transition between the flat
0
5000
10000
15000
20000
t, s
Fig. 1. Conversion during the start-up of the process, as a temperature profile before the function of the pre-heating temperature. Overall heat-transfer start of reverse-flow operation, coefficient: 0.18 W/m2K; cycle period: 1050 s: inlet and the typical temperature concentration: 1200 ppm (methane).
700 pseudo-steady-state profile, with a hot part in the center of the reactor and two lateral cold part. Before the final profile is attained, the maximum temperature of the reactor passes through a minimum, due to the heat losses at the wall, not efficiently compensated by the heat of combustion. This drop in the maximum temperature produces the temporary decrease of conversion shown in Fig. 1. The extent of this effect is strongly influenced by the pre-heating temperature as well as by the cycle period and the fraction of inert material on both sides of the bed. By increasing the cycle period, the hot zone can be concentrated in the central part of the reactor since the first cycles, making faster the transition. The same effect can be obtained by increasing the fraction of inert material at the sides of the bed: the reaction zone is forced to stay in the central zone of the reactor. 3.2. Design of the bench-scale reactor The main concern in the design of the bench-scale reactor has been finding the proper devices to make its behaviour similar to that of a large-scale industrial reactor. The effect of changing the reactor diameter have been investigated deeply, finding out two main parameters that influence the reactor performance as a consequence of the change in the reactor size. They are the heat conduction in the tube and the heat losses at the wall. Both these variables depend on the ratio between the reactor surface and the reactor volume, and therefore on the reactor diameter, and both are very important for the energy balance of the reactor. Their global effect on the temperature profile is shown in Fig. 2: decreasing the size of the bed, the fundamental capacity of the reverse-flow reactor to keep the heat of combustion inside the bed becomes much poorer. In order to reduce the heat conduction in the tube, two 700 ways can be followed: to 100" 600 decrease the tube thickness or to adopt a less conductive material. .... 10" 500 The former solution is easier, o 400 ~ but it is not very efficient and, in addition, the tube thickness can 300 not be reduced below a certain value. The latter method, on the 20O contrary, proved to be very 100 efficient adopting a quartz tube. Its thermal conduction is very 0 low (less than 1/20 of stainless 0 0.2 0.4 0.6 0.8 1 steel), its resistance to high axial coordinate, adimensional temperatures is excellent and the mechanical properties are good Fig. 2. Temperature profile as a function of the reactor diameter enough for a laboratory plant. (in). Bed length: 500 mm; reactor diameter: 50 mm; cycle The goal to reduce, or even period: 60 rain; inlet concentration: 5000 ppm (methane). to eliminate, the heat losses at the wall is much harder to reach. The effect of different kinds of insulation has been simulated, but no satisfying solutions have been found. The uniform thermal compensation proposed by some authors (see the introduction) has been simulated, too, evidencing that its main effect is .......................
701 to by-pass the heat from the central hot zone to the lateral cold zones, leading to inadequate temperature distribution. Finally, a new device has been tested by numerical simulation: a set of seven independent band heaters, controlled by a PID system that keeps the temperature 500 of each heater continuously equal to that of the bed in the axial 400 coordinate corresponding to the central point of the heater itself. The elements are heated o 300 electrically and cooled by an air r flow. Seven thermocouples are 200 placed in the proper positions on the axis line of the bed and seven 100 on the surface of the bend heaters: the heating and cooling of the devices is regulated by the 0 0 0.2 0.4 0.6 0.8 difference between the two axial coordinate, adimensional temperatures. An extensive investigation has been carried out Fig. 3. Temperature profiles for the adiabatic configuration on the minimum number of and the band-heater configuration. Bed length: 500 nun; elements that allows to reach reactor diameter: 50 mm; cycle period: 60 mira inlet satisfying results and on the best concentration: 1600, 2400 and 3200 ppm (methane). configuration for these devices. The system is of course complex, but according to the numerical simulations, the results are really very interesting. In Fig. 3, a comparison between the temperature profiles given by an ideal adiabatic reactor and that given by the reactor equipped with the band heaters is shown for three different inlet concentrations of pollutant. The agreement of data is really very good in comparison with other solutions. The designed reverse-flow reactor is presently being assembled in the University of Oviedo, under the framework of a joint European Research Program
3.3. Optimisation of the catalytic pellets Working on the design of the reactor, not only the characteristics of the reactor itself must be considered, but also the characteristics of the pellets must be optimised. The most important parameter for the performance of the reactor is the thermal capacity of the bed. Our previous investigations showed a strong influence of the thermal capacity of the bed on the performance of the reactor: increasing this parameter, the minimum inlet VOC concentration for autothermal operation decreases considerably. Thus, the possibility to improve the performance of the reactor by increasing the thermal capacity of the fixed bed must be tested. Assuming that the properties of the catalyst are fixed and can not be changed, the only way to increase the thermal capacity of the bed is to mix the catalyst with an inert material with a higher capacity. Naturally, this means a reduction of the global catalytic activity of the bed: an optimisation must be found. We have considered three possible configurations: I) Pellet made by a mixture of catalyst and inert material II) A mixture of pellets made by catalyst and pellets made by inert material III) Pellet made by an inert support washcoated by the catalyst
702 The configurations I and II are 5O equivalent for this investigation, * r=l assuming that the thermal ~,. ~ r = 1.33 45 conductivity in the bed is high -* r= 1.66 enough to prevent, for the case II, 0040 _ ~ 9 r=2 relevant differences of temperature ~ ~ -" _r_-_3.33_ between the catalytic and the inert 35 pellets. The configuration Ill, on the contrary, is different, for the 3O influence of the effectiveness factor (degree of exploitation of the catalyst), that, in this case, is higher 25 than in cases I and II. As it is 0 0.2 0.4 0.6 0.8 1 known, the effectiveness factor catalyst mass fraction depends strongly on the diameter of the pellets. Fig. 4. Minimum ATad for autothennal operation (methane) The results of the simulations as a function of the catalyst mass fraction in the pellets and regarding configurations I and II of the ratio r = (pCp)inert/(pCp)cat.. Bed length: 5001ran: reactor (uniform mixing) are summed up in diameter: 50 iron: cycle period: 60 min. Fig. 4, where the minimum adiabatic temperature rise required for autothermal operation (methane) is given as a function of the fraction of catalytic material in the bed, for different ratios between the thermal capacity of the inert and of the catalyst. As shown in the figure, the replacement of part of the catalyst by inert material with the same thermal capacity (upper line) worsen strongly the pertbrmance of the reactor, due to the reduction of activity. But, if the thermal capacity is higher enough, the drop of activity is compensated. If the percentage of catalyst in the bed is 50 %, the performance of the reactor is better than in the case of 100% catalyst, on condition that the thermal capacity of the inert is ahnost twice that of the catalyst. If the percentage is l0 % the ratio must be slightly higher. The situation is still better if the catalyst is washcoated on the surface of the inert pellets, as in this case the effectiveness factor is higher. Another consideration is very important: a bed with an higher thermal capacity is of course able to dump better the periodical variations in the feed and to tolerate longer periods of stand-by. REFERENCES
Cittadini M., Vamli M., Barresi A. A. and Baldi G. (1999). Efficient design and scale-up of reverseflow catalytic combustors. AIDIC Con/; Series 4, in press. Hindmarsh A. C. (1983). ODEPACK, a systematized collection of ODE solvers. Scientifing ('omputing (Stepleman R. S. et al., eds.), North Holland, Amsterdam, 55-64.7 Matros Y. S. (1996). Forced unsteady-state processes in heterogeneous catalytic reactors. ('an. J. ('hem. Engng. 74, 566-579. Nieken U., Kolios G. and Eigenberg G. A. (1994). Fixed-bed reactors with periodic flow reversal: experimental results for catal~ic combustion. Catal. Today 20, 335-350. van de BeN B., Bonnan R. A., Derkx O. R., Von Woezik B. A. A. and Weterterp K. R. (1994). Removal of organic compounds from polluted air in a reverse flow reactor: an experimental study. Ind. Eng. (-'hem. Res., 33, 2946-2956. Zufle H. and Turek T. (1997). Catalytic combustion in a reactor with periodic flow reversal. Part 1. Experimental results. Chem. Eng. Proc., 36, 327-340.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
703
Intent and Rationale in the Design of Chemical Processes: A Case Study Alejandro Guzrrfin-Reyna a and Ren6 Bafiares-Alc~intarab'* aSchool of Chemical Engineering, University of Edinburgh, EH9 3JL Scotland, United Kingdom. Email: [email protected] ~Department of Chemical Engineering, ETSEQ, Universitat Rovira i Virgili, 43008 Tarragona, Spain. Email: [email protected] This work presents a framework that integrates the representations of design rationale, intent and artifact, and shows how it can help improve the documentation of the design activity. In order to demonstrate the advantages of the framework, the use of a computer aided design support system prototype is proposed and then used to document a design case study for the production of ethyl acetate via esterification. 1. INTRODUCTION Design of chemical processes is a complex activity involving the collaboration of multiple designers to achieve a set of predefined goals. The design process is oriented to define or modify a physical entity or artifact (such as a chemical plant or a plant section). During this process a large amount of information is produced in the form of manuals, blueprints, diagrams, CAD files, etc. This information is the result of extensive calculation and decision making processes that contribute to define the physical structure and operating conditions of the chemical process. In the design process great effort is made to produce a clear documentation of the structure and operating conditions of the plant. However, almost all the information related to the reasoning of the decisions made (i.e., design rationale) and to the evolving goal structure (i.e., design intent) is implicit or, at best, partially documented. As a result, most of the design intent and rationale is lost, leading to imprecision, inconsistent assumptions, reconsideration of issues already discussed within the project, deviation of the original intent of the design, and an inefficient use of scarce and valuable resources during the design process. Several works related to the adequate representation of design rationale and design intent have been presented in recent years with applications to software design and mechanical design among others (Nagy et al., 1992; Pefia-Mora et al., 1995, 1997). A number of problems have been identified that make difficult or limit the use and applicability of such representations: 9 Most of the designers admit the value of recording design rationale, however, they express concern about the time it takes them to record it. 9 Most research in design rationale has focused on capturing design rationale without concern for its later use.
704 9 Many of the models proposed lack of an explicit representation of the artifact being designed, therefore, it is impossible to represent its evolution. 9 The representation and importance of design intent are frequently disregarded. In some cases intent and rationale are scarcely differentiated. We believe that a complete and adequate way to capture knowledge associated with the design activity is to use a more natural, straightforward design representation that accounts for the different knowledge structures and associations that integrate the design process. Such a representation must be simple and easy to understand in order to encourage its use among the designers, and consist of a number of predicates that clearly capture the following aspects: (1) what one wants to achieve or design intent; (2) the constraints or restrictions imposed on the design; (3) what is being done to ensure the fulfilment of the initial goals of the design; (4) the design plan or strategy and its ordered set of actions that enable the achievement of the goals of the project; (5) the design issues and unplanned events in need of solution; (6) the alternatives or solution proposals to the unresolved issues and the body of arguments associated with each alternative; (7) the decisions made and the criteria taken into account to adopt such resolutions. 2. THE REPRESENTATION
A framework that provides a unified representation of design rationale, intent and the artifact for the conceptual design of chemical processes is presented in this work. The framework is shown in Figure 1 and uses the OMT's graphical notation (Rumbaugh et al., 1991). Its aim is to improve the documentation of the design process by making explicit the elements of knowledge that conform the design activity.
I Data,ba~H
Chemic'I ~ l ~ro~e",I
I Intent J
J11 Agent I
1
J Resource
r
J
Fig. 1. Integrated Intent-Rationale-Artifact framework
I,
J Object,veJ
I .......
I Con,,t,',intj
I'G!I al
705 The framework was developed taking into consideration the following aspects: 9 Importance of the intent space: Regardless of the nature of the design activity (e.g., in the areas of chemical, mechanical, software development, etc.) the decision to undertake the design process as such originates from a number of high order needs and motivations that lead to the formulation of the final ends or intent of the design. Our representation of design intent is integrated by goals (what is to be achieved), objectives (features to maximise, minimise or optimise), and constraints (restrictions or limitations imposed to the design). Since the goal structure of the design is liable to change through time we allow the introduction of new intent entities to the hierarchy as well as their evolution and reformulation. 9 Importance of the rationale (argumentation) space: The deliberation process that leads to the resolution of design issues about the artifact involves the assessment of multiple alternatives raised by the designers. Each designer provides arguments for or against one or more alternatives and, after careful analysis of the advantages and disadvantages of each proposal, one or more decisions are made. The Issue-Position-Argument structure involved is known as the IBIS model (Kunz and Rittel, 1970). This model was developed to record the process by which decisions are made and its advantages and limitations have been widely discussed by several authors (Conklin et al., 1991; Shum, 1991). As a starting point, we adopted an extended version of the IBIS model that acknowledges the existence and evolution of an artifact (Pots and Bruns, 1988) However, in the context of the conceptual design of chemical processes the evaluation process is often represented by a cross-examination of the design alternatives against a number of criteria (e.g., economic, environmental, safety, etc.). The resulting structure is equivalent to the Question-Option-Criteria (QOC) model proposed by Maclean et al (1991). The notion of discriminating criteria is acknowledged within the framework, thus a similar representation to the QOC model is also attained. It is necessary to point out that IBIS and QOC are not mutually exclusive models but complementary to each other. IBIS is more adequate to capture deliberations as they happen while QOC structures result as an act of further reflection and analysis where it is possible to compare how the alternatives trade off against each other. 9 Importance of design plans in the achievement of goals: In this work, a design plan specifies a precompiled sequence of actions formulated to achieve design goals and objectives. The implementation of the plan requires the allocation of resources (e.g., money) and the delegation of tasks to the participant agents (e.g., project engineers, CAD operators, etc.). Similar to goals and objectives, complex tasks can be decomposed into sub-tasks which, in turn, may introduce new goals or, strictly speaking, new context-dependent sub-goals (as they are important only in the scope of the task and not in the global context of the design), and their corresponding sub-plans. 9 Representation and evolution of the artifact: The artifact is the physical realisation of the design. As the design activity carries on the complexity and level of detail of the artifact increases. The refinement of the artifact structure also implies a functional specialisation of the different components of the artifact. In order to document its evolution it is necessary to maintain a record of the changes -both structural and functional- occurring along the design history. Theoretically, it is possible to maintain a continuous record of the evolution of the artifact, that is, an incremental record of each single alteration made, however, the number of possible events and their concurrence would make this task almost impossible, and no additional benefit would derive from such a procedure. Instead, a discrete evolution approach is used where the state of the design is captured only when the number and importance of
706 occurring events produce a relevant modification to the artifact. In our representation, the state of the design at a specific moment is named scheme and consists of a number of elements that represent the structure and connectivity of the artifact. The functional aspects of these elements are not explicitly represented in Fig. 1; however, they are expressed in the computer implementation of the framework. It is important to note that even though the artifact representation presented in this work focuses only on conceptual process design, the scope of design intent, rationale, and the plan structure is valid for most of the activities associated with the life cycle of process plants. 3. THE ETHYL ACETATE CASE STUDY In order to demonstrate the advantages of having a design representation that accounts for intent, rationale and the artifact, a computer implementation of the framework described in the previous section was used to document a design case study for the production of ethyl acetate via liquid esterification. The activities carried out include a rigorous study of the vapourliquid equilibrium of the chemical species involved (determination of azeotropes, liquid immiscibility regions, distillation residue curves and distillation boundaries), the sequential synthesis of the process on the basis of the phase equilibria constraints, and a computer simulation, optimisation, and economic analysis of the resulting process using the ASPEN PLUS T M process simulator. The results of these activities constitute the design memory of the conceptual design of a chemical plant to produce 60,000 tons per year of ethyl acetate on a continuous basis. The simplified flow diagram of the process is presented in Figure 2. Water
Ethanol
~._
Acetic Acid
,
Sulpfuric Acid DECANTER
ALCOHOL COLUMN MIXING
DRYING COLUMN
Ethyl Acetate EACTIVE
Waste Water
Fig. 2. Ethyl acetate process via esterification
707 3.1. Analysis of the ease study The accumulated data were used in a number of both real and invented examples aimed to demonstrate the importance of knowledge reuse in chemical plant design. In one of the examples we wanted to investigate how difficult it is for a design engineer, unrelated to the ethyl acetate case study, to familiarise with the design memory in order to use this existing knowledge in the development of a similar design project.
The designer is provided with a complete set of data of the plant: the equipment list, operating conditions, flowrate and composition of each stream, etc. In order to determine if some of this information is applicable to the new design project, the designer needs to have a deep understanding of the design philosophy of the plant. Specifically, the following issues are of prime importance: - Why is the process designed the way it is? - How were the reaction and separation sequences o f the process determined? - W77at assumptions were made during the conceptual design o f the plant? It is important to note that, though the designer is able to retrieve detailed information about the structure and operation of the plant, it is extremely difficult for him or her to infer any information about the facts that led to its current state unless this information had been originally recorded. Under normal circumstances the engineer would spend considerable time learning the insights of the process by referring to a number a technical reports of the plant and cross examining them with the process flow diagram, the process stream table, etc. This examination process may become extremely difficult to follow for the larger and more complex plants. Thus, it would be convenient to count on some kind of design support system (DSS) from which it would be possible to retrieve information on how the design process evolved and led to establish the actual structure of the plant (that is, its design history). By using the DSS, the designer would be able to replay the design activity as it originally happened: how assigned tasks were carried out, what assumptions were made, and what was the outcome of such activities. In order to ensure this functionality, the DSS should support a number of features and operations such as: 9 Search capabilities: fred one or more records related to a specific topic, date, etc. 9 Multiple format storage: the program must support the storage of complementary information or at least be able to point to external information sources such as process simulation files, costing information, email messages, etc. 9 Concurrent design support: the design support system would be of very limited use if only one designer could use it at a time 9 Evaluation o f the state o f the design: In order to avoid deviations from the original intention it is necessary to provide a means to verify how the current activities contribute to achieve the goals of the design. A design support system prototype was developed using the proposed framework representation. The prototype was used as an experimental vehicle to evaluate the adequacy of the framework presented. The system was named Intent and Rationale Object Builder as it uses a simple browser-like object editor to link new objects to the existing network The design support system was useful for capturing unexpected facts and events that are normally forgotten after the decision process. From the study of similar esterification systems it was
708 also possible to learn aspects related to the structure of the process. We carried out a number of studies on the phase equilibrium and esterification reaction of iso-propyl alcohol and npropyl alcohol with acetic acid. The similar phase behaviour of these systems made it possible to synthesise reaction and separation sequences for these systems using the ethyl acetate case study as starting point. 4. CONCLUSIONS The aim of this work was to demonstrate how an integrated representation of design intent, rationale and artifact can help improve the documentation of the design of chemical processes. Some of the benefits derived from a better documentation of the design activity are: 9 Verification of the current state of the design. 9 Better understanding of the issues discussed and the decisions made. 9 Reduction in design costs. 9 A basis for a more efficient computational support in design. The extent of the benefits to design depends largely on the richness and expressiveness of the representation as well as the quality of the computational support of the implementation. There must exist also a balance between the richness of the representation and its simplicity in order to provide a robust documentation of the design without interfering with the designer activities. REFERENCES
Conklin, E.J., and Burgess-Yakemovic, KC. (1991). A Process-Oriented Approach to Design Rationale. Human-Computer Interaction. 6, 357-391. Kunz, W., and Rittel, H. (1970). Issues as Elements of Information Systems. Institute of Urban and Regional Development Working Paper 131, University of California, Berkeley, Berkeley, California. Lee, J., and Lai, K.Y. (1991). What is Design Rationale?. Human-Computer Interaction. 6, 251-280. MacLean, A., Young, R.M., Belloti, V.M.E., and Moran, T.P. (1991). Questions, Options, and Criteria: Elements of Design Space Analysis. Human-Computer Interaction. 6, 201-250. Nagy, R.L., Ulman, D.G., and Dietterich, T.G. (1992). A Data Representation for Collaborative Mechanical Design. Research in Engineering Design, 3, 233-242. Pefia-Mora, F., Sriram, R.D., and Logcher, R. (1995). Design Rationale for ComputerSupported Conflict Mitigation. Journal of Computing in Civil Engineering, 9, (1), 57-72. Pefia-Mora, F., and Vadhavkar, S. (1997). Conflict Mitigation System for Collaborative Engineering. AI EDAM, 11, 93-108. Potts, C., and Bruns, G. (1988). Recording the Reasons for Design Decisions. Proc., 10thInt. Conf. On Software Engng. IEEE, New York, N.Y., 418-427. Rumbaugh, J., B laha, M., Premerlani, W., Eddy, F., and Lorensen, W. (1991). ObjectOriented Modeling and DesigrL Prentice Hall, Inc., Englewood Cliffs, N.J. Shum, S. (1991). A Cognitive Analysis of Design Rationale Representation, Ph.D. Thesis, Department of Psychology, University of York.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
709
Energy Efficient Distillation by Optimal Distribution of Heating and Cooling Requirements Torben Ravn Andersen l, Gino Siragusa JCrgensen 1.*
2, Bjarne Andresen 3, Peter Salamon 2 and Sten Bay
1 CAPEC, Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark 2 Department of Chemistry, San Diego State University, San Diego, CA. 92182 ,USA 30rsted Laboratory, University of Copenhagen, DK-2100 Copenhagen, Denmark
Abstract In this paper investigations regarding optimal placement and duty distribution of internal heat exchangers in distillation columns have been carried out based on simulations. In particular the effect of adding 2, 4 and 19 internal heat exchangers to a 19 tray distillation column separating 2-propanol and methanol was studied. It is concluded that the minimum entropy production placement of the heat exchangers depends strongly on the degree of separation. The optimum is however rather flat, which indicates that optimal placement for a nominal operating point may remain nearly optimal in terms of entropy production even for significant changes in feed and product compositions. When compared to a conventional column run under the same operating conditions, entropy production for an optimally operating column with four thermally active stages (4TA) is reduced by 25 % as compared to 37 % for a column with six thermally active stages (6TA) column and 49% for a column in which all the trays are equipped with heat exchangers.
1. Introduction In spite of its relatively low energy efficiency (typically a few percent), distillation is still one of the most widely used separation techniques in the chemical industry. This is illustrated by the fact that distillation accounts for more than 3 % of the total energy dissipation in the USA, Mix et al. (1978), Humphrey and Siebert (1992). Energy consumption attributed to distillation has been reported as high as 5-6% within the industrialized countries by Pilavachi (1996). An immense amount of work has been carried out in the field of heat integration within distillation trains. Most of this work had the purpose of minimizing the use of external utilities in heat exchanger networks. The major tool for this work has been pinch-analysis which can minimize the external duties when integrating conventional columns, Linnhoff et al. (1982). From a 2 nd law efficiency point of view, conventional distillation performs very poorly in general. One way of improving this efficiency is to distribute the heat added and removed over the distillation column via heat exchangers. Various methods for optimizing the 2nd law efficiency (minimizing the entropy production, minimizing the exergy loss) for distillation process design have previously been suggested: equipartition of entropy production, Tondeur and Kvaalen (1987); equipartition of forces (EOF), Ratkje et al. (1995); equal thermodynamic distance (ETD), Salamon et al. (1998). Assuming a distillation column with N stages and a heat exchanger on each stage, all methods predict that the entropy production in the column, ASI4x, goes to zero as N goes to infinity (see * Corresponding author. Tel.: + 45-4525-2872; fax: + 45-4588-2258
E-mail address: sbj @kt.dtu.dk (S.B. JCrgensen)
710 appendix). In the limit of "large" (but finite) N, the predictions however diverge. This divergence has been outlined in the coffee cup example of Salamon et al. (2000). From economical/engineering point of view, an issue that needs to be addressed is how well can one do with only a few more thermally active stages compared to conventional columns. Optimal tray locations for 4 thermally active stages (condenser, 1 heat exchanger in rectifying section, 1 heat exchanger in stripping section and reboiler = 4 thermally active stages = "4TA") have previously been investigated by Koijer et al. (1999), and Siragusa et al. (2000). 2. Economic vs. Thermodynamic O p t i m u m The net heat input to a distillation column, Qnet,input- Qadded-- Qremoved = ho + h8 - hF, for a specified separation is practically constant. This is true for binary distillation, and applies also to a large extent to multi component distillation with specified product concentrations. The amount of high grade energy added to a distillation column is however not constant, but depends on the sizing and design as well as the operation of the column. The "true thermodynamic optimum" minimizes the entropy production or, equivalently, the exergy cost. This is carried out by minimizing the quality of the heat input to the column and at the same time maximize the quality of the heat output. This makes it necessary to add heat exchangers on each plate. Possible design procedures for such processes can include the approaches of EOF or ETD. For industrial applications where the objective is operating at the economical optimum, one can imagine using a "Poor man's" ETD/EOF column. Since the introduction of internal heat exchange on trays both affect capital investment and possibly also separation efficiency the economic design optimum is likely to be an approximation to an ETD/EOF distillation column with only a few extra heat exchangers. In this paper we investigate a "6TA" column, i.e., a column which is equipped with an additional 4 internal heat exchangers. In the following section we describe the model used in the simulations. We compare our results to a conventional column, a 4TA column as well as a column where each plate is equipped with a heat exchanger. Finally the optimal results are discussed. 3.
E n t r o p y P r o d u c t i o n in D i s t i l l a t i o n
Figure 1 illustrates the thermodynamic transitions in distillation: 1) temperature equilibration of the feed from the feed temperature to an arbitrary reference temperature (e.g. the boiling point of the low boiling component) 2) unmixing of the ideal feed mixture to the two products D and B at the reference temperature and 3) temperature equilibration from the reference temperature to the product temperatures (e.g. saturated liquid). From a thermodynamic efficiency point of view, it is most interesting to look at the entropy production due to internal irreversibilities, ASirrev. If the subpaths traveled within figure 1 are
D xo
F T~ X~
a&
D L T ~ ,-
X~
-
ASr I AS.nm
Qreml
F T~ X~
F
T~er ~ T , X, X, Fig. 1. Thermodynamic transitions in distillation.
" Qcon~..... .... D - 1
..................................... Qrem2 ...................... ..................................... "~ Qadal ......................
Qadd2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
~
ireml Nrem2 Nfeea Naddl N add2
19 Qreboiler
9B Plate # Fig. 2. The 6TA distillation column.
711 reversible changes ASirrevc a n be expressed as:
(1)
A S i.... : A S HX -- A S rev -- A S HX -- ( A S ref 31- A S unmix "1- A S D -at-A S B )
The change of entropy in the universe for a distillation column can with the given assumption be formulated as: (2)
A S u -- A S Hx --!-z~kSunmix "1- A S D Jr" A S B
Where ASHx is the entropy change carried by heat exchange on the TA-plates, in the reboiler and in the condenser: NT
ASHx = ~ q.__L+ q___o_o+q___&
rj
r~
(3)
r.
ASunmix is the entropy change of reversible unmixing in the separation. ASo and AS8 are the entropy changes due to cooling and heating the distillate respectively the bottom product. The detailed explanations of performing entropy calculations have been outlined in Brown and Siragusa (2000). 4. Simulation Model The simulation model consists of a distillation column with 19 ideal stages, a partial reboiler (1 ideal stage) and a total condenser, separating methanol and 2-propanol. A schematic of the 6TA distillation column model can be seen in figure 2. The vapor-liquid equilibrium model used is Soave-Redlich-Kwong. Feed and product specifications are given in table 1. The model was implemented in Matlab | The procedure for minimizing the entropy production has been outlined in the appendix.
Table 1 Feed and product specifications. Specification Value Unit XF 50 mole% MeOH XD 95 mole% MeOH XB 5 mole% MeOH F 1 Mole
Table 2 Investigated parameter space. TA plate # Parameters Nreml 2:4 Nrem2 2:8
Naaal Naaa2
11:14 14:17
5. Results An entropy minimization of the 6TA column was carried out for different TA plate positions. The initially investigated integer-parameter space is given in table 2. In figure 3 a plot has been made over the performed entropy minimization. For each heat exchanger combination in the stripping section, the entropy change carried by heat exchange is plotted as a surface against the heat exchanger combinations in the rectifying section. It is seen that the optimal combination for minimum entropy production within the investigated parameter space is: [Nreml, Nrem2, Naddl, Nadd2] = [2, 6, 12, 17]. As can be seen from figure 3, the minimum entropy change does not take place at an interior point but rather it lies on the boundary of the set of allowed parameter values. Thus it was decided to expand the parameter range to also include the tray one position (Nreml=l). This was not originally included because it corresponds to a model column which differs significantly from the one shown in figure 1; with Nreml-'l, n o reflux is used from the condenser. In figure 4 it is shown that the entropy production is reduced significantly when NremJ=1 is used in combination with the best TA plate choices in the stripping section. Table 3 shows the heat input of high grade heat, the net heat input
712
Fig. 3. Minimum entropy production [J/(mol K)] for investigated thermally active tray positions. Main x-axis: lowest positioned heat exchanger (Nadd2). Main y-axis: highest positioned heat exchanger in stripping section (Naaal). Minor x-axes: highest positioned heat exchanger in rectifying section (Nreml). Minor y-axes: lowest positioned heat exchanger in rectifying section (Nrem2). Minor z-axes entropy change carried by heat exchange (ASHx). (constant), the entropy change carried by heat exchange and the entropy production due to internal irreversibilities. Table 4 shows the optimal TA plate positions with the corresponding duties as well as the duties for reboiler and the condenser. Table 3 Comparison of absolute heat input, entropy change carried by heat exchange and entropy production due to internal irreversibilities for conventional, 4TA, 6TA and ETD column cases Column type Conventional 4TA (optimal) 6TA (optimal) ETD ("optimal")
Absolute heat input [kJ/mole feed] 50.0 54.0 57.1 62.5
Net heat input [J/mole feed] 419 419 419 419
A&IX
ASirrev
[J/mole K] 5.64 4.95 4.63 4.31
[J/mole K] 2.71 2.02 1.70 1.38
*) Optimal is in quotes for the case with heat exchangers on each tray since ETD does not predict the true thermodynamic optimum well for small systems,but it does so for larger systems, Salamon et al. (1998)
713
Fig. 4. Entropy change [J/(mol K)] carried by heat exchange, ASI-IX, for [Naddl, Nadd2]=[11,15]. Table 4 Optimal positioning and duties for the thermally active plates. Column Type
4TA (optimal) 6TA (optimal)
Rectifying Section TA plate TA plate Condenser position(s) duty(ies) duty [kJ/mol feed] [kJ/molfeed] 4 -16.4 -37.2 1 6
-44.6 -11.8
-
Stripping Section TA plate TA plate Reboiler position(s) duty(ies) duty [kJ/mol feed] [kJ/molfeed] 13 19.0 35.0 11 15
13.9 12.6
30.3
6. Conclusions The overall integration of heat integrated (4TA, 6TA .... ) distillation columns will increase the resolution of the temperature interval for supplied/removed heats and lower the need for external utilities, provided that the new higher cooling temperatures and lower heating temperatures can be made use of elsewhere in a given plant. It is shown that in order to perform the separation at the minimum entropy production it is necessary to add 8% and 14% more heat in the stripping section for the 4TA and the 6TA column respectively when compared to the conventional case (table 2). It is also shown that entropy production can be significantly reduced by adding only 2 extra heat exchangers, namely 25% compared to conventional distillation. The reduction in entropy change carried by internal irreversibillities for the addition of 4 extra heat exchangers is 37%. The optimal tray locations for the thermally active trays are for the 6TA: 1 , 6 , 11 and 15. In comparison, the 4TA optimal placements are trays 4 and 17. The optimum is rather flat, which indicates that optimal placement of TA's for a nominal operating point may remain nearly optimal in terms of entropy production for even significant changes in feed and product compositions. 7. Discussion It would be more reasonable to investigate the true optimum for a "symmetric" column. The column would be more "symmetric" if the condenser was partial instead of total. In the present simulation model the reboiler constitutes an equilibrium stage whereas the condenser does not. A thermodynamically improved structure would be a deflegmator which only condenses the necessary reflux followed by a total condenser which condenses only the distillate product stream. With the present setup a significant amount of entropy is produced by entering the subcooled reflux which also have a different composition to the top tray. One important aspect yet to be analyzed is operability and control of thermally integrated columns.
714
Appendix A. Nomenclature Q - Heat irrev - irreversibility D - Distillate HX - heat exchange B - Bottom product rev - reversible F - Feed u n m i x - unmixing x - Molar ratio N - Thermally Active tray location # T - Temperature addi - the i'th addition ref - reference state remj - the j'th removal h - enthalpy A - change S - entropy Appendix B. Entropy minimization procedure The procedure below was used for minimizing the entropy production for each specified set of TA-plate positions. 1) Make an initial guess at the duties of the intermediate TA-plates (using simplex method), 2) Make an initial guess for the condenser duty (using secant method), 3) Carry out balancing of column by plate to plate calculation, finding the reboiler duty and optimal feed plate location, 4) If balancing is not converging goto (2) otherwise, 5) Evaluate entropy production, 6) If convergence of entropy production not obtained (minimum entropy production) goto (1), otherwise 7) Repeat the procedure for new TA-plate positions. References Brown, Danielle and Siragusa, Gino. (2000) Calculating total entropy production for a distillation process. Manuscript in prep. Humphrey, Jimmy L.; Siebert, A. Frank. (1992) Separation technologies. An opportunity for energy savings. Chem.Eng.Prog., 88 (3), pp. 32-41. Koijer, Gelein M.; Kjelstrup, Signe; Koi, Hedzer J. van der; Gross, Bernd; Knoche, Karl F. and Andersen, Torben R. (1999) Positioning Heat Exchangers in Binary Tray Distillation Using Isoforce Operation. Proceedings for ECOS'99, Tokyo. Linnhoff, B.; Townsend, D.W.; Boland, D.; Hewitt, G.F.; Thomas, B.E.A.; Guy, A.R.; Marsland, R.H. (1982) A User Guide on Process Integration for the Efficient Use of Energy. Rugby, England: The Institution of Chemical Engineers. Mix, T.J.; Dweck, J.S.; Weinberg, M.; Armstrong, R. (1978). Chem.Eng.Prog., 74(4), pp. 49. Pilavachi, P.A. (1996) Systems Modelling as a Design Tool for Energy Efficiency-Research within the European Union. Comp. Chem.Engng., Vol.20, Suppl., pp. 467-472. Ratkje, Signe Kjelstrup; Sauar, Erik; Hansen, Ellen; Lien, Kristian Magnus and Hafskjold, BjCrn. (1995) Analysis of entropy production rates for design of distillation columns. Ind.Eng. Chem.Res., 34, pp.3001-3007. Salamon, Peter and Nulton, James. (1998) The geometry of separation processes: A horsecarrot theorem for steady flow systems. Europhys. Lett., 42, pp. 571-576. Salamon, Peter; Nulton, James; Siragusa Gino; Andersen, Torben R. and Limon, Alfonso (2000) Principles of Control Thermodynamics. Submitted for publication in Energy. Siragusa, Gino; Andresen, Bjarne and Salamon Peter (2000) Optimal heat integration on a 19 plate distillation column, utilizing two additional heat exchangers. Manuscript in prep. Tondeur, Daniel and Kvaalen, Erik. (1987) Equipartition of entropy production: an optimal criterion for transfer and separation processes, lnd.Eng. Chem.Res., 36, pp.50-56.
EuropeanSymposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V.All rightsreserved.
715
Optimal Design of Heat-Integrated Multipurpose Batch Facilities Ana Paula F. D. Barbosa-P6voa *a, Tfinia Pinto b and Augusto Q. Novais b a
CEG-IST, SAEG, Instituto Sup. T6cnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
b DMS, INETI, Est. do Pa?o do Lumiar, 1649-038 Lisboa- Portugal This paper presents a mathematical formulation for the detailed design with heatintegration of multi-purpose batch process facilities. This appears as an extension of the design model proposed by Barbosa-Pdvoa and Macchietto (1994). Economic savings in utility requirements are studied while considering the possibility of having direct plant heat integration with associated costs of the auxiliary equipment. The heat-transfer equipment structures are designed simultaneously with the plant processing and storing equipment as well as the associated connectivity. The problem is formulated as a Mixed Integer Linear Problem (MILP) where, binary variables characterise operational and topological choices. The applicability of the proposed model is shown via the solution of an illustrative example. 1. INTRODUCTION Multipurpose batch plants are general purpose facilities where a variety of products can be produced by sharing the available resources- equipment, raw materials, man-power, utilities - over the time of planning. These type of plants due to their in-built flexibility towards meeting real market conditions are very important at the industrial level. However, a high level of complexity is present when designing such plants. Thus the need of adequate tools to help the decision-making at the design level is identified. Papageorgaki and Reklaitis (1991) addressed the general batch plant design. Later on, Barbosa-Pdvoa and Macchietto (1994) studied the detailed design problem where plant topology was considered. Although detailed formulations were proposed, an important point has been left o u t - the heat integration problem at the design level. As pointed out by Reklaitis (1989), Corominas et al. (1993) and Papageorgiou et al. (1994), heat integration is frequently present in the operation of batch facilities and is often important for the plant economics. If a close interaction exists between design choices and heat integration costs, considering heat integration in early design stages can lead to more efficient designs. This paper presents a mathematical formulation for the detailed design of multi-purpose batch process facilities with heat-integration. This appears as an extension of the design model proposed by Barbosa-Pdvoa and Macchietto (1994). Consumption of external utilities as well as the possibility of having direct heat integration within the plant are addressed. Economic savings in utility requirements are studied while considering alternative modes of plant heatintegration characterised by different heat-transfer equipment structures (i.e. heat-exchangers, serpentines) and associated design costs. Capital and operational costs are considered. The problem is formulated as a Mixed Integer Linear Problem (MILP). Author to whom correspondence should be addressed, e-mail: [email protected]
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2. T H E M O D E L L I N G AND H E A T - I N T E G R A T I O N C H A R A C T E R I S T I C S Batch facilities are characterised by three main components: the transformation process, the p l a n t and the operation. The need for appropriate representations to describe the interactions between these components is crucial. In this paper two main representations are used: the simple product State-Task N e t w o r k (STN, Kondili et al., 1993) to describe batch transformations; and the maximal State-Task N e t w o r k (mSTN, Barbosa-P6voa and Macchietto, 1994), which represents process and equipment interactions. This is generated automatically from the product STNs, the plant flowsheet and the equipment suitability. In this paper a direct form of heat-integration when designing a plant is assumed. At least one pair of eTasks can be coupled to each other performing a direct heat transfer. These eTasks (i'/j) have, within the mSTN, the correspondent non-integrated eTasks (i"/j) which have the same equipment allocation, thus can never occur simultaneously. For each possible direct form of heat-integration two steps are defined - an exothermic and an endothermic eTask with a possible time offset. Each heat-integration option assumes that at least one of these coupled eTasks is self-sufficient in terms of heat involved. However, and knowing that the rates of heat consumption and production of each eTask may be different we consider the possibility of using external utilities in order to provide a supplement of heating or cooling to the correspondent task. Additionally, in design terms, a set of possible equipment structures that guarantee the required heat transfer, between the heat-integration steps, is defined. Finally, the problem in study optimises the plant heat-transfer policies considering the trade-offs between the production and plant costs accounting for all operating restrictions. Thus we may have, within the same plant, heat transfer policies that might range from the only utilisation of external utilities to only direct plant integration and/or a mixture of both. 3. M A T H E M A T I C A L F O R M U L A T I O N The formulation to the design of batch facilities with heat-integration is developed using as basis the model proposed Barbosa-Pdvoa and Macchietto (1994).
eTask i E T = set o f p r o c e s s i n g a n d s t o r a g e tasks p,i,/p,.,.out =proportion of material of state s entering/leaving e Task i a x / 0 } m i n = maximum/minimum utilisation
S, m S, ~ ={s." set of intput/output states to eTask i} K,={/.set of processing units suitable for eTask i} ={u." set of utilities u used by eTask i} c~,,o/fl,,o = fixed/variable demand factor of external utility u by eTaski at time 0 / proc Pi.,.lag/Pi.,. = lag time of states s ES, entering Aeaving e Task i relative to its start p, = max {p,.[roc}, duration of eTask i
factor of eTask i 1 O C ~ / O C !/=fixed/variable operating cost of eTask i in unit j
State s e S = set o f m S T N states K,. = {j ." set of dedicated storage vesels suitable for storing eState s} OC,. = operating cost of storage of state s v,./p., = value/price of state s Or=size factor of estate s in storage unitj tTasks 7ce H = set o f transfer tasks S~'" / S ] 'u' = {s: set of input~output states of tTask 7c}
TS~ T, i" ={i ." set of eTasks producing/receiving material to state s} H,m/H,~ ={7c.'set of tTasks transferring material to/from state s} SP / SF = {s." set of product/feed states}
717
U n i t j E J = set o f units ~ = { k 9set of k types of unit j} ~ = { i." set of eTasks which can be performed in unit j} S = { / 9set of dedicated storage vessels} ,Q = {/" ." set of processing units j} CC~ = fixed/variable capital cost of unitj Connection c E C = set o f connections cj / cj = ( c: set of connections to which Ic = {n.. set of tTasks which can be performed in unitj is a sink/source } connection c} Xc = {k 9set of k types of connection c} CC~ = fixed/variable capital cost of connection c of type k ~b~= size factor of tTask rc in connection c Utility u ~ U = set o f utilities Uu t max maximum utility u availability at time t OC, = operational cost of utility u cp~= heat capacity of utility u ATe-- temperature difference for utility u Heat-Integrated eTasks - pair o f e Tasks i" and i " l"'={(i "/j, i " / j , ~, pint)., set of heat-integrated Ira~ I TM = {i." set of e Tasks in the first/second level of integration } e Tasks i "/j and i"/j" with time offset, ~' and l~,,={i . the non-integrated corresponding eTask} integration time, pm~} AT,.,.. = mean temperature difference occurring in qs,.,,.= { h: set of auxiliary equipment structures suitable to perform the heat-integration of the heat-integration of e Tasks i" and i " eTasks i" and i"} A ={i. set of self-sufficient eTasks i E 1'~} z-n SinK
/ ~
source
r
A u x i l i a r y h e a t - t r a n s f e r e q u i p m e n t structures, h ~ F CUb- heat transfer coefficient of structure h Ahmoz/A~ mm - maximum/minimum heat-transfer area of structure h B i n a r y variables E; =1 if unitj is installed," 0 otherwise E~,k -1 if connection c of type k is installed,' 0 ~ k = l if unitj of type k is installed," 0 otherwise otherwise Eh=l if auxiliary equipment structure h is Wvt-1 if unitj is performing eTask i at time t; 0 installed,' 0 otherwise otherwise. Continuous Variables ~-capacity of processing equipmentj D,,.t- amount of material delivered from the STN BT~.-capacity of connection c state s at the beginning of period t, B,,/.,-batch of eTask i in unitj in period t R,. , -amount of material received from the BT~.rthe amount of material transferred by tTask outside into the STN state s at the beginning of period t lr at the beginning of period t U~.,-utilisation level of external utility u at the Ah-transfer area of the heat-transfer equipment beginning of period t structure h. UI~,i.rutilisation level of external utility u by the Q,. ,.. rheat transferred in the heat integration of self-sufficient e Task i in period t e T a s k s / ' a n d / " a t the beginning of period t. QNZi,t-heat produced/received by the nonQh,~-heat transferred through equipment structure integrated eTask i at the start of period t h at the beginning of period t U,,,,.ff tra - consumption of external utility from an Qi.,cz~ra-heat transferred by the external utility u to the integrated eTask i at the beginning of integrated eTask i in period t. period t S,.ramount of material in estate s in period t Constraints
As in B a r b o s a - P 6 v o a and Macchietto (1994) for the detailed design we have: Z k~X
E j, k ,
-
Ej
Vj
(1)
718 V j e ~ . j , t =1 ..... H
(2)
Vi, j ~ K , , t = 1,..., H
(3)
Vi, j ~ K , , t - 1 , . . . , H
(4)
vj
(5)
0 _< S.,., _< V,~.,.,
Vs, j e K , , t = O , . . . , H + I
(6)
0 < BT,~,, < BT c "O,~,c
Vc, rc ~ I c ,t = 1,..., H + 1
(7)
Vc
(8)
Vc
(9)
Y~ ~ W ,,/,,, i6l jt'=t- p, +l>_l
0
,.
< E~
< V .... W
,
--
,i
t,j,t
ominv/-vmax(1-<jt)
i,,l
.
ke X i
Z
keX c
.i
gminEik i,k "'
9Ec,k
BT"in ~ c,k
max
, ,
< (~i,.l V I ~ .
~
I < ~ V .... E ' ~ k a j,k ,I ,k
<~ B T c <-- Z
R T m~,k ax ----
keX c
.
Ec ,k
Y~ Ec, k <1 kcX~. k:SsSTN/") rain _<
=
S,.,
. . . .
(Ss,H+I-S,.o)Tt-Ds.H+I. . ,
Z
,~'E,~,S's
/--) max -'( k5 s.s.7:v
V S ST?V
{-) max ] k:5 ,.,sT.v > 0
(10)
*" B ,,;,~_<,-p,':~_
-
ieT,5.... jei
+ Z
BT~,,-
~renU
(11) Z B T ~ , , - D , v +R,v
Vs, t = l .... , H + I
eren~."
When considering the Heat-Integration aspects the following constraints must be added: Heat-Integration Time Offset Constraints: in order to ensure that a given pair of heatintegration eTasks start within a pre-defined time offset r we have" W., =W t,j,t i",j',t+~
V ( i ' / j , i " / j ' , ~ ) e I int t=l,. H ' ""
(12)
Batch Relation Constraints: for each heat-integrated option a operational batch relation exists" rain max /,tcc, Bc.j.,+r
V ( i ' / j , i " / j ' , O e I mt "i'e(I'" ~A),t =I,...,H
(13)
mmB ,./,, < Bc, j,,+~ _/at;.. < . . . .z~,,,j,, . /ace.
V(i'/j,i"/j',~)eI mt "i"~(I 2"J ~ A ) , t = l .... ,H
(14)
lai.~,,max is the enthalpies ratio and/a,,; ,,mmtranslates the minimum required batch relation. Heat- Integration Global Heat Constraints - the global heat in each heat-integration option is
calculated using the external utility hypothetical consumption for the non-integrated eTask: First level o f integration is self-sufficient. pim_l
UI~,.,+~ - 2 (a,,oWc,i,, +/5',,oBc;,,) Vu,(i'/j,i"/J'~,Pmt) eIInt "i'e( Ilst ~A),i e I~,,t =I,...M ''
~
(15)
0=~
Q,s,, =UI,,c,,'cP,'AT~
V(i',i")eI m~"i'e(I ~'' ~ A ) , u e ~ , , , t = l , . . . , H
(16)
Second level o f integration is self-sufficient. pint - I U I u , i",, =
E '~uiOWi",j',z+fluiOBi",j',t)Vu,(i'/j,i"/J,(, 0=0
pint)ellnt"i"e(
Qc,r.,, = UI~,,.,,cp~AT~ V(i',i")~ Imt "i"~ (I 2"a ~ A ) , u ~ ~,..,t -1 ..... H
I2n"
~A),
ieIc
t
(17)
(18)
Utility Consumption Constraints. for the non self-sufficient e Tasks a consumption of an extra amount of utility to compensate its requirements is assumed.
Extra External
719 Q ,,, ~x,ro = Q,,,sv~ - Q ; . ; , , , + r
v (i " i") ~ I im "i'~ A , i ~ I,.,c t = 1.... , H
(19)
+A,oBc,/,,_o)~(i'/j)~Iint'i'~A,i~I~ t , u~vP~ , t = l ,...,H
(20)
Pi - I
QX'=cp, AT~-'(a,,oWc,j,,_o,,, 0=0
U~X' . . =. .Q i ' t ,to /(cp u,~,,
u
g ( i ' ) ~ I m' "i'~ A ' u e ~ i
ATu)
'
t = 1' ' ' "
H
(21)
External Utilities C o n s t r a i n t s - the global external utilities demand considers the demand from the non-integrated and the non-self-sufficient heat-integrated eTasks: Pi -1
U ~ , , - Z E Z (c~,, o.w,,j,, _o+fl,,o.B,/,,_o)+,. iffI int j ~ K i 0 = 0
0
u
~-~g/~,r,~,,.,
gu, t = 1,..., H
(22)
i E ] int :i~A
Vu, t = l ,. .. , H
, , _< [~f m,a,x,
(23)
Heat-Integration Equipment Design C o n s t r a i n t s - the choice of the required heat-transfer auxiliary equipment structures to be installed in the plant and their heat-transfer areas are: Q,,;,,
-
~-'~Qh,,
g ( i ' , i " , ~ ' , p ~n~)E Iint, h E (D,,i,,, t - 1,..., H
(24)
h
CUhAhAT,,, ,, > Qh,,
g(i',i",~:,p~nt) ~ iint,h e ~,,,,,,t = 1.... ,H
Ami. E~ < A h < ~th.... Ee h . . . .
Vh
(25) (26)
Finally, the Objective Function - maximisation of the plant profit given by:
(27)
ji
s
'
where HoursYr is the number of annual working hours while CCF is the charge factor. The non-linear terms are easily linearized. 4. E X A M P L E
A multipurpose batch plant must be designed at a maximum plant profit so as to produce products $4 and $5 respectively with 200 to 250 and 350 to 400 material units (m.u.) over a time horizon of 8 hours. The product recipe and the plant flowsheet are shown in Figures 1 and 2. Two alternative auxiliary equipment structures are represented to perform the heatintegration between Tasks T1 (endothermic) and T2 (exothermic). Combining the STN, the plant flowsheet and the operational characteristics the mSTN, is derived, where the possible heat-integration between Tasks T1 and T2 is p r e s e n t - eTask HI_T1/R1 (self-sufficient with a processing time greater then the non-integrated form (3 hr.-2 hr.)) and eTask HI_T2/R2. Tasks" T1 and T2 if not integrated require respectively vapour and water with a rate of 1.25+ 0.4 B t/h and 1.25+0.45 B mu/h.
720
~--
( S4 ,..
T3
i
/
;'
I v3 v4 ....
Fig. 1. Product Recipe STN.
....
~R2
i
/
.....
.
. _
Fig.2. Equipment Flowsheet.
The results show that units R1, R2 and R3 are chosen respectively with a capacity of 120, 150 and 200 m.u. Also, all the storage vessels are installed. In operational terms R1 and R3 present a multipurpose operation (RI: H1-T1,T2 and R3: T1,T3), while R2 is allocated to the single Task T2. A mixed heat-integration policy is obtained where Tasks T1 and T2 are operated in an integrated and non-integrated form This corresponds to the choice of the heatexchanger structure with an optimal transfer area of 1.107m 2. The objective function was 27240xl 03 currency units. 5. CONCLUSIONS The design of multi-purpose batch facilities with direct heat-integration has been studied. The optimal plant configuration and operation are optimized as well as the optimal plant heattransfers policies and the heat-transfer associated areas. Important aspects were considered such plant topology - the choice of the plant equipment and the associated connections - and plant operation where the final scheduling is performed accounting for a set of heat-transfer policies. These policies may range from the only consumption of external utilities to only direct plant integration as well a combination of both. The problem was formulated trough a Mixed Integer Linear Programming were binary variables define equipment choices and operability. The present model seems promising although, further important generalisations are now been undertaken by the authors. In particular, economic savings in utility requirements are being studied while considering simultaneously not only the cost of the auxiliary structures through their transfer area but also the design of the utility circuits and associated piping costs.
Acknowledgments - This work has been supported by grant PRAXIS/2/2.1/TPAR/453/95. REFERENCES
Barbosa-P6v0a, A.P.F.D and S. Macchietto (1994), Detailed Design of Multipurpose Batch Plants. Compt. Chem. Eng., 18, 11/12,1013-1042. Corominas, J., L. Puigjaner and A. Espufia (1993), A new look at the Energy Integration in Multiproduct Batch Processes. Compt. Chem. Eng., 18S, 15-20. Papageorgaki, S. and G. V. Reklaitis (1991), Optimal Design of Multipurpose Batch Plants 1.Problem Formulation, Ind. Eng. Chem. Res., 20, 10, 4852-486. Papageorgiou, L. and C.C. Pantelides (1994), Optimal Scheduling of Heat-Integration Multipurpose Batch Plants. Ind. Eng. Chem. Res., 33, 12, 3168-3186. Reklaitis, G.V. (1989), Progress and Issues in Computer Aided Batch Process Design. In third Intl. Conf. on Foundations of Computer-Aided Process Design, Snowmass, 241-276.
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V.All rightsreserved.
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Plant-independent Process Representation Kevin Wall+, Paul N. Sharratt +, Naheed Sadr-Kazemi + and John N. Borland *l +Dept. Chem. Eng., UMIST, PO Box 88, Manchester, M60 1QD, UK *BRITEST Project Manager Abstract- Conflicting perspectives create a language barrier between engineers and chemists that potentially constrains process development. A representation of chemical processes has been devised that focuses on the process-controlling physical and chemical phenomena- the Process Def'mition Diagram (PDD). The PDD grows in detail through the process design activity and represents the designers' understanding of the process. It is particularly useful in the early stages of process development. Application of the PDD to facilitate process development is illustrated. Introduction
The accepted approach to process design adopted by chemical engineers is predicated on the concept of "unit operations". These are discrete pieces or groups of equipment designed to carry out distinct generic tasks. The term was coined by Arthur D. Little in 1916 although the concept was clearly recognised by George E. Davis, as can seen from his "Handbook" (Davis 1901). Most process developers now think in terms of equipment when they are developing conceptual process designs (Siirola 1994). Prior to the concept of unit operations, each process was represented by drawings and descriptions of the equipment, again emphasising the mechanical aspects of the plant (see for example Knapp, 1848). Of the traditional process representations commonly used by chemical engineers, flow diagrams (or in detailed design Process and Instrumentation Diagrams) represent the interconnections between plant items, while mass and energy balances present the flow, composition and conditions at the inlets and outlets of the plant items. There have been many attempts at pictorial representation of processes, although these have usually been devised to address rather narrow issues. Examples include representations of the mass balance (Flower et.al. 1993), the cost structure (Douglas and Woodcock 1985), energy flows (Linnhof et.al. 1983), Hazop information (Snirivasan and Venkatasuramanian 1996) and process/operating sequence structure (Naka, Lu and Takiyama (1997). Mahalec and Motard (1977) did present a more general representation in their analysis of the process of synthesising separation flowsheets, although even this was equipment-, rather than operation-focused. Block diagrams are commonly used (for depicting mass and energy balance data for example (Cremer and Davies (1956)), although the assignment of the blocks is almost always by unit operations. The block diagram has been developed into a State Task Network (Kondili, Pantelides and Sargent 1993) for the analysis of the scheduling of batch process operations. l [email protected], [email protected], [email protected], [email protected]
722 For a chemist, process design is more likely to be focused on the composition and conditions of laboratory experiments. For a batch process, a completed process chemistry description is in effect a recipe, detailing quantities and order of addition and the temperatures and pressures to which the chemicals are subjected. In adopting a process rather than equipment-based representation it is recognised that the selection of a particular piece of equipment for a duty will impose some limitations on the process. However, by determining the needs of the process, the mapping of plant onto the process may take place in the light of knowledge of the potential performance-limiting phenomena. Likewise, the decision to operate the process in a batch or continuous mode may compromise performance. The use of a structured representation facilitates an explicit evaluation of any trade-offs between process performance and plant or operation mode - even if these are qualitative in nature.
Plant-independent Process Representation A good pictorial process representation should present the operations performed and the associated flows of mass and energy. Key variables that either must be controlled for safety, quality or environmental reasons or else would potentially limit the performance of the process should be readily identified. This should reduce inefficiency in development arising because key experimental data are not shared. State Task Networks (STNs) are a representation of processes that consist of discrete transformations. For chemical processes, states are the composition and condition of processed material while tasks are the reactions, separations etc. that transform states into different states. Thus, the STN can conveniently be used throughout process design. In the "Process Definition Diagram" (PDD) representation, several levels of detail are used. At the lowest level, the PDD represents the stoichiometry of the process. At the next level, separations and all major streams are identified. At the more detailed levels, the controlling physical and chemical phenomena are represented, these are the "fine detail" of the processing tasks.
Illustrative Example Application of the PDD is illustrated by the following example. The example starts when the chemist first devises a route to the product. All that is explicitly known initially is the reaction stoichiometry (equation 1), and some information about the products and raw materials. A+B
--)C+D
............................... (1)
In equation 1, A, B and C are solids and D is a gas. C is the desired product. The whole process can be represented by a Energy @ ~ Gaseous Byproduct D single task box Raw material A ~ . (Figure 1), React where the circles and Separate Raw material B Solid Product C represent the initial and final states of the materials. Figure 1 First Level Process Definition Diagram
723 The diagram is accompanied by a table (Table 1) that gives the conditions, amounts, composition and any other relevant data on the "states" of the materials at those points. Table 1 First level PDD data 1 State ID: Form: Solid Material wt. A (base) B C D Heat A T (~ P (atm) Max size (~tm)
mol 302.9 138.1 404.5 36.5 kJ/kg
2 Solid
3 Solid
4 Gas
5 energy
Stream mass fraction/amount (kg/kg of A) 1 0 0 0
1 0 0 0
0 0 1 0.456 0 0 0 0
0 0 0 0 1 1.335 0 0
0 0 0 0 0 0 1 0.121 -2000
Amb 1
Amb 1
Amb 1 200
Amb 1
At this level of resolution, the diagram itself has only limited value, but the accompanying data permits the calculation of the lowest possible operating cost and specifies the product requirements that must be met. Note that the heat of reaction is significant, with impact on both the final reactor design and the plant infrastructure.
Figure 2 Second level PDD- diagram only
As process design proceeds, more detailed representation is needed. The next stage of development in the example introduces a solvent S, and an excess of raw material B. Traces of water in commercially available material B would cause a loss of yield and so would need to be removed. Environmental constraints impose limits on the gaseous emissions of D.
724 The PDD at this stage (Figure 2 shows the graphical part) looks like an expanded mass or energy balance block diagram. The task boxes represent process steps that are recognisable both to the chemists and chemical engineers and may not be normal unit operations. Different shading is used in Figure 2 to signify the different phases present. Of course, colour would be the usual way to show this. The energy flows on this diagram are usually only available as estimates at this stage. The composition of many of the streams may also be estimated, but nonetheless provide the engineer and chemist some basis for discussion. The PDD provides a means to store experimental data as well as identify missing information that still needs to be measured. The development work in the laboratory provides the information for the next level of PDD, where each of the main tasks is sub-divided into its major physiochemical sub-tasks. Here it is necessary to define the phenomena of interest and the symbols used to represent them. The phenomena needed to represent a processing task have been divided into nine categories (Figure 3). Within each category there are only a relatively small number of combinations for example there are only four types of phase separations possible (assuming that supercritical fluids do not need to be identified explicitly). Bulk mass and energy flows are generally only considered when representing specific instances of a process step, for example where the optimum mechanical design features are sought, although bulk mass flow may be important in some membrane processes.
Figure 3 Categorisation of process sub-tasks and associated symbols The constraint symbols represent the action of monitoring and controlling process variables. These must be measured during the process to ensure proper operation, although some may be inferred from other measured variables. The condition profile symbols represent the imposition of temperature and pressure profiles on the process step. Their "action" may extend into the subsequent sub-tasks and they may be simple or complex functions of time. Several of the sub-tasks in Figure 3 all may involve significant energy flows. When the PDD is first drawn with the sub-tasks included,
725 most of the variable values will not be known accurately - refmement of the sub-tasks by experiments or modelling would determine the values to use. The PDD at the sub-task level for the reaction step of the example would be represented as in Figure 4 This representation should be interpreted as follows: Stream 9 (dry B in solvent S) is subjected to a temperature profile in the reactor and has solid A dispersed in Figure 4 PDD Representation of Reaction Task it. A dissolves and immediately reacts to form product C in solution and an additional gas phase that must be separated from the liquid phase. The concentrations of B and C in the liquid phase must be constrained (the former to provide some excess of B, the latter based on the required product concentration in the reactor outlet). There are thermal energy flows associated with the maintenance of the temperature profile, the phase change (heat of dissolution) and the reaction exotherm. In addition there are mechanical energy flows associated with the dispersion of the solid in the liquid and possibly for the maintenance of the correct pressure totalling 5f). Even without entering quantitative data into the PDD, it is possible to apply qualitative arguments to design experiments to test the validity of the model. This allows the user quickly to identify and refine the parameters that are vital to effective functioning of the process. Clearly, if a good understanding of the process exists at the sub-task level, then identification of coupled phenomena that are likely to lead to scale-up problems is straightforward, and steps can be taken to quantify and mitigate the problem. It is also evident that the semi-quantitative model represented in Figure 4 would be a useful step on the way to developing a more detailed simulation. Discussion and Conclusions
The application of this representation is under test against a range of (confidential) industrial examples from the free and effect chemicals sector. A clear representation of process understanding provides a common ground between the empirical approach of chemists and the quantitative and computer-based approach of process engineers. The method is being developed to link with other tools appropriate for use in the early stages of process design. As well as the obvious interface with computer representation of processes and simulation, there is a simple connection with experimental design. Use of the knowledge
726 embodied in the PDD can help to target experimental work, as well as representing the current state of understanding during process design. The methodology does require some changes in approach fi'om companies. The provision of quantitative data is sometimes problematic, not because of the intrinsic difficulty of measurements, but because laboratories are often not equipped up to measure the relevant parameters in the form that is required for this representation (rates of dispersion for example). It is recognised that this tool is in its early stages of development and this is one of many aspects to be resolved. Another task to be addressed is the quantification of the process performance that standard pieces of process plant can deliver so that the process-need may be mapped onto equipmentcapability (Frampton and Sharratt 1998). The novel representation presented here has significant potential benefits in the early stages of process design. It provides a medium for discussion that is viable at the low levels of data availability common at that time in the design activity. The representation highlights the important process variables, driving forces and constraints in a pictorial form, but this is backed-up by tables of data that presents information such as material specifications and mass and energy balances.
Acknowledgements The contribution of EPSRC (Grant number GRJL65956) and the industrial sponsors of the Britest Project (Batch Route Innovative Technology Evaluation and Selection Techniques. See: http:/www.britest.co.uk/) is acknowledged. Britest aims to develop tools to achieve step change improvements in process development time, manufacturing time, working capital, plant capital cost, occupancy and flexibility over traditional approaches. References Cremer, H.W., and Davies, T., (Eds) (1956), Chemical Engineering Practise Vol 1. Pg 438, Butterworths, London. Davis G.E. (1901), A Handbook of Chemical Engineering; Davis Bros. Manchester. Douglas, J.M., Woodcock, D.C., (1985) Ind. Eng. Chem. Proc. Des. Dev. 24, 970-976. Flower, J.R., Bikos, S.C., Johnson, S.W., (1993) Trans L Chem. E. 71 (B), 194-202. Frampton, S.H., Sharratt, P.N. (1998) "Versatility in Batch Processes", Paper 59a, AIChE Spring Meeting, New Orleans, March 1998 (Topical conference on batch processing) Knapp, F. (1848), Chemical Technology; or Chemistry applied to the arts and manufactures, Hippolyte Bailliere, London. Kondili, E., Pantelides, C.C., Sargent, R.W.H., (1993), Comp. Chem. Eng. 17, (2) 211-227. Linnhof B., Townsend, D., Boland, D., Hewitt, G.F., Thomas, B.E.A., Guy, A.R., Marsland, R.H., (1983). User Guide on Process Integration for the Efficient Use of Energy, The Institution of Chemical Engineers, Rugby. Mahalec, V., and Motard, R.L., (1977), Comp. Chem. Eng. 1, 57-68. Naka, Y., LU, M.L., Takiyama, H., (1997), Comp. Chem. Eng. 21 (9), 997-1007 Siirola, J.J. (1994). "An Industrial Perspective on Process Synthesis". Proceedings of FOCAPD '94, Snowmass, Colorado, July 10-15. Srinivasan, R., Ventatasubramian, V.,(1996), Comp. Chem. Eng. 20, $719-$725
European Symposiumon ComputerAidedProcess Engineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
The
design
and
management
of material
727
cycles
- towards
a functional
specification for an awareness-tool
E.V. Verhoef ~, G.P.J. Dijkema b and M.A. Reuter ~ ~DIOC Infiastructures, Delft University of Technology, Rotterdamseweg 145, 2628 AL Delft, The Netherlands; [email protected] ~Depaltment of Technology, Policy and Management, Delft University of Technology, Jaffalaan 5, 2628 BX Delft, The Netherlands; [email protected] ~Department of Earth Sciences, Delft University of Technology, Mijnbouwstraat 120, 2628 RX Delft, The Netherlands; [email protected] Material cycles include primary production, specialist incineration facilities, as well as other industrial facilities capable of using waste as a resource. In the management of material cycles, the emerging number of possible processes and options for system structure complicate the design problem. Where procedures for the design of individual industrial units are well developed, the problem to determine the best arrangement for these interconnected units is more difficult, because both public bodies and private companies must decide on specifications. Notably, regulators or policy-makers shape the playing field by regulation that may inhibit optimal design of material cycles, for instance regarding the fate of toxic materials or residues. To let decision-makers become aware of the many degrees-of-freedom and unanticipated consequences of a single-plant oriented approach, we have sought to develop an awareness tool that visualizes the information on complete material cycles. A case study on co-incineration was conducted for the Dutch Waste management Council to extract the functional specifications for such a tool. In order to provide a transparent overview of the system, the tool must be able to (i) make the limits and scope of the subsystems understandable to the regulator or policy-maker, and (ii) translate this information to the system level of the regulator or policy-maker. I.
INTRODUCTION Material cycles include systems for primary production, secondary production, or recycling, and waste abatement [1]. Sustainable waste management -waste management with the o[2jective to close material cycles- therefore, includes all three subsystems. This implies that where waste regulation and policy were focused on the waste abatement sector, these now must extend their focus to include industrial production systems. This may result in a large, composite system, in which traditional waste disposal plants must cooperate with industrial resource processors. In the case that per system the players are given the proper incentives and boundary conditions, in theory the market mechanism will resolve the design and management of the large composite and interdependent systems involved. Under the regime of the market each
728 player will optimize his own subsystem, which is small and less complicated than the total system. It may be seen that the market mechanism can reduce the design and management problem to the design of a proper incentive structure. This is precisely the task of the regulators and policy-makers, whereby the incentive structure or playing field is represented by regulation and legislation, which act as the boundary conditions and incentives both per subsystem and for the complete system simultaneously. Regulator and policy makers who ol:ten have a good overview of the system, however, often lack in-depth information on the subsystems, or players. It is thus of utmost importance this information is presented to regulators in understandable logic. An idea of the complexity involved is illustrated in Figure I, where it is shown that only three important industrial operations, viz. glass, steel and cement production are connected to the waste management infrastructure. We conjecture that sustainable waste management basically is a design problem: the design of prol~er incentives and regulatory framework. Regulators and policymakers in waste sector require a tool, which provides transparent overview of the system, in order to be able to make the design. In this paper we explore the system for sustainable waste management, and on the basis of a case study, co-incineration in the Netherlands, we extract functional specifications for such a supporting tool.
Fig. 1. : Industrial activities integrated with waste processing (after [2]).
729
2. CO-INCINERATION IN THE NETHERLANDS A research study was conducted for the Dutch Waste management Council (the AOO) conducted into the technological options and the regulatory constraints on co-incineration of waste fiom a EU perspective [3]. Co-incineration is the use of waste in industrial furnaces to replace fuel and raw materials. Increasingly, however, Dutch legislation must fit within the EU legal fiamework. A crucial question for the Netherlands, therefore, is how EU regulation influences the co-incineration potential of industrial processes and what other constraints limit the solution space to sustainable waste management. Technology, economics, regulation and public acceptance determine the scope for coincineration. In the study, technology and regulation where combined to paint the spectrum of possible options under present and developing EU regulation. The latter was analyzed with respect to environmental and public health quality and its effectiveness to achieve sustainable waste management. 2.1. EU-regulation The general design of the waste regulation depends on the definition of waste, which provides the distinction between regulations for primary resources and products on the one hand, and regulation for waste on the other. Apart from the gaps in waste regulation and the associated uncertainty of future regulatory requirements, regulatory constraints to coincineration concern operational specifications and waste availability. The proposed Directive on Incineration (Amended Commission Proposal COM (1999)330 final) is taken to be a good measure for EU regulations on waste co-incineration. The EU Directive on Incineration prescribes several requirements with respect to waste treatment, licensing, handling, accepting, operating and monitoring, for example, a minimum operating temperature of 850 ~ for non hazardous waste, and 1100 ~ for hazardous waste containing more than 1% halogens, and a minimum overall 6% oxygen concentration. A minimum residence time of 2 s is required. Stringent emission limits must be observed. Further to these limits a mixing rule applies to co-incineration plants. Limits will apply that are a weighed average of the upper limits for final emissions that originate from normal operation (e.g. from their operating permit) and the upper limits for final emissions of dedicated waste incineration plants. The weighed average will be based on the ratio between the two operations performed, primary production and waste incineration. 2.2. Candidate-process characteristics Under EU Regulation high temperature processes only may qualify for the co-incineration of waste. In primary production, however, operating temperature is very much process dependent, as it is often determined by thermodynamics and operability, while residence time and oxygen content sometimes can be adjusted. Furthermore, under EU Regulation only processes with an oxidative atmosphere can be considered. This excludes, for example, the use of iron production, and some brick making processes that can co-incinerate waste at environmentally sound conditions, because they require a reducing atmosphere. Thus, it may be seen that European regulation is stringent to ensure health and environmental quality and focuses on enforceability. In this manner, however, regulation can also inhibit the use of waste in industrial processes. EU Regulation controls process input and output, as well as process conditions. Prescribing process conditions dramatically reduces the degrees of fi-eedom for a process to meet its input and output specifications. To design
730 regulations for proper input and output specifications of the subsystems alone (reception and transport procedures, maximum emission loads, byproduct and ash compositions etc.) are sufficient for the achievement of the objectives. Industrial processes, therefore, must not be used because they qualify certain process conditions, but rather must be used if waste is COml)atible with the application. If process and waste are not compatible, the use of waste is likely to result in a reduction in process efficiency, (by)product quality, operability or reliability and a decrease in process profitability.
2.3. Compatibility of processes and waste Faced with the context of stringent EU regulation and technological design specifications iJl l~rimary production, the question has been raised how to determine the mutual coHq)atibili O, of t)rocesses and wastes. A precondition for co-incineration is that the process allows for the use of waste. A process is incompatible with some kind of waste when process output specifications are violated in waste co-incineration, e.g. by deteriorated product quality or exceeding maximum emission levels), or in case process reliability or efficiency are reduced. In primary production, many industrial processes are interconnected, as illustrated in Figure 1. As a consequence, for many a process an evaluation as single independent or isolated unit is insufficient. In such interdependent systems, it is of crucial importance use l~roper system boundaries, when comparing or evaluating processes, and to determine the effects on the entire enveloping system of concern. The assessment of compatibility with the system as a minimum must involve the specification of connected processes. A change in the leed of the steel plant, e.g. via the co-incineration of plastic waste, can result in a change in product quality, slag composition or properties, or flue dust quality, because plastics have a difl:erent composition than cokes, and may contain additives or pollution. If the slag, flue dust or the pig iron quality respectively is reduced through co-incineration in such a way that it violates industrial specifications, these cannot be used in the cement plant (slags), dust treatment plant (flue dusts), or be further processed into steel (pig iron). In addition, coincineration inevitably produces some new kind of waste or by-product). When these materials do not match the specification of any other process, co-incineration may not represent a net contribution to the closing of material cycles, reduction of fuel and other advantages. The cost of final disposal of these material may even exceed the economic benefits of co-incineration, and largely determines the economics of the entire operation. 2.4. Tile feedforward control concept Decisions, regulations and policies therefore must be made according to a chemical process control principle viz. feedforward control, which is based on anticipation of the effects of disturbances. Feedforward mechanisms use direct measurements of input (disturbances) to adjust the values of the manipulated input values, which results in ideal process control because the disturbances are measured and changes in control objectives are anticipated, hnplicitly, however, non-measured o1 non-modeled disturbances are not compensated fbr. If we apply this control principle to co-incineration, the input disturbance translates into a change in the system by introducing co-incineration. This may result in changes in (by-) products and wastes, which in turn may effect the system control objectives, such as sustainability, environment, and public health. According to the feedforward principles effects of disturbance are anticipated using a model of the system in order to get or keep the control
731
o/!iectives at the desired values. Application of these principles to co-incineration teaches, that in evaluating a process for co-incineration (i) possible residues arising from co-incineration of (certain) waste should be anticipated, as well as (ii) possible processing, recycling and/or reuse options for these residues. Subsequently, it should be possible to investigate how and/or if these disturbances can be metabolized by the system directly, or engineered of wastes in such a way that they can be recycled or reused. To allow efficient regulation and policy-making, some model of the system of industrial activities is required, to visualize both primary production, reuse and recycling options, and the materials that may required final disposal. This model should visualize available technologies for processing materials, constraints of these technologies as well as gaps in the available technology. It must, furthermore, be capable of communicating this information to the regulators and policy-makers. The awareness-tool should help in the analysis of complex systems, for example the one illustrated in Figure 1. In this particular example, the initial system that included glass, steel and cement production is expanded to link with zinc and copper production. A number of secondary materials or wastes are processed, and a number of specialist treatment facilities had to be included. 3. I M P L I C A T I O N S FOR THE A W A R E N E S S T O O L By understanding of the subsystems, the regulator or policy-maker can adjust can design the playing field in such a way that optimization of each subsystem will result in the achievement of the objectives. In order to provide a transparent overview of the system, the tool must be able to (i) make the rules of the subsystems understandable to the regulator or policy-maker, and (ii) translate this information to the organizational level of the regulator or policy-maker. 3.1. Heuristics In the design of chemical processes, the approach based on the hierarchical structure of design decisions makes use of both models and heuristics [4]. These rules help to order large and complex problems, by providing good starting points, or by reducing the solution space. Whilst no optimality of the resulting solution can be guaranteed, these are useful. "Rather than denigrating heuristics, we should observe that they are everywhere. Heuristics are in our normal everyday decisions when selectilig the technology we will use to solve a problem. Thcy are there when we choose to use a form of simplified model to characterize a batch process when developing a scheduling model for it. In other words, they are there to aid in choosing" [5]. Some of these heuristic rules can be obtained by analyzing constraints and Ol~portunities, others are based on physical or chemical laws, or experience. Similar to the approach in the designing of chemical plant a hierarchical skeleton based on heuristic rules and (simplified) models will be used for constructing the awareness tool. Simple heuristic rules must be extracted using a solid technological base and a systems approach, and built into an understandable logic. Recognizing the importance of making regulators understand technology. Two decisive parameters for the recycling-potential of waste materials are molecule size and 'actual potential hazard for biosphere' [6]. It is argued that direct recycling, resloring a material to its original properties, becomes increasingly difficult with the molecule size and complexity. If a material cannot be recycled, it should be down-cycled or eliminated (e.g. incinerated). Down-cycling is the case if the market conditions do not allow for the energy or money to be spent required to returning the material to its original properties, but
732 only to be turned into products with inferior properties. Elimination of materials must occur when the potential hazard for biosphere is too large for the material to be reused.
3.2. Functional specification The awareness system that provides a useful overview must include a simple generalized linear system model of the industrial system, in addition to a representation of the ecological, technical, economical and regulatory constraints of each subsystem. Since process systems engineers act at a different organizational level than the regulator or policy-maker, the tool, thcrelore, is envisaged to translate information at the level of processes to the organizational level of the regulator or policy-maker. Regulation, for example, excludes a number of environmentally sound industrial options for co-incineration. Consequences of regulations must be made transparent by demonstrating how these affect the system. The tool, must provide this information at the system level of the regulator or policy-maker, it must show the solution space based on regulatory constraints and technological opportunities and constraints. The tool must facilitate communication between regulators, decision-makers and system designers and use an understandable logic originating from a solid technological base using the interconnected material cycles as leading concept. 4. C ( ) N C L U S I O N The study into options for waste co-incineration has clearly indicated that the technological playing field is increasingly determined by EU Regulation. At present, not only process inputs and outputs or emissions are controlled, but also process conditions. Industrial primary production increasingly is interconnected with secondary or waste processing. Regulation forthcoming, however, may not only limit the degrees-of-freedom in systeln design, but also progression towards sustainable resource management by the closing of material cycles. The leedforward control principle provided a useful analogy to draw preliminary Sl)ccifications of an awareness tool, whereby it may be ensured that industrialists, lechnologists, process system engineers, regulators and policy makers alike can become fully aware of the consequences of their decisions, and the opportunities available in material-cycle m ;,t l l a g e m e n t .
REFERENCES I. G.P.J. Dijkema, M.A. Reuter, Comp. Chem. Eng. 23, $795-798, 1999 2. J. Szekely, ISIJ International, 36( 1): 121-132, 1996 3. E.V. Verhoef, G.P.J. Dijkema, M.A. Reuter, H. Huisman, Exploring co-incineration, AOO/DIOC Design and Management of Inflastructures, Utrecht, the Netherlands, 1999. 4. J. M. Douglas, Conceptual design of chemical processes, McGraw-Hill, New York, 1988 5. A. Westerberg, Part H-A Systems view, in: Process Engineering, Institute of Complex Engineered Systems, Carnegie Mellon University, 1999. 6. U.M.J. Boin, Recycling- Traum und wirklichkeit, Heilbronner Umweltforum September 29, Berzelius Umweltservice AG, 1993.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
733
A S T R A T E G Y F O R THE G E N E R A T I O N OF R O B U S T A C C I D E N T S C E N A R I O S IN QUANTITATIVE RISK A S S E S S M E N T USING M U L T I - C O M P O N E N T ANALYSIS Ku Hwoi Kim, Ji Ho Song, Dongil Shin and En Sup Yoon School of Chemical Engineering, Seoul National University Shillim-Dong, Kwanak-Gu, Seoul 151-742, Korea A strategy for producing robust accident scenarios in quantitative risk assessment, which is performed in the process design or operation steps, is proposed. Most governments over the world require industrial companies to submit proper emergency plans through the off-site risk assessment. However, there have been no systematic approaches or criteria for generating reasonable virtual accident scenarios, and it is very difficult to get the unified or coherent assessment result. To get over these shortcomings, this study proposes a strategy for analyzing process elements and then selecting and generating robust accident scenarios that simulate the worst accidents most likely to happen and should be foremost considered. The result through the proposed analysis enhances the reliability of the produced risk scenario and prevents the risks from being overestimated. The obtained result should be more helpful in the proper process design and emergency planning.
1. INTRODUCTION Chemical industries are composed of complicated processes with many recycle streams of energy and materials, regulated by environmental and safety considerations. As concerns about protection from accidents and environmental problems increase, we need better process technology and safety management systems that can deal with process safety more efficiently in real time [1]. Worldwide chemical processes are in need of off-site risk assessment as well as the on-site one. Most governments over the world require industrial companies to submit proper emergency plans through the off-site risk assessment; Korea is also preparing for executing IRMS (Integrated Risk Management System) along with PSM (Process Safety Management) and SMS (Safety Management System). These kinds of analyses are helpful in determining appropriate safety devices, capacity of safety facilities, and the minimum distance from residential areas. Therefore, more and more petroleum and oil/gas companies are adopting these technologies to improve the safety and productivity. However, there have been no systematic approaches or criteria for generating reasonable virtual accident scenarios, and it is very difficult to get the unified or coherent assessment result. Robust accident scenario selection is considered essential for the success of offsite consequence analysis because analysis results may vary significantly according to the selection of scenarios.
2. RISK MANAGEMENT PROGRAM The off-site consequence analysis technology is a world-widely accepted method, which aims to establish an appropriate emergency planning for the off-site area. But due to the limitations of conventional hazard analysis techniques, consequence analysis has the same drawback; analysis results are different according to the individual analyst's view. In the United States, EPA (Environmental Protection Agency) has announced 'RM (Risk Management) Program' in 1996,
734 and every industrial company is supposed to submit reports about the off-site consequence analysis of the release of specific substances [2, 3]. The most notable feature of RM Program is that it asks to carry out consequence analysis for a WCS (Worst Case Scenario) and an ACS (Alternative Case Scenario) for each hazardous material. The worst case scenario is defined as the release of the largest quantity of a regulated substance from a vessel or process line failure that results in the greatest distance to an endpoint. The endpoint is the concentration, explosion overpressure or radiant heat at which serious human health effects or environmental damage could occur from exposure to a release of that substance. Usually, for regulated substances, the release distance that impacts off-site area is fairly long. Parameters required in modeling the scenarios for WCS and ACS are similar. 2.1. Drawbacks in the establishment of accident scenarios The most important part in a consequence analysis program is to determine accident scenarios that are likely to occur in a process. Generally, there are three kinds of methods in deciding accident scenarios: qualitative methods, quantitative methods, and methods using past accident data. HAZOP study and What-If analysis are examples of qualitative methods. Event Tree Analysis (ETA) is an example of the quantitative method [4-8]. Accident data of five years in a similar process are analyzed and used as an imaginary scenario in past-accident data based methods. Each method has its own fortes and drawbacks, and it's difficult to apply these methods in real consequence analysis, because there is no systematic selection criteria for the scenarios among the above methodologies. In qualitative methods, only kinds of accident results are presented and they cannot be applied in ranking or selecting accident scenarios. In quantitative methods like ETA, results change according to the selection of the initial event. In the RM Program, WCS is calculated only using the maximum capacity (i.e. not using the state information of the process or operational condition), and the result tends to be overestimated than the real one. Therefore, to overcome these drawbacks, a method based on the qualitative result, considering the process condition, material property and equipment behavior, etc., and can apply the result in a quantitative manner is required. The result of off-site consequence estimation is presented as toxicity level, heat radiation or overpressure, and used as the basis of emergency planning [9]. When an accident occurs, we can analyze the unit or equipment that affects the surrounding area. Existing methods for calculating the risk depend heavily on the individual analyst's view in generating accident scenarios; the calculation represents so variety of results. Sometimes, heavier risk in a process is overlooked because it would not consider the status of the process. Therefore, we should consider chemical property, meteorological condition and the equipment behavior to achieve calculating the accurate effect to the surrounding area in the off-site consequence estimation.
3. REASONING A L G O R I T H M FOR THE ACCIDENT SCENARIO SELECTION In this study, we propose a new reasoning algorithm through process partition and process component analysis to improve the reliability of accident scenario selection. Process elements are analyzed and then the proposed strategy selects and generates the robust accident scenario of a worst case that is most likely to happen and should be foremost considered. (This concept is being extended using statistical methods, but that part is not included in this paper.) The scenario reasoning algorithm consists of the following steps: macro decomposition, micro decomposition, equipment analysis, scenario selection, and the consequence analysis (see Figs. 1 and 2). In the macro decomposition, process units are selected according to their functions and the meteorological condition around the area. For the decomposition, the chemical plant is classified into the feed system, reaction system, separation system, storage system, and utility system.
735 STEP I : MACRO DECOMPOSITION [ I STEP II : MICRO DECOMPOSITION I
+
Topography~1 Equipment1I Componentb
] STEP III : EQUIPMENT ANALYSIS I RELIABILITY ANALYSIS 4,
InferenceEnginej.[ Library I
RISK
"~"~
No
Yes
[ STEP IV : SCENARIO SELECTION I
ScenadoSelec~on
I
Fig. 1. Structure of the scenario selection mechanism
+
STEPv : EFFECTANALYSIS
I
Fig. 2. Flow diagram of the algorithm
Meteorological characteristics and the surrounding condition are also considered: the main unit is defined, and meteorological characteristics and the topography of the selected unit are considered. The procedure for selecting units through functional decomposition is represented in Fig. 3. We can conclude that unit 2, unit 4, and unit 6 give potential hazards resulting to damages in the surrounding area. Considering unit 2 and unit 4 are utility units with minor risk compared with the main unit, we can determine that unit 2 and unit 4 from the macro decomposition may cause major damage to the surrounding area. For the second step, we propose ESM (Equipment Screening Method) analyzing the process condition and selecting the process equipment with higher priority risk ranking. Equipment characteristics such as material property, operating condition, flow-rate, capacity, safety devices, age and accident history are analyzed using ESM, which is a sequential reasoning method. In case of material property, we use NFPA (National Fire Protection Association) code to confirm the flammability and toxicity; the criterion of this property is more than 3 NFPA rating. In the next stage, we consider whether the equipment is operated in high pressure or temperature, and equipments with high flow-rate or capacity are determined. In the fourth stage, we decide whether the selected equipments have safety devices. In the final stage, we consider the age and accident history for individual equipment using the sequential screening method.
w,o
r Operating~ Condition ~
t c...-~ ]
?, Utility
v
Fig. 3. Unit selection using process partition
Priority 1
Priority 2
Fig. 4. Reasoning scheme for ES Method
736 Table ! Example of failure modes for equipment behavior Equipment Valve Pump Fail on Failure mode Open Transfer off Close Seal leak/rupture Rupture Pump casing leak/rupture Leak
Heat exchanger Leak/rupture (tube to shell) Leak/rupture (shell to tube) Plugged Fouling
The analyzed process elements are ranked and risk grades are determined. According to the grades, risk assessment is performed. Fig. 4 shows all five stages of the reasoning algorithm. In the equipment analysis, the effect estimation for the selected equipment in the step II (micro decomposition) is accomplished: equipment with high severity is researched to find a detailed accident scenario. We use effect analysis method for the failure mode of the selected equipment to identify single equipment failure modes and each failure mode's potential effect on the system and the plant. This mode describes how equipment fails and is determined by the system's response to the equipment failure (see Table 1). In the scenario selection, we infer possible effects depending on the failure mode of the equipment. Possible scenarios for each failure mode are so variable that risk rankings are assigned according to the potential hazard of material and the magnitude of abnormal situation. Table 2 shows the failure mode and scenario selection procedures.
4. CASE STUDY: LPG STORAGE FACILITY
This system is one of typical LPG transportation and storage facilities, including propane underground cavern, propane coalescer, propane dryer, and a propane storage tank, illustrated in Fig.5.
Table 2 Example of scenario selection for a failure mode .
.
.
.
Identi.fication
Mode
Effect
Valve A on the chlorine line
Fail open
Excess flow of chlorine to the heater May cause a high level in the cleaning bed
Fail closed
No flow of chlorine to the cleaning bed Excess water flow to the cleaning bed
Material Chlorine
Risk Ranking C
Excess chlorine and water Water
D
Minor
Water
Minor
737
: ...... {~..i
P
t
minor hazard
[~'orZ
............................................
.
~t
:
}, Propane tans
r
Priority 2 Priority 1 Fig. 6. ESM application for LPG facility
Fig. 5. PFD for LPG storage facility
4.1. Accident scenario selection using the proposed algorithm In step I (macro decomposition), the entire process is decomposed into unit processes and process units are selected according to their functions and the meteorological condition around the area. Step II is the micro decomposition step. Through this step, ESM is applied to the 5 valves, 3 pumps and 1 heat exchanger, which have been selected as the most influential process components to the off-site area when an accident occurs. The result is shown in Figure 6; an accident due to pump A and valve C is the most hazardous one. In step III (equipment analysis), the analysis is performed to the process elements chosen in the step II, and the elements are ranked and risk grades are determined. Table 3 shows a part of the analysis. In step IV (scenario selection), according to the result of step III, propane releases due to the rupture of valve E or the open of valve E caused by the failure and rupture of pump sealing are selected as the most suitable accident scenarios. Step V is effect analysis: a consequence analysis is performed using the scenarios chosen in the step IV. An analysis program is being developed as part of this research, and that program may be used in this step (See Fig. 7 and Fig. 8 for the screenshot of implemented system). An analysis result can also be obtained using one of commercial software packages available today. Table 3 Scenario selection for LPG facility (in case of Pump A)
Identtfication Pump A on the liquefied
Mode Fail open
Effect Excess flow of propane to
Material
Risk Ranking
Propane
Minor
the propane underground
propane line
cavern Fail transfer
No flow of propane to the
off
propane underground
Minor
cavem Seal rupture
Large release of propane to the surrounding area
Propane
A
738
Fig. 7. Screenshot of main menu and input data
Fig. 8. Result of equipment analysis
5. CONCLUSION A strategy for producing robust accident scenarios in the quantitative risk analysis, which is performed in the process design or operation steps, has been proposed and tested to one of the LPG facilities and the DAP process (not shown in this paper). The obtained result of the systematic analysis enhances the reliability of the generated risk scenarios and prevents the risks from being overestimated; the result should be more helpful in the proper process design and emergency planning. The proposed strategy is being implemented as a part of the government-supported, quantitative process hazard analysis system, and expected to be successfully applied to most of mandated off-site consequence analyses in Korea. REFERENCES
1. Greenberg HR and Cramer JJ, Risk Assessment and Risk Management for the Chemical Process Industry, Van Nostrand Reinhold, 1991. 2. EPA, RMP Offsite Consequence Analysis Guidance, EPA, 1996. 3. Murphy JF and Zimmermann KA, "Making RMP hazard Assessment Meaningful", Process Safety Progress, 17(4), 238-242, 1998. 4. Hoist S, Hjertager BH, Solberg T and Malo KA, "Properties of Simulated Gas Explosion of Interest to the Design Process", Process Safety Progress, 17(4), 278-287, 1998. 5. Khan FI and Abbasi SA, "Techniques and Methodologies for Risk Analysis in Chemical Process Industries", Journal of Loss Prevention in the Process Industries, 11(4), 261-277, 1998. 6. CCPS, Vapor Cloud Source Dispersion Models, CCPS of the AIChE, 1989. 7. CCPS, Guidelinesfor Evaluating the Characteristics of Vapor Cloud Explosions, Flash Fires, and BLEVEs, CCPS of the AIChE, 1994. 8. CCPS, Guidelinesfor Hazard Evaluation Procedures, 2 nd Ed., CCPS of the AIChE, 1992. 9. Amaldos J, Casal J, Montiel H, Sanchez-Carricondo M and Vilchez JA, "Design of a Computer Tool for the Consequences of Accidental Natural Gas Releases in Distribution Pipes", Journal of Loss Prevention in the Process Industries, 11(2) ,135-148, 1998.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
739
Simulation of Blowout Events: Integration of Different Modelling Approaches Within Industrial Risk Assessment and Management Tools N. Mancini", F. Podenzani", M. Bonuccelli b, P. Andreussi b, P. Blotto c, R. Galinetto ~ aEni Te cnologie, f,Ta Mari tano 26, 20097 San Donato Mi lanese (Italy) e-mail: nmancini [email protected] bTEA Ceuter - ConsotT, io Pisa Ricerche, P.za D 'Ancona 1,56127 Pisa (Italy) e-mail: m.bonuccelli, [email protected] ~l';uiAgip Divisiou, ~2a Emi#a 1, 20097 San Donato Milanese (Italy) e-mail: [email protected] J77,;MARS, ~7a Medici del Vascello 40/D 20138 Milau (Italy) e-mail: rit a. g_aljne_t.to@temars, it
One of the most critical incidental events associated both to the exploration and exploitation of hydrocarbon fields is the uncontrolled well blowout. In particular, during blowout of onshore wells, the associated oil and gas dispersion in the atmosphere and land requires the quickly definition of a contingency plan aimed to reduce damages. To this aim, the Agip Division of Eni has undertaken a specific R&D project, with the technical and scientific support of EniTecnologie, TEA and TEMARS companies. 1. INTRODUCTION The objective of the project is to develop tools for the simulation of blowout event (fig. 1), at different level of accuracy, obtaining the following main information: 9 single phase flow-rate at wellhead during a blowout; 9 extension of the area interested by the dispersion of both gas plume and oil droplets under assigned meteorological conditions; 9 gas plume and oil droplets concentration distribution. Different predictive approaches and tools are provided within the project, with the scope of: A l.defining a procedure that, using a 1-D dynamic multiphase flow simulator (OLGA), validated for some critical situations with a 3-D Computational Fluid Dynamic (CFD) code (FLUENT), could supply high accuracy estimations of the blowout effects; A2.developing a users-friendly PC software, easily usable by the safety operators for a quick quantitative evaluation of the output information. A3. obtaining a parametric system, using tables a/o graphs, Fig. 1 Blowout phenomena to immediate qualitative evaluations.
740 2. BLOWOUT MODELLING APPROACHES All the blowout aspects are considered by the modelling activity, following different approaches at various stages of the research project 9 multiphase tran,sportatiou in the well. The transportation in the well streams (drill pipe, annulus, casing and tubing) has to take into account the multiphase and multicomponent flow; the OLGA code will be used in A l a simpler code (WELL) will be used in A2. 9 discharge flow. The discharge from the well to the environment is regulated by a critical or sub-critical flow. Both in OLGA and WELL different critical flow models are included a selection/validation of the available critical flow models is performed with FLUENT. 9 droplet.fi~rmatiou. To describe the droplet formation mechanism, a specific experimental activity investigates the multiphase discharge (jet) at the typical blowout conditions, with the scope of generating specific correlations to be integrated in A1 and A2 also in this case CFD simulations are used to support their development. 9 Gas plume and drot?let di,spersion. The gas plume and liquid droplets dispersion are calculated with 3-D FLUENT including wind, turbulent flow and buoyancy effects; the results will be also used to validate a simpler A2 model (CALPUFF). 3. RESULTS
3.1 Multiphase transportation in the well and discharge flow 3.1.1. OLGA and WELL simulations The characterisation of the source term for the dispersion in the environment have to include the simulation of the following phenomena, present in the well: 9 I'brmation perfi~lwJance. Is the definition of the inlet boundary conditions to the well. 9 Multiphase ttzmsportation #1 the well. The multiphase and multi-component flow in the well can be correctly simulated taking into account the various two-phase flow regimes (single-phase, slug/bubbly and annular typically) that can be present in the well. 9 1)ischargeflow. The discharge from the well to the environment is regulated by a critical or sub critical flow. A critical flow model have to be necessarily coupled with an in-well transportation model being this last its outlet boundary condition. A specific analysis has been carried out in order to correctly simulate the multiphase transportation and discharge flow. The output data, necessary for an exhaustive source term characterisation, are the single-phase flow-rate at the well outlet, the outlet fluid temperature, and, if critical flow occurs, pressure and velocity in the throat. In agreement with the project requirements, three different approaches are defined by decreasing the results accuracy but increasing simplicity and rapidity: I. Approach A1. The most refined approach foreseen the use of the OLGA multiphase transient code. This commercial code is a widely validated tool (1,2) for the simulation of multiphase hydrocarbons flows. The application of the code of the 35 project reference scenarios was performed, generating an exhaustive database necessary for the development and testing of the models integrated in the more simple approaches. 2. Approach A2. Upgrading a steady-state one-dimensional code developed by TEA (3) a tool integrated in a PC software, has been developed and validated with the OLGA results. 3. Approach A3. A zero-dimensional model useful for a first evaluation of the source term,
741 defined and tuned with the OLGA results. Calculations with OLGA for all the simulated scenarios show a wide range of variation of the output parameters (oil flow-rate from 0 to 16000 m3/d, gas flow-rate from 18260 m3/d to 5.9 Mm3/d, outlet velocity from 80 to 400 m/s). The typical flow regime in the upper part of the well is the annular flow, with the most part of the liquid phase transported in form of droplets. Very high outlet temperature (near to the reservoir value) are present in the crude oil wells due to the high heat capacity of the mixture and the high velocity in the well, while a strong cooling characterises the gas wells. Temperature varies from-55 to 162 ~ In the most cases a critical flow is present in the discharge section with the throat pressure varying from 1 to 44 bara. 3.1.2. CFD Activities In order to validate the discharge model used in the previous OLGA simulations a CFD activity has been performed with the commercial code FLUENT, considering the compressible Navier-Stokes equations. The comparison between FLUENT and OLGA results have been done in term of gas flow rate, pressure and temperature after the discharge section. The calculation domain is axisymmetric and include the last part of the well Oust a few meters) and a plenum, representing the discharge into the atmosphere. As inlet boundary conditions gas upstream pressure and temperature have been imposed: such values are those calculated with OLGA in the last mesh-point before the outlet; the inlet liquid flow rate is also assigned according to OLGA simulations. In the well investigated have been considered flow in the drillpipe, in the annulus, and both. The hypotheses adopted for the simulation activity are the following: 9 steady state conditions; 9 compressible flow; 9 liquid flow calculated with the Discrete Phase Model (Lagrangian model); 9 turbulence model RNG k-~ (more suitable for compressible flows); 9 gas and liquid properties calculated with the PVTSIM simulator The Lagrangian discrete phase modelling tracks the trajectories of spherical droplets, dispersed in the continuous phase, taking into account the coupling between the phases; the trajectory prediction is accomplished by integrating the ordinary differential equation representing the force balance on the particle (drag, gravity and inertia). The model assumes that the dispersed phase is so diluted that particle-particle interactions and the effects of the particle volume fraction on the continuous phase are negligible; therefore not all the cases simulated with OLGA could be carried out, but only the ones with a volumetric liquid flow rate less than 12% of the total flow rate. In this model it is necessary to define the initial position of the particles and also their velocities, size and temperature, along with their physical properties. To calculate the inlet liquid velocities from the inlet flow rate it has been assumed that the areas occupied by the two phases are proportional to their volume fractions. Uniform droplet diameter equal to 301am has been considered, but a sensitivity study has been carried out between 10 and 100 ~tm. The liquid inlet temperature is set equal to the gas one. In all the simulations the fluid coming out from the well is supposed to enter a domain full of a fluid with the same physical properties. Such hypothesis implies a small density variation between the fluid and the external air and reduce the computational effort, because the conservation equations have not to be solved for all the chemical species. To check the validity of such a
742 hypothesis a simulation of a jet expanding into air has been carried out. A very small difference in the gas flow rate (about 0.65 %) has been obtained. Pressure equals to atmospheric value is the boundary condition of this part of the domain. As shown in fig.2 the geometry of the well caused slight differences in the velocity field near the discharge section. From table 1 it can be seen that the gas flow rate calculated by FLUENT is in all cases greater than the values obtained with OLGA. The differences vary between 10 and 30%. Tab. 1 Simulation results for all the cases examined. ......................................
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Case
Geometry
Pressure
Temperature
OLGA oil flow rate (nra/d)
Case I Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case 9 Case 10 Case 11
l)rillpipe Tubing Both Drillpipe Annulus Both Both I)rillpipe Alumlus Drillpipe Ammlus
2.15 ().67 1.50 1.69 0.83 0.69 (1.21 0.13 0.19 1.3 0.14
1 ! 9.5 117.3 124.9 50.3 41.1 37.4 66.5 80.1 85.7 46.2 69.6
4914 1274 8398 2389 6613 7563 3503 841 4448 365 3841
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OI ,GA gas l]ow rate (m3/d)
FLUENT gas flow rate (In3/d)
36158 48427 1423(-)1 30442 211144 293501 73626 30307 129211 66511 204768
40952 53642 188720 420(112 294715 395589 95711 34458 164335 97546 218112
Fig. 2 Gas velocity field in three cases: annulus+drillpipe (on the left), drillpipe (on the center) and annulus (on the right) 3.2. Droplet formation
3.2.1. Experimental Activities From the fluid dynamic analysis of the discharge flow the following phenomenological aspect can be identified: 9 The annular flow upstream the discharge section is characterised by thin liquid film and high speed droplet and gas field. 9 Critical multiphase flow with throat conditions is characterised by high pressure and, relatively, high multiphase speed of sound.
743 9 Sonic jet downstream the discharge section with high droplet density and velocity. 9 A droplet field generated in the upper part of the jet with a relatively small droplet diameter that strongly affects the oil dispersion phenomena. The above listed phenomena have not counterpart in literature; therefore an experimental activity is doing in the TEA laboratory reproducing the real blowout discharge conditions. The objectives of this work are the following: 9 To validate the two-phase flow models and the critical flow models implemented in OLGA and in the simpler approaches. Obviously these experimental activities are performed on a small scale, but could be scaled-up with the help of CFD. 9 To develop an atomisation model for the description of the oil droplet field. 9 To develop a multiphase sonic jet model for the evaluation of the main jet parameters (in particular height, spread and outlet velocity distribution). In figure 3 a simple scheme of the experimental apparatus is showed. The jet is generated mixing gas and liquid flows in a pipe (three different diameters- 10, 20 and 3 5 mm- are used to evaluate the scaling effect). The jet and the dispersion zone are contained in a 900 mm diameter pipe connected to a separator. A fan blower allows to vary the air velocity in the dispersion flow.
Fig. 3 Simplified scheme experimental apparatus
of
the
Fig. 4 Pressure in the mixing comparison between measured calculated values.
pipe: and
The droplet field granulometry is measured with ultra-rapid photography. In the containment pipe the liquid oil fraction not involved in the dispersion can be measured. The experimental tests are in progress. From the first tests a validation of the OLGA annular flow model has been made. The absolute pressure measured in three different locations of the mixing pipe, are compared with the ones calculated, obtaining an acceptable agreement (fig.4). 3.3. Gas plume and droplet dispersion The main requirements of the approach A2 are the following: 9 calculation of the extension of the area interested by the dispersion of both gas plume and oil droplets under assigned meteorological conditions; 9 calculation of the gas plume and oil droplets concentration distribution in the relevant area.
744 Besides the computation have not to be time expensive and the input data easy collected. The model used in this approach is CALPUFF (4); it is a multi-species, non-steady-state Gaussian puff model, able to simulate near-source effects and long range effects as dry and wet pollutant removal. With reference to the our requirements, CALPUFF has suitable alghorithms to compute the deposition flux (based on the velocity gravitational settling, computed using the Stokes law), the "calm winds" effects and the plume rise (using the Briggs formulations). The main inputs of the model can be summarised as follows: 9 Control parameters (starting date, run length, number of species emitted, computation and sampling grids, etc.) 9 Wet deposition parameters (scavenging coefficients for each pollutants and precipitation type) 9 Dry deposition parameters for species treated as particles (geometric mass mean diameter, geometric standard deviation) 9 Meteorological variables (wind field) 9 Point source parameters (source location, stack height, exit velocity, etc.) Some tests have been performed on a real blowout QmphLegend CALPUFF dry (lop case, using the OLGA simulations results for the 10 TEA Model emission data (temperature, velocity, droplets diameter, * Measured Dam dep F etc.). The CALPUFF results in terms of soil contamination show an acceptable agreement (fig. 5) with the measurements performed on the well area after the blowout.The soil contamination has been computed averaging the concentration values lying at a constant distance from the source. In the diagram the results o.o1obtained with another Lagrangian approach (TEA Model) are represented. A sensitivity analysis has been 0.| ~-!:!,-- ~---~--F-~-~--~--~ r-q--~ 1 performed on the main parameters of the model (i.e 0 400 800 1200 1 6 0 0 2000 2400 geometric mass mean diameter, exit velocity) and the Distance from the well (m) Fig. 5 Comparison among CALPUFF results, results are under evaluation. TEA model results and measurements of soil 4. CONCLUSIONS As shown in the paper different modelling approaches and experimental activities are integrated in order to simulate the blowout event. The work is in progress, but the results obtained shown that this integration can be successful. REFERENCES 1. Villa, M., Bonuccelli, M., De Toma, G. Mazzoni, A, Multiphase Trasportation III, Roros (Norway), 1992. 2. Aprile G., Bonuccelli M., Ghiselli P.W., Mazzoni A., Granato M., OMC99, Ravenna, 2426 March, 1999. 3. R. Basana, M. Bonuccelli, L. Giacometti, C. Ferretti, I. Negri, M Buelli, M. D'Amato, D. Berti, G. De Ghetto, Deep Offshore Technology 1998. 4. Scire J. S., Strimaitis D.G., Yamartino R.J, User guide for the CALPUFF Dispersion Model (1999)
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rightsreserved.
745
Fault diagnosis system support for reactive scheduling in multipurpose batch chemical plants Diego Ruiz, Jordi Cant6n, Jos6 Maria Nougu6s, Antonio Espufia and Luis Puigjaner* Chemical Engineering Department, Universitat Polit6cnica de Catalunya, Av. Diagonal 647, E-08028 Barcelona, Spain In this work, a simple strategy for the development and implementation of a Fault Diagnosis System (FDS) that interacts with a schedule optimiser in batch chemical plants is presented. The proposed FDS consists in an Artificial Neural Network (ANN) structure supplemented with a Knowledge Based Expert System (KBES) in a block-oriented configuration. The information needed to implement the FDS includes a historical database of past batches, a Hazard and Operability (HAZOP) analysis and a model of the plant. A motivating case study is presented to show the results of the proposed methodology. 1. INTRODUCTION The complexity of process control in present batch chemical plants affects the performance of the supervision tasks making it very difficult. Therefore, operators need a support for decision-making when a deviation from the normal operating conditions occurs. This support is also necessary at the upper levels in the decision-making system as is the planning and scheduling level. Due to its inherent flexibility, batch plants can operate efficiently under different scenarios if the consequences of abnormal situations can be anticipated. A robust Fault Diagnosis System (FDS), that timely provides the fault information to the scheduling level, allows to improve the efficiency of the reactive scheduling, to update the schedules in the most effective way. In this work, a simple strategy for the development and implementation of a FDS that interacts with the schedule optimiser is presented. 2. FAULT DIAGNOSIS IN BATCH PROCESSES Currently, the use of pattern recognition methods based on Artificial Neural Networks (ANNs) and the use of statistical techniques are matter of research. The problem of the traditional ANNs related to totally capture the space and time characteristics of process signals is overcome with the use of wavelet functions. With respect to the use of statistical techniques, Multiway Principal Component Analysis (MPCA) has shown good results in batch process monitoring [ 1]. However, it has some drawbacks like the difficult isolation and localisation of the fault. Finally, in order to combine the strengths of both pattern recognition and inference methods, adaptive neuro-fuzzy systems are being developed. The idea is to obtain an adaptive learning diagnosis system with transparent knowledge representation. Some combinations are subjects of current research. *To whom correspondence must be sent
746 In all the above cases, the main problem is the complex strategy of implementation that delays their application in real industrial plants. It is important to take into account that the information given by the FDS of a batch plant has to be used at different levels in the decision-making hierarchy structure, including the advanced control system and the scheduling system. While developing and implementing a FDS, this important aspect must be considered. This work is focused on the implementation of a robust FDS support for the reactive scheduling system in multipurpose batch chemical plants. 3. FAULT DIAGNOSIS SYSTEM STRUCTURE The proposed FDS consists in an Artificial Neural Network (ANN) structure supplemented with a Knowledge Based Expert System (KBES) in a block-oriented configuration. The system combines the adaptive learning diagnostic procedure of the ANN and the transparent deep knowledge representation of the KBES. It has been successfully applied to complex steady state chemical plants [2]. Figure 1 shows the Fault diagnosis system structure. M1 is the subset of the direct and indirect measurements and/or observations from the plant, and is selected as input of the ANN approach. N1 is the set of q "pre-faults" diagnosed by the ANN approach. The values Nl(i), i from 1 to q, are usually between 0 and 1. They are the input of the Fuzzy Logic System (FLS). M2 is a set of r direct and indirect measurements and/or observations from the plant, which is selected as input of the FLS. The inference engine of the FLS has the knowledge base, expressed in a set of if-then rules. These rules are of two types: those containing process deep knowledge and those that are built from experience of the ANN's performance. In Figure 2 a scheme of the set of rules is presented. The outputs F (/'),j from 1 t o f are theffaults considered. The information needed to implement the FDS includes a historical database, a Hazard and Operability (HAZOP) analysis and a model of the chemical plant. The historical database that includes information related with normal and abnormal operating conditions can be used to train the ANN structure. The on-line measurements from the plant are the ANN's inputs. The ANN's outputs are the signals of different suspected faults. These outputs are a subset of the set of KBES's inputs. The ANN, not only has the advantage of a classifier but also it can be retrained during use. By this way, the changing operating conditions of chemical plants should not affect the FDS's performance.
Plant Mll
FLS t
LI~ ANN[-~ Fuzzificatioengi n//IJnnfeere/\ nce
~F
Based on experience with ANN performance
IF N1 (I) is ... A N D
(M2)... THEN/70") (M2)...THEN
F(/) is...
(M2)... T H E N
F(j) is...
. . .
IF N1 (q) is ... A N D
IF M2(1) is ... A N D . . . T H E N F(/) is...
Based on process deep knowledge
. . .
IF M2(i) is ... A N D ...THEN F(/) is... . . .
IF M2(r) is ... AND... T H E N F(/) is...
Defuzzification
Fig. 1. Neuro-fuzzy FDS
is...
. . .
IF N1 (i) is... A N D
Fig. 2. Scheme of the set of rules.
747 The HAZOP analysis is useful to build the process deep knowledge base (KBES) of the plant. This base relies on the knowledge of the operators and engineers about the process and allows formulating artificial intelligence algorithms. A model of the plant can be used to obtain plant operation experience through simulation. The simulation can provide data on infrequent faults because in the cases of faults that rarely occur it is not possible to test the FDS using only plant data. In addition, by testing the ANN with the model, the development of the rules based on experience with ANN performance is straightforward. The model is also useful for testing and validating the FDS. The methodology can be summarised in the following steps: 1)Model the process;
2)Define the faults; 3)Determine measurements; 4)Simulate the faults; 5)Train an ANN, 6)Design the KBES using fuzzy logic; 6)Test the new system by simulation," 7)Design the adaptive method; 8)Test with the model; 9)Implementation in the real plant. 4. H A Z O P A N A L Y S I S IN B A T C H P L A N T S
HAZOP is one of the most powerful hazard identification methods available and has been well described in the literature. In the case of batch processes the HAZOP analysis examines every stage of the batch process sequence. Table 1 shows an example of a line in the HAZOP analysis of a batch process (Stage: Reactor charge; Unit: Tank n~ Node: Pipe from tank 1 to pump 1; Objective: To provide reactant to the reactors; Variable: Flow) Table 1. An example of a HAZOP analysis line Guideword Causes Consequences LiSV~..... Pumpl Time needed ibr reactor malfunction charge increased
Corrective actions Switch to pump 2
Safegu~ds Maintenance tests
Extending the HAZOP method to fault diagnosis characterisations provides a more "down to earth" approach for implementing an operator support system. The rules are kept simple to avoid the general problems of large rule-based knowledge systems, such as contradictory rules, large amounts of irrelevant information and complex tree structures [3]. The HAZOP analysis allows to: a) Generate the if-then rules for the KBES; b) Determine the information to be sent to other levels in the information system. 4.1. Generation of if then rules from H A Z O P analysis Not all the variables considered in the HAZOP analysis are measurements from the plant. Some variables can be observed (estimated). Regarding the generation of IF-THEN rules, only the measured and the observable variables should be considered. In general, the term "Causes" in the HAZOP analysis corresponds to the root cause of a deviation and it can be designated by the term fault. Only the causes that have been defined in the set of faults are considered at this step. In the Example, the fault is "pump 1 malfunction". Consequently, the conversion to an if-then rule is: IF Flow IS Low THEN "Pump malfunction" IS HIGH. The adjustment of the membership functions is the final step. 4.2. Determination of the information to be sent to other levels The FDS receives sensor data from the plant and the control signals. They can be continuous signals (temperatures, flowrates, pressures) or discrete signals (valves open or close, pumps on or off). The outputs are a set of suspected faults. The signal corresponding to
748 each suspected fault is considered binary (0 or 1). This output can be used by the advanced control module in order to take control actions, or by the operators who have to make decisions or by other levels in the computer system as the scheduling system. The output of the FDS has different forms according to the level of information. Table 2 shows the information at different levels from the FDS output when the fault "Pump 1 malfunction" is diagnosed. This evaluation corresponds to the explained Example. Note that the construction of that table is straightforward from the HAZOP analysis. Table 2. Information to different levels based on HAZOP analysis Module Translation from the FDS .....Control system Switch to pump 2 Scheduling system Time needed for tank charge increased Operators' console Check pump 1 5. REACTIVE SCHEDULING When a deviation from the predicted schedule in a multipurpose batch plant is diagnosed, the FDS activates the reactive scheduling module to minimise the effect of this deviation on the remaining schedule. The planning and scheduling system uses event operation networks (EON) modelling system [4]. The EON model has proved to be an efficient way to describe time constraints between operations in a complex production structure, as it is the batch process. The EON model is built using a general recipe description and other guidelines from ISA $88 specification. On the first step, according to the present situation of the plant, and the client orders, a first batch sequence is generated using the information provided by the recipe and the stage levels. Then, an EON graph is generated using the information located at the operation level description of the recipes, and the information generated in the previous step about the unit assignments and task sequence. Finally different methods can be used to adjust the proposed solution under the constraints imposed by the different resources required. Once the schedule is running on the real (or simulated) plant, the control system communicates to the planning system the deviations detected from the proposed plan. With this information, the scheduling system generates new information that is sent back to the control system. This feedback closes the loop between the planning and scheduling system and the control system. 6. CASE STUDY: MULTIPURPOSE BATCH CHEMICAL PLANT Figure 3 shows the flowsheet of the considered multipurpose batch chemical plant used as case study. It is constituted by three tank reactors, three heat exchangers and the necessary pumps and valves to allow changes of configuration. Equipment of this plant is fully interconnected and the instrumentation allows configuration changes by software. Two recipes with two stages each one have been considered. Figure 4 shows the representation of recipes in a Gantt chart performing two batches. Table 3 shows the operation description and the operation times corresponding to the two recipes considered. In both recipes the time of the operation with code 5 are different depending on the reactor chosen to perform the second stage.
749
....
R2
[iiiiiii~l~iiiiiiii~iiiiiiliii}iiiiiiiiiiiiiii~iiii]iiiiii!il}t iiiii!iiiil]
i il
iiiiii!iiiiNl
.....
Time
Fig. 4. Gantt chart performing two batches Fig. 3. Flowsheet of the multipurpose batch chemical plant Table 3. Operation description and operation times (hours) Operafibn ........ Stage Descriptidi] Unit code code 1 1 Load tank 1 T1 2 1 Stirring / Homogenising T1 3 1 Discharge to R1 / R2 T1 4 2 Load reactor R1/R2 5 2 Reaction R1 5 2 Reaction R2 6 2 Discharge of final product R1/R2 7 2 Reactor cleaning R1/R2 .
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Recipe 1
Recipe 2
0.066 0.084 0.066 0.066 0.25 0.33 0.066 0.167
0.066 0.167 0.066 0.066 0.33 0.416 0.066 0.066
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Two different faults have been considered to test the implementation strategy. The first one corresponds to a delayed operation caused by a pump malfunction. Figure 5 shows the tank level profiles, comparing a normal batch, an abnormal batch without FDS and an abnormal batch with the FDS implemented. It can be observed that the rapid diagnosis and communication to the supervisory control level allows reducing the delay. The whole schedule considered in this study consists of 5 batches of product A (recipe 1) and 6 batches of product B (recipe 2). The makespan of the initial schedule is 4.37 hours. The second abnormal situation considered is the unavailability of a piece of equipment, in this case the Reactor 2. It has been simulated that the unavailability lasts for fifty minutes since the beginning of the schedule. Figure 6 shows the Gantt charts. Table 5 summarises a comparison of the results taking into account the plant functioning without a FDS and with the FDS and reactive scheduling for the two considered abnormal situations 7. CONCLUSIONS A simple strategy for the development and implementation of a FDS that interacts with the schedule optimiser in multipurpose batch chemical plants has been presented. Besides, two examples of abnormal situations have been shown in a multipurpose batch chemical plant. The proposed integration of the FDS in the information system shows promising results by significant improvement in the production efficiency. Industrial applications of the proposed system are straightforward because of the simplicity of implementation. Therefore, future work includes the implementation of the system in real industrial scenarios.
750
Fig. 5. Tank levels profiles, Normal batch; Abnormal batch without FDS; Abnormal batch with FDS support
Table 4. Abnormal situation management performance comparison (makespan). Abnormal situation
Delayed operation Unavailability (R2)
Without the FDS support 4.43 h (+1.4%) 4.83 h (+10.5%)
With FDS & reactive scheduling 4.39 h (+0.5%) 4.70 h (+7.6%)
Fig. 6. Abnormal situation management comparison: a) Initial schedule, b) Realised schedule without the FDS support and c) With FDS and reactive scheduling.
ACKNOWLEDGEMENTS Financial support from the European Community is gratefully acknowledged (projects IC18CT98-0271 and IMS 26691). Nouguds was sponsored by Generalitat de Catalunya, II Pla de Recerca, TDOC Grant. REFERENCES
1. P. Nomikos and J.F. MacGregor, AIChE Journal, 40 (1994) 1361-1375. 2. Ruiz, D., Nougu6s, J. M. and Puigjaner, L, Computers & Chemical Engineering, 23S (1999) $219-222. 3. Wennersten, R., Narfeldt, R., Gr~infors, A. and Sj6kvist, Computers & Chemical Engineering 20S (1996) $665-670. 4. Graells, M., Cant6n, J., Peschaud, B. and Puigjaner L., Computers & Chemical Engineering, 22S (1998), $395-402.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
751
Improving on chemical process safety through distributed computer assisted knowledge analysis of preliminary design Bastiaan A. Schupp 1'3, Saul M. Lemkowitz 3, Louis Goossens 2, Hans J. Pasman 3, Andrew R. Hale 2 1. TU Delft, Faculty of Technology, Policy and Management, Safety Science Group, Kanaalweg 2b, 2628 EB Delft, The Netherlands. Phone: +31 (0)15 278 36 42 Email: [email protected] 2. TU Delft, Faculty of Technology, Policy and Management, Safety Science Group 3. Faculty of Applied Sciences, Chemical Process Technology, Julianalaan 136, Delft, The Netherlands 1
ABSTRACT Because accidents cause delays that have become increasingly expensive, the chemical industry should strive to improve safety by shifting from secondary protection to primary protection. The current design approach, that is basically reactive, is not fully adequate, as opportunities for technology are most efficiently realised in early design. A proactive framework, using the concept of 'anomalies', is proposed to identify desired as well as undesired relations in a system being designed. The proposed methodology also presents a strategy to deal with system anomalies. It efficiently prevents incidents that cause delays, helping to ensure profitability. 2
INTRODUCTION Presently, during design chemical engineering uses a number of formal methods, indexes, and software tools to ensure safe operation. However, the application of most tools is limited to a relatively small group of specialists, while both safety itself, as well as the input and use of outputs of these tools, are influenced by the activities of others. Many tools are applied only at one specific moment of time (a 'snapshot') rather than continually; often application occurs so late in the design process that changes are difficult and expensive to incorporate. Furthermore, these tools often just partially evaluate a design and offer limited possibilities to communicate with others important for achieving and maintaining safety.
2.1
The importance of safety management
A number of major industrial accidents and environmental scandals in the early seventies led governments to impose increasingly stricter regulations. The chemical industry has complied with this new legislation, and its processes are now among the safest activities carried out by man. Though public concern about industrial safety has become less pressing, a new challenge for safety management is emerging: profitability. The business environment is changing, and profitability is increasingly dependent on the ability of a market player to efficiently manage its technology [1]. Consequently, the chief motive for building a safe plant may well be shifting from ethical and legislative considerations to more direct economic incentives. Failure of a safety management strategy can be rather expensive. In the bulk industries, even a small incident can cause losses exceeding $ 1M per day. A larger event may require selling a product produced by a competitor, in order not to lose the market. As technological development in the bulk industries has slowed, the industry is actively seeking new sources of profit. One method is to be the first player on the market. Speed is therefore essential, and a delay that follows a relatively small mistake in production or in process design may well lead to a total loss of
752 a market. Even if this is not the case, the likely result of delay will be a relatively large decrease in revenue. It can be shown that when a 4% delay is experienced in a 48-month market window, the lost revenue can be as high as 12% [2]. Extra investments necessary to solve the problem are not included. In a real-life case, it is shown that a 33% loss was caused by a delay of a half-year, while a 50% increase in the development budget would have reduced this loss to only 4% [3]. As lack of safety or ineffective implementation of a protection strategy will likely cause delay, it is easy to see why a shift from ethical and legislative considerations to economic rationality is imminent. In the future, small incidents will be even less likely to cause major accidents, but such incidents are nevertheless more likely to cause financial disaster! For reasons of profit it is thus imperative that a company has an effective safety management strategy.
2.2
Management Strategies
Four different management strategies exist that can be concurrently applied to achieve a high level of safety. These are: 9 9
Primary protection / prevention Secondary protection
9
Mitigation
9 Knowledge Management The concept of primary protection was developed by Trevor Kletz in 1977 [4]. Since then this design philosophy, usually called 'Inherently Safer Design (ISD)', has matured. Textbooks (e.g. [4]) give a useful overview of this philosophy. In principle, ISD is the best method of protection, since avoiding risks prevents accidents from ever happening. Although the basic wisdom of ISD is widely recognised, industry is still in a transition state in applying it [5,6]. Important barriers that slow the introduction of ISD include inadequate awareness and information by managers and designers. Another major problem that impedes the application of ISD might well be the poor appreciation of safety in R&D. This is unfortunate, since in the R&D phase ISD has its greatest possibilities for application. It should also be emphasised that ISD is a design philosophy rather than a concrete design tool.
Secondary protection is the traditional and still very common way in which the chemical industry achieves its high level of safety. It employs analytical tools, such as QRA, and HAZOP, as an aid to design protective systems. These protective systems function as 'Layers of Protection'. In the [ rers of Protection approach, Mitigation systems form the outer layers, to be engaged if
753 secondary protection fails in the inner layers (fig. 1). While the Layers of Protection approach works well to reduce risk, it is a poor protector of profitability! First, the layers present extra costs and complexity. Second, if during design a hazard emerges that cannot be dealt with, redesign is necessary, which is very inefficient in terms of time and money. Third, such layers are reactive rather then preventive. Thus they do not remove the possibility for an incident to occur. Fourth, if the layers need to function, an expensive delay in production is likely, even if no damage is done. In addition to safe design, Knowledge Management (KM) can also protect people and assets from risks. Though KM is not usually regarded as a protection strategy, its use offers opportunities to increase safety through improved communication, corporate memory (learning from accidents!), and worker education. KM helps to overcome barriers to introduce ISD, it complements the usefulness of currently available tools, and clarifies the why of procedures. Furthermore, while safety management can be achieved in four ways, only ISD and KM are additionally effective in improving profitability. 3
THE DESIGN PROCESS Because of decreasing possibilities to modify a design as it becomes more detailed, it is crucial to maximise the effectiveness of the design process. In order to efficiently achieve primary protection, designers must have the appropriate information available at the fight moment. Innovative activities in the chemical industry can be divided in product design, process design and plant design. Each of these activities offers different opportunities to ensure safe operation. Westerberg points out that process design is often a very complex, poorly understood human activity [7]. Necessary are models that help us understand the design process, information pathways, objectives and constraints of designers, and design error. Unfortunately, such models hardly exist. Chemical engineering textbooks of [e.g. 8] basically employ a model based on a scheme conceived by French (Fig. 2a) [9]. Few will argue that this is the way that design rationally should take place, but it is the question whether in practice design actually follows this logical progression. In any case, this model does not consider organisational aspects (e.g. which teams are involved) and crucially important activities like R&D. How quality factors (objectives) are evaluated is not discussed [ 10]. Thus, the current model provides at most a bird-eye's view of the actual process. In contrast to the manufacturing industry, where methods such as concurrent engineering are well developed, the chemical process industry does not appear to show much interest in understanding its own design processes.
754 Taylor [ 11] suggests that a chemical process design in practice never proceeds in a top down approach, but in terms of subcomponents, designed more or less independently (see Fig 2b). Modifying an existing technology or a good idea offers a starting point at a low level, in contrast to the model of French. Adding components afterwards completes the system. Like this, the analysis of need is in terms of existing solutions, from a local and partial perspective. Moreover, the authors of this paper consider it possible that a department or team is limited by the 'paradigm' that fixes the technologies it applies and the 'worldview' that filters its communication. Different groups may have different perceptions of what 'facts' are to be selected from available information. This cognitive process, in which the concept of a 'paradigm' is central, is similar to that introduced by Kuhn for the progress of science in general. Wei[ 12] defines paradigm clearly.
4
A F R A M E W O R K TO AID DESIGN FOR SAFETY During early design, it is easy to notice a basic hazard, e.g. a substance is toxic or flammable, though assessing the risk will be far more difficult, as many details are yet unknown. When comparing technologies, different alternatives will present dissimilar hazards, and these will be difficult to quantify as detailed information is unavailable. This problem is exacerbated by poor dissemination of information due the absence of a good knowledge system. Typically, an experienced designer will have little difficulty to make a choice, but this choice may well impede accommodation of a potentially better technology, new knowledge, or design objectives other then those that were always important to the designer. Furthermore, the worth and availability of experience is decreasing rapidly with the increase of the development rate of technology. Thus, devoid of a good method, a design team will have inadequate insight to select the most favourable option in an early design stage. The proposed framework increases this insight, establishes a strategy to deal with hazards, and captures both the mechanism by which an hazard is created and its relationships with the rest of the process system: it can identify explicitly what must be eliminated or prevented and how that is done best. 4.1 Anomalies When designing, a designer establishes a system with certain attributes that has a desired function, for instance the production of polyethylene. In doing so, other functions will be included as well, for example the ability to poison or to burn. Such an undesired function is the central concept of the proposed framework, and is called an anomaly. Identifying anomalies will subsequently enable a strategy to deal with these. The word anomaly has several meanings in natural language; here it is used to mean a deviation from a desired state, a deviation from a design intention. A design anomaly is defined here as an undesired attribute of a system, inseparably associated with a desired attribute needed for the function of the system. Figure 3 shows an example. An anomaly has relations and a life cycle. Its relations link it to a system. First, it is part of a
Figure 3: An object is introduced in a system for its primary function. It will also have undesired functions that can make if fail. In this case a valve has strength, for it is made of metal. Metal can corrode, which is undesired because corrosion introduces a shape-change, making the valve become stuck. Thus with the Cunction 'strength', an anomaly is introduced.
755 pathway or scenario by which harm reaches a victim. Second, a desired attribute or function associates it to the system. The life cycle of an anomaly will start at the moment a specific subsystem is designed to which it is associated. When an anomaly is identified, a strategy can be established to deal with it. Once the design has materialised, the anomaly can eventually alter the desired state of the system; it becomes active as soon as it starts 'functioning' in the failure scenario. This can happen throughout the life cycle of the system, or in a specific stage such as start-up. An anomaly will usually be latent, only able to do damage when other factors in the scenario are present as well. Observing it during operation can be difficult, as the system still appears to function in the intended state. It becomes a potential incident trigger.
4.2
Protection strategies and anomalies
Figure 4- Primary protection is achieved in three manners in terms of anomalies: 1. eliminate anomalies, 2. Change the anomalies, 3. Change the sensitivity of the system. Secondary protection adds an auxiliary reactive system.
Now that the concept anomaly has been introduced, it is possible to discuss the four protection strategies in terms of this concept and to use it to obtain a protection strategy. Primary protection can affect anomalies in three distinct ways. It can either achieve a reduction of the number of possible anomalies, reduce the capability of the anomaly to change the state of the system, or in reduce the sensitivity of the system to a selected anomaly. (Figure 4). Examples of these three strategies include: a design that does not needs a subsystem (therefore the associated anomalies disappear), the use of a material that is less likely to corrode, and a system that is less likely to fail when corroded. Secondary protection functions in different way. It neither changes the nature of an anomaly nor removes it, but reacts when a scenario unfolds. It is an add-on feature to the system, i.e. not related to the desired function. Thus, it will alcontrols: Purpose, History ways present additional complexity. The shaded area in Figure 1 gives examples of such systems. 4.3 A systems approach to anomalies The idea behind the introduction of anomalies is that these make undesired functions of a system visible. However, in order to be cognisant that anomalies and scenarios are present, as well as to communicate effectively without information overload to individual designers, the system must be modelled as a whole to provide the necessary insight. The Structured Analysis and Design Technique (SADT) [e.g. 13] provides a comprehensive, understandable, and well-established way to gain insight. In the suggested approach, the input and output structure of the SADT model is modified. Desired and undesired inputs and
Desired
Probabilistic
Subsystem
Deterministic Undesired
(caused by
anomalies)
Mechanisms:
Knowledge Domains,Technology,Operation Figure 5: Modified SADT box, input and output are split in two parts, desired and undesired, to make both visible. Undesired relations are due to anomalies. Deterministic relations are fully understood, probabilistic result from QRA.
756 outputs are made explicit (Fig. 6) and are subsequently divided into fully deterministic and probabilistic relations. Anomalies associated to the system account for the undesired relations. Apart from the input and output structure, each box has associated controls and mechanisms. Controls instantiate the system with a set of needs that describe its functionality. Subsequently, a mechanism is chosen to achieve that functionality. Since one or more anomalies are associated with a given mechanism, undesired functions are introduced as well. During the design process the SADT model of the system becomes increasingly complex. At the design onset, only the basic control and the desired output are defined, while during the design process the mechanism is introduced that defines other input and output relationships. The explicitness of this methodology improves knowledge exchange, since the structure is passed on to a large number of specialists, each of whom is working on a specific subset. The power of this approach is that visualising the consequences of anomalies facilitates the design of preventive strategies.
5
CONCLUSION: A DISTRIBUTED APPROACH In the Introduction it was argued that, due to financial reasons, in the future it will become increasingly important to prevent delays in design or production. Secondary protection is no longer sufficient, and a shift to primary protection becomes necessary. This can only be achieved when designers already in an early stage are fully cognisant of which anomalies are present and the technological means to overcome them. Information must be gathered from multiple knowledge domains to obtain this insight, while decisions must be disseminated over many specialists. For this reason, the approach is distributed in that it establishes a design infrastructure that continually keeps the appropriate experts in contact with one another, thus facilitating knowledge transfer. The paper proposes a backbone for establishing such an infrastructure. The framework is, however, still incomplete as it does not suggest how to represent the information it contains. In addition, generic structures, such as generic anomalies, scenarios and accidents, must be added to complete the framework and to prevent information overload to its users. Another challenge is to develop a user-interface that the intended users are able and willing to use. Otherwise, the insight provided remains incomplete. Also differences in jargon must be dealt with to prevent inconsistencies. REFERENCES 1. c.s. Syan (editor), Concurrent Engineering, Chapman & Hall, London, (1994). 2. Carter, D. and Baker, B., Concurrent Engineering: the product environment for the 90s, AddisonWesley, (1992) 3. A. Korbijn edt., Vernieuwing in de productontwikkeling (in Dutch), Stichting toekomstbeeld der techniek, The Hague, (1999). 4. R.E. Bollinger edt., Inherently Safer Chemical Processes, CCPS, New York, (1996). 5. D.P. Mansfield, Viewpoints on Implementing Inherent Safety, Chem. Eng., 103(3), pp 78-80, (1995). 6. N.A. Ashford and G.I.J.M. Zwetsloot, Towards Inherently Safer Production, TNO Work & Employment R990341, Hoofddorp, the Netherlands, (1999). 7. A.W. Westerberg et. al., designing the proc. design process, Comp. chem. eng. 21, pp. S 1-$9, (1997). 8. L.T. Biegler, Systematic Methods of Chemical Process design, Prentice-Hall, New Jersey, (1997). 9. M.J. French, Conceptual Design for Eng., The Design Council, London, UK, (1985). 10. P.M. Herder, Proc. Design in a changing env., Ph.D. thesis, TUDelfi, Delft, the Netherlands, (1999). 11. J.R. Taylor, Design Error, unpublished, (1997). 12. J. Wei, A century of paradigms in chemical engineering, ChemTech, May, pp. 16-18, (1996). 13. A. Kusiak, Eng. Design: Products, Processes, and Systems, Academic Press, San Diego, (1999).
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
757
P l a n t Monitoring and Fault Detection 9Synergy between D a t a
Reconciliation and Principal Component Analysis. Th. Amand a, G. Heyen a, B. Kalitventzeff b a L.A.S.S.C., Universit6 de Liege, Sart-Tilman B~timent B6a, B-4000 Liege (Belgium), [email protected] b Belsim s.a., 1 Allre des Noisetiers, B-4031 Angleur - Liege (Belgium) Data reconciliation and principal component analysis are two recognised statistical methods used for plant monitoring and fault detection. We propose to combine them for increased efficiency. Data reconciliation is used in the first step of the determination of the projection matrix for principal component analysis (eigenvectors). Principal component analysis can then be applied to raw process data for monitoring purpose. The combined use of these techniques aims at a better efficiency in fault detection. It relies mainly in a lower number of components to monitor. The method is applied to a modelled ammonia synthesis loop. 1. INTRODUCTION Measurements are needed to monitor process efficiency and equipment condition. Model based statistical methods, such as data reconciliation, are provided to analyse and validate plant measurements. The objective of these algorithms is to remove any error from available measurements, and to yield complete estimates of all the process state variables as well as of unmeasured process parameters. Besides that, algorithms are needed to detect faults, i.e. any unwanted, and possibly unexpected, mode of behaviour of a process component (Cameron 1999). Fault detection and diagnosis is generally accepted to occur in three stages: 1. Detection - has a fault occurred? 2. Identification - where is the fault? 3. Diagnosis - why has the fault occurred? The goal of our study is to examine how two well accepted techniques, data reconciliation (DR) and principal component analysis (PCA), can work in conjunction to enhance process monitoring. 2. PRINCIPAL C O M P O N E N T ANALYSIS Any process is affected by variability, but process variables do not fluctuate in a completely random way. These are linked by a set of constraints (mass and energy balances, operating policies) that can be captured in a process model. Even if a rigorous mathematical model is not available, statistical analysis of the process measurement time series can reveal underlying correlation between the measured variables (Kresta et al, 1991). If measurements related to abnormal conditions are removed from the analysed set, the principal components of the covariance matrix (i.e. the eigenvectors associated with the largest eigenvalues) correspond to the major trends of normal and accepted variations. Most of the variability in the process variables can usually be represented by the first few principal components, which span a subspace of lower dimension corresponding to the normal process states (Snedecor G., 1956). Projection of any new measurement point in this subspace is
758
expected to follow a Gaussian distribution, and can be checked for deviations from their mean values using the usual statistical tests (example in figure 1). With this approach, one can verify whether a new measurement belongs to the same distribution as the previous sets that were recognised as normal. If not, a fault is likely to be the cause; it can result from an excursion out of the normal range of operation, or from an equipment failure breaking the normal correlation between the process variables. It can also result from normal operation in conditions that were not covered in the original set used to determine the principal component basis, 3. DATA RECONCILIATION The underlying idea in DR is to formulate the process model as a set of constraints (mass and energy balance, some constitutive equations). All measurements are corrected in such a way that reconciled values do not violate the constraints. Corrections are minimised in the least square sense, and the measurement accuracy is taken into account by using the measurement covariance matrix as a weight for the measurement corrections. Sensitivity analysis can be performed and is the basis for the analysis of error propagation in the measurement system (Heyen et al, 1996). With this technique, variations of some state variables can be linked to deviations in any measurement. A drawback of DR is the presence of a gross process fault (e.g. a leak): since the basic assumption of data reconciliation is the correctness of the model, it is efficient to detect and correct failing sensors, but it may be less adequate to detect process faults. In this case, the DR procedure will tend to modify correct measurements while in fact there is a mismatch between the model and the actual process.
4. PRESENTATION OF A COMBINED DETECTION METHOD 4.1. Projection Matrix The PCA orthogonal projection matrix is the key point of the fault detection method. In order to determine it, one might use either a raw data set, or reconciled values. The number of significant principal components (PC) obtained by way of the projection matrix determined with both data sets is very different. Taking into account more components allows explaining a higher fraction of the total process variability, as shown in figure 2 for an example with 186 variables. When using raw measurements to determine the projection matrix, a large number of components is needed to explain most of the process variability (upper limit is the number
i lliitlii=iiiiiii.l',il,
1 "
>'0.8
"
> 0.6 9
"~ 0.4"
t~ 0.2'
0:l
, ....
1 i
|
,
Data
i
|
set
Figure 1 : evolution of component #2 raw measurements
21
41
61
81
101
121
141
161
number of principal components
Figure 2 : explained process variability vs number of components
1;
759
of original variables). When using the reconciled data sets, the number of significant PC tends to a much lower number than the number of state variables (upper limit is the number of degrees of freedom of the data reconciliation model). This reduction in the problem size allows the system monitoring with only a few PC. The DR technique applied to the raw data sets has somewhat filtered them and reduced the noise. Furthermore, since data reconciliation enforces the strict verification of all mass and energy balance constraints, the linear combinations captured in the projection matrix are truly representative of realistic models. For example, the mass balance of a simple mixer is represented by the equation: F1 § with F1 and F2 as the inlet and F3 as the outlet flowrates, while statistical analysis of noisy measurements would lead to a correlation such as: a . F1 § b . F2 = F3 where coefficients a and b would probably differ from 1. Using DR as a preprocessor will ensure that the PC represent the proper process behaviour. For later analysis, we will thus use the projection matrix obtained from the reconciled values. 4.2. Confidence fimits The number of measurement sets for normal conditions has to be large enough in order to span all causes of process variability and to avoid spurious fault detection (i.e. in case of conditions that are acceptable, but were not available in the training set). A confidence limit has to be chosen to determine the number of PC to be used : choosing a 100% confidence level would be equivalent to use all the PC. Confidence limits are calculated on the base of a normal distribution for each of the PC. This limit will be used for fault detection purpose. A lower confidence limit on a component increases the sensitivity: potential faults are detected earlier, but erroneous detection is more frequent. The measurement precision must also be taken into account for the determination of these limits. When measurements are affected by large random errors, a fault can be detected even when there is no actual problem on the process. Thus repeated excursions of the component beyond the fixed limits will be needed to flag a fault. 5. CASE STUDY : A M M O N I A SYNTHESIS LOOP 5.1. Primary model and reference data sets A simulation model of an ammonia synthesis loop has been used to generate pseudo measurements. The model involves 186 state variables. A set of 210 reference operating conditions has been obtained by solving the simulation model repeatedly with varying specifications. The specifications were generated randomly from a Gaussian distribution. Variables assumed to be measured were then corrupted with a random noise of known distribution. Each raw data set generated in this way was also processed using VALI III data reconciliation package ~elsim 1999). In the present case study, we selected a number of PC allowing to represent 95 or 99% of the process variability. This corresponds to the first 20 and 27 PC (figure 2). Figure 1 shows an example of 90% and 95% confidence limits for the second principal component obtained from the raw reference data set, compared to its contribution in each of the 210 test cases. The null mean value of the component is also represented. A similar graph can be obtained from reconciled data. 5.2. Simulation of fault in process operation Data sets corresponding to process faults of variable severity have also been generated with a model modified to represent an equipment fault. The altered model is first set with
760
2O
\
10 C
lsg121cji
f~102o_21
Isg12oc~Ij
10
~
2O
Scoreon
~0
#4
50 30 7O 0
Figure 3 : modelling an internal leak in a heat exchanger
5
Data 10
15
21)
s 25
set
Figure 4 : evolution of component #4, Raw measurements
specifications leading to no fault at all. A data set is generated by solving the simulation model with random variations of the specifications, but with the fault effect becoming progressively larger. The measured variables are corrupted with normally distributed noise. A data reconciliation procedure, based on the primary, fault free model, is also applied to the data. Different fault types were generated for this study. For instance, to simulate the appearance of a leak in a heat exchanger, we used the model in figure 3. The exchanger is split in two halves, and part of the tube-side stream is split and mixed with the shell-side stream. A leak was also considered between process and utility flows in a heat exchanger, or a direct leak to the ambient and a deactivation of the catalyst in the conversion reactor. 5.3. Internal leak in a heat exchanger
A faulty data set is composed of time series of measurements of all the state variables. In the case of a leak, the leak flow is initially set to zero for the first element of the series, and its value is gradually increased with time. In this way, it is possible to evaluate the PCA fault detection method by analysing how quickly and how trustfully it responds to a problem arising in the ammonia loop. As a single excursion of a component out of its limits is not enough to detect safely the presence of a fault in the process. An error is only flagged when a component remains out of bounds during several consecutive steps. In this study, we identified the time at which a fault had been detected as the time a PC left its confidence limits to remain permanently out of bounds. The confidence limit for each component has been set to 99%, because the 95% value proved to be too sensitive and lead to spurious fault detection. In the case of a leak in the reactor product to feed heat exchanger, analysis of ten different time series gives the results in table 1. Principal component number 4 is the first to drilY out of its confidence limits, both when using raw or reconciled values. Figure 4 shows the evolution of that component and the confidence limits for one of the cases with raw data. The analysis of the correlation matrix between the state variables and the principal components allows locating the fault on the process. The fourth component is much correlated to the temperature of one the streams coming out of the exchanger. The NH3 composition of the other stream is also modified. On the five first variables correlated to this component, two
761 more are related to flows connected to the liquid ammonia separator. In the average, the fault has been detected for the seventh data set, and it corresponds to a leak of 6% of the gas flow. Table 1
Raw Data Meantime 6.8 9.5 11.8 11.9 12 Component 4 3 1 5 2 Reconciled data Meantime 6.6 11.6 15.6 19 19.8 Component 4 1 5 2 3
13.8 16.9 17.9 18.1 18.4 23 24 27 8 25 20.7 20.9 20.9 20.9 21 8.... 1 1 23
21 6
5.4. Exchanger leak between process and utility flows In this case, the leak is simulated in exchanger cooling the synthesis gas with liquid ammonia. This exchanger has been selected because it operates in parallel with another branch of the main loop. To model the leak, the original exchanger has been split in two smaller units performing the same duty when the leak vanishes. Table 2 shows that 4 principal components drift out of their confidence limits almost immediately. The analysis based on reconciled data flags the error somewhat earlier. The major PC implied are numbered 3, 7, 8 and 4. The large number of PC makes it difficult to locate precisely the cause of the fault. The detection method has been modified to get more information from the data sets. For each time step t, an index It is determined for each measured variable. It is obtained by summing up over a time horizon the score of all principal components drifting out of bounds, weighted by the variable contribution in the corresponding eigenvector. This index reveals any repeated departure from average for the variables mostly correlated with the drifting components. Applying this test in the previous example indicated that suspect variables were the partial flows of nitrogen and argon in the ammonia produced and the partial flow of hydrogen in the reactor feed and the reactor effluent.. Table 2.
Raw data Mean time Component Reconciled data Mean time Component
2 3
2 7
2 8
2.1 4
2.2 2
2.2 6
2.8 1
3.4 23
3.6 11
3.8 19
2 3
2 4
2 7
2 8
2.1 2
2.1 6
2.7 1
3.1 23
3.6 19
3.9 11
Monitoring the sign of the measurement deviations is also indicative. In our example, nitrogen and argon flowrates appear to decrease in the separator outlet, while hydrogen is rising in the reactor effluent. The temperature of the recycled flow is decreasing. Overall, the hydrogen partial molar flow is increasing in the reactor as well as the temperature. From all these observations it seems difficult to locate the fault in the process. However, a leak to the ambient can be considered, to explain the decrease of the inert (argon) flow rate in the loop. The mean detection time varies between 3 and 4, which corresponds respectively to leaks of 0.1 and 0.15 % from the synthesis gas flow. A fault is quickly detected but identification of the exact cause is not easy with the available data.
762
5.5. Other tests
The fault detection technique has also been applied to the detection of a leak to the environment, or to the analysis of catalyst deactivation. The method is able to detect the occurrence of the fault but the location of its cause is not always straightforward. 6. CONCLUSION The fault detection method exposed here, by combining data reconciliation with PCA, is promising. The first benefit is to reduce the number of variables needed to monitor the process. The monitoring of the process is made easier and the computing demand is decreased. The PCA method does not require any type of distribution of the original variables. However, the detection method is based on confidence regions estimated from a normal distribution. The proposed method of fault identification is a two step procedure. First a fault is detected when the score of a new measurement set results in some components lying out of their confidence region. In a next step, the effect of the latter components on the original variables is calculated. This allows sorting out the more altered variables, which are likely to be linked to the primary cause of the fault. Data reconciliation is recommended as a preliminary filtering step before the determination of the PCA projection matrix and the PCA correlation matrix between the PC and the original variables. These matrices are determined only once on the base of the reference data set. If the raw data were to be used for the determination of these matrices, a larger number of components would be needed to represent the data variability at the same confidence level, but the least significant components would be much affected by noise. However raw data should be used when using the method to detect faults. DR allows filtering of some data such as faults from measuring devices, and is likely to impede detection of a fault by correcting measurements. The quality of the model used is here very important. Overall, the fault detection method by principal component analysis is effective in all the cases studied here. The sensitivity of the method can be adjusted thanks to the confidence regions for each component. This method is a priori relevant to any kind of process. The precise identification of the fault location is not always possible. But one should remind that this study is only based upon simulated data. Taking into account the process dynamics and dead times in fault propagation might improve the capabilities of the method if measurements are sensitive and fast. REFERENCES
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) Heyen G., Mar~chal E., Kalitventzeff B., Sensitivity calculations and variance analysis in plant measurement reconciliation, Computers and Chemical Engineering, vol. 20S, pp 539544 (1996). Kresta J.V., MacGregor J.F, Marlin T.E., "Multivariate statistical Monitoring of Process Performance", The Canadian Journal of Chemical Engineering, 69, 35-47 (1991) BELSIM, VALI HI Users Guide, BELSIM sa, All6e des Noisetiers 1, 4031 Angleur (Belgium) (1999) Snedecor G., "Statistical Methods (fifth edition)", The Iowa State College Press (1956)
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
N o t e on v a p o u r
disengagement
dynamics
763
modelling
A. Sogaro, M.L. Caldi, D. Franchi, G. Biardi Dipartimento di Chimica Industriale e Ingegneria Chimica " G. NATTA" Politecnico di M i l a n o - Milano
ABSTRACT Aim of this paper is to gather some suggestions about the most consolidated tool in vapour disengagelnent dynamics modelling represented by the DIERS (Design Institute for Emergency Relief Systems of AIChE)) procedure (Fauske et al., 1983a; Forrest, 1992) in order to favour a more approachable handling by an unskilled user. After a brief comment on the experimental range of reliability and on the classification of the flow regimes foreseen, the following aspect are analysed: 9 the evaluation of the available correlations based on the drift flux theory; 9 the choice of the vessel flow regime and of the particular vent model to solve the problem aimin~, at the computation time saving" 9 the care in adopting a suitable time step of integration; All these aspects are connected with the obvious, but absolutely rigid constraint for the user: to be (in design phase) or to check to be (in simulation phase) within an acceptable bound of conservative options. To support this analysis, the simulation of a saturated water depressurization is reported. INTRODUCTION When an emergency relief occurs, the spectrum of (Fisher, 1992):
possible situations is the following
9 all vapour flow: generally such an assumption will yield a minimum relief area; 9 all liquid flow: it usually yields a oversized area, only in few cases, such as two-stages runaway reactions, the relief area can be understimated; 9 homogeneous (no slip) flow: the fluid entering the relief system is a homogeneous mixture characterized by the same void fiaction present into the vessel. Excepting some situations (such as in all liquid cases), regarding gassy or hybrid systems, this assumption will always lead to overconservative vent sizing; 9 paltiaily disengagement (slip) flow: the fluid entering the vent has different vapour quality from the one into the vessel, because of the slip between the phases and coalescence of the vapour bubbles. The vent size will lay between the two bounds represented by all vapour anct all liquid flow. DIERS procedure, after the check on the existence of two phase flow and after the exclusion of the homogeneous flow assumption, on the basis of the experimental conditions and fluid properties, works adopting the vapour partial disengagement assumption; the mass and energy
764 balances of the integrated vessel-vent system are solved by means of the so called "coupling equation", representing a vapour mass balance written at the vent entrance. The procedure is applicable to top-vented, vertical, cylindrical vessels with constant cross area; it properly refers both to nonreacting systems (externally heated or not) and tempered reacting systems. This body of considerations, focused on the simulation phase, applies, obviously, for the design phase too. T H E O R E T I C A L BACKGROUND The DIERS analysis ( Fauske eta., 1983a; Fisher, 1992)) is founded on a drift flux approach starting from the following simple function derived from the initial analysis of Wallis (Wallis, 1969): Jgf= (x(1- c~)nU~176 (1) 1-a m where: Jgf is the superficial drift velocity, (x represents the local void fraction and U oo (bubble rise velocity) is given by: U oo= k [crg(pf
_pg)~.25(pf)-0.5
(2)
where: pf= liquid density at stagnation temperature;
pg= vapour density at the stagnation
temperature; cr = surface tension; g = gravity. For the flow regime taken into account, Bubbly (B) and Chum Turbulent (CT), DIERS proposed, for what concerns the coefficent k, m, n, the following values: k=l.18, m=3, n=2 (B) and k=1.53, m=oo, n=0 (CT); the fluid properties appear in eq. 2 The flow regime, depends substantially on fluid viscosity, in any case B regime always represents the most conservative assumption. Assuming open systems, volumetric vapour source and steady state conditions (liquid volume flux Jf=0) the following approximated vapour holdup relation is obtained:
Jgoo B
V
=
- -
Uoo
CT
~(1- ~) 2 -
-
(3)
(1 - ~3)(1 - C O~)
Jgoo 2or V : -- - - - _ Uoo (1- Coa)
(4)
where Jgoo stands for the vapour superficial velocity at the liquid surface, tg for a dimensionless superficial velocity, ot for the average void fraction and Co for a parameter introduced to account for the radial distributions effects (Zuber and Findlay, 1965). Its value, depending on flow regime expected and on a more or less conservative choice, lies between 1 and 1.5 (Fisher, 1992).
765 A submodel of CT regime allows for the presence of a nonboiling volume fraction due to hydrostatic head and recirculation effects (Fauske et al., 1983a). This results in a vertical temperature profile, otherwise speaking, a top-biased vapour generation occurs and the vapour disengagement increases (Forrest, 1992). In this case, the relationship between c~ and the average void fraction in the boiling region, &, is given by: A
= [1-(Hnb/H0)]~
(5)
A
1-(Hn /H0) Where Hnb is the nonboiling height and H 0 is unaerated rest height of liquid. A further step, under the same assumptions, allows the computation of tXmax, (void fraction at the swollen liquid surface), for which DIERS proposed the following empirical expression: B
(x max=0~
CT
tx
2C~ = ~ max 1 + Cor
(6) (7)
Now, the extension of the previous drift flux analysis from open systems to close, vented systems is assumed. The coupling equation, assuming a non-zero value of the liquid flux, can be written down and represents a vapour mass balance at the vent entrance, as previously mentioned; the formulation proposed by Forrest (Forrest, 1992) is:
Xe.A v .G= Jgoo "Pg "Acr + Xmax(Av" G-Jgoo "Pg "Acr)
(8)
where" Acr and Av are respectively vessel cross-sectional area and vent area; Xe represents the vapour mass fraction enterig vent line; G is the critical specific flowrate; Xmax is the vapour quality at the liquid surface evaluated by the following equation:
Xmax=
(XmaxPg ~maxPg+(1-a;max)Pf
(9)
The unknowns in the coupling equation are Xe and G: these two terms are evaluated, at any given stagnation pressure, by solving simultaneously the coupling equation with an appropriate two-phase vent model. DIERS procedure has, schematically, the following structure: Step 1: starting from an initial all vapour assumption and using an appropriate vent flow model, a mass flow rate is evaluated. n
Step 2: from equations 1,2 and 3, or 4 (depending on the regime assumed), a value of r is obtained and it is compared to a vessel void fraction, etves , found from the liquid inventory.
766 If a < ~ves all vapour flow occurs and the procedure is stopped. If a >ave s two phase flow occurs and the procedure continues, that is: a = (Xves and J goo can be updated by equations 3 or 4. Step 3: the void fraction at the liquid surface O~max is calculated by means of equations 6 or 7.
Step 4." Xmax is evaluated by equation 9. the unknowns being Xeand G, the system constituted by the coupling equation and a suitable appropriate vent two phase model can be solved; as for the detail of coupling equation rooting, a sufficiently deep discussion has been developed by Fauske et al. (Fauske et al., 1983a). At the Step 4 X e and G are known at the vent entrance Step 5." for a complete and explicit description of the dynamics, the energy balance (Fauske et al. 1983a) is to be solved in order to update the stagnation temperature:
aT q EXe d-~ = C--~-
M Cm
(10)
where: q=possible specific heat source, C m=constant volume liquid specific heat Cd=discharge coefficient, T=stagnation temperature, M--total mass, hfg=latent heat, Vfg=liquid/vapour specific volume difference. At this point, due to the time evolution of M and T, it is possible to update the global stagnation conditions and to restart from Step 2 till an all vapour condition is reached (Xe:l). NUMERICAL APPLICATION The reported application is the case Tlc of DIERS Phase III Large Scale Integral Test (Fauske et al., 1983b); on such an experimentation, SAFIRE code has been validated (Fauske et al., 1984). The example refers to the simulation of a saturated water depressurization so characterized: vessel volume: 2.190 m3; vessel cross sectional area: 0.656 m2; relief initial pressure: 505. kPa, nozzle diameter: .0508 m; Cd=l. The time step adopted for integrating equation 9 is 20 seconds. Two different vent models in solving the integrated vessel-vent system have been compared: Homogeneous Equilibrium Model (HEM) (Huff, 1985) and f2-method (Leung, 1986). The flow regime assumed is chum turbulent. The condition of complete vapour disengagement is represented in each figure by the last simulated point (Xe= 1). Figure 1, reporting c~ versus time, shows a comparison between the two vent models adopted considering both the pure chum turbulent regime and its modified form allowing for nonboiling height. Figure 2 reports vent mass flux versus time adopting f2-method and comparing the pure churn turbulent regime and its modified form allowing for nonboiling height. Figure 3 shows HEM and f2-method performances adopting only the churn turbulent nonboiling height form.
767
AVERAGE VOID FRACTION vs TIME
........
0.5
0.3
............
,. . . . . . . . . . . . . . . . . ,. . . . . . . . . . . . . . . .
i........................
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: ......
, ...........
~__i.~.
....
9
.
.
.
.
0.2
0.1
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50
100
150
200
250
300
350
"I]ME s F i g u r e 1. VENT ~ S 3500 3000
VESSEL PRESSURE vs TIME
FLUX vs TIME 550O00
! :
' 'Om~' 'Om~ nonbo~g'
......... i............ ',............,"...........~............:...........~,............
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..............
"~!
450O0O ......... i
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~ :. . . . . . . . .
........ i .......i
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'
i ........................
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: :
"dem nonboilil{g ---,-'Omega nonboing ---+-
i
zxp~m~' o
i ........... :, ........... :,..........
i i
i
i
i
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!
!
i
:
',
:
:
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i
~;
',
',
*' 50
i
i
i
i
i
100
150
200
250
300
o
250O0O 2O00OO 5OO
150OO0 ,
i 50
,
,
100
150
200 TIMEs
Figure 2
i 250
i 300
350
1OO00O 0
350
TIMEs
Figure 3
T h e n o n b o i l i n g h e i g h t a s s u m p t i o n d e s c r i b e s in a v e r y m o r e realistic w a y the d i s e n g a g e m e n t d y n a m i c s , r e d u c i n g the risk o f v e n t o v e r s i z i n g (see F i g u r e 1 and 2); it loses partially its effectiveness as d e p r e s s u r i z a t i o n p r o c e e d s b e c a u s e o f the d e c r e a s i n g o f h y d r o s t a t i c h e a d and, accordingly, the n o n b o i l i n g v o l u m e fraction disappears, as can be o b s e r v e d in F i g u r e 2.
768 FINAL CONSIDERATIONS On the basis of the results, it is possible to say: 9 HEM and O-model, from the point of view of experimental data fitting, give essentially the same performances, but the choice of O-model becomes very attractive because of the significant save in computation time (ratio 1/14). This consideration becomes more and more important in describing the complete dynamics (simulation phase) whilst, in design phase, the attention is focused only on the discharge onset. A warning is needed in adopting f2-method: its extension to multicomponent mixtures with a broad range of boiling temperatures as well as to more complex organic molecules is still a research object. 9 The choice of the time step deserves special considerations: as it decreases, a better description of the dynamics can be obtained (the constant flowrate assumption holds for a smaller interval time at each integration step) but, obviously, the computation time becomes heavier. As a further effect, a numerical instability in computing Xe in the last simulated points (low pressure and semiempty vessel) can arise. Such a problem, even if without any macroscopic effect on pressure and c~ evaluation, would require a deeper investigation. 9 The approximated drift flux correlations adopted by DIERS seem to be largely acceptable at least for constant cross sectional area geometry (cylindrical vessel). The synthesis effort here proposed in describing the essential characteristics of such complex problems, doesn't exempt the unskilled user from taking extreme care in applying this modelling tool in view of the extreme variability of the real possible scenarios.
REFERENCES:
1. Forrest H. S., 1992, The coupling equation and flow models, Emergency Relief Systems Design using DIERS Technology, AIChE Pubblication, p.5. 2. Fauske H. & Associates, 1983a, Emergency relief systems for runaway chemical reactions and storage vessels: a summary of multiphase flow methods, DIERS Report FAI/83-27. 3. Fisher H. G., 1992, Vapor disengagement dynamics, Emergency Relief Systems Design using DIERS Technology, AIChE Pubblication, p. 1. 4. Wallis G.B., 1969, One dimensional two phase flow, Mc Graw-Hill Book Company, Chapter 4. 5. Zuber N. and Findlay J.A., 1965, Average volumetric concentration in two phase flow system, Trans. ASME, J. Heat Transfer, Vol. 87, series C, p. 453. 6. Fauske & Associates, Inc. , 1983b, Phase III Large Scale Integral Test, six reports numbered DIERS III-1 to DIERS 111-6. 7. Fauske & Associates, Inc. ,1984, SAFIRE Users Manual Vol I-VII, DIERS Report FAI/ 84-5 8. Huff, J.E., 1985, Multiphase flashing flow in pressure relief systems, Plant Operations Progress, 4, pp. 191-199. 9. Leung, J.C., 1986, A generalized correlation for one component homogeneous equilibrium flashing choked flow, AIChE J., 32, No 10, pp. 1743.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
769
Computer aided transportation risk assessment Roberto Bubbico", Sergio Di Cave b, Barbara Mazzarotta b "Gruppo Nazionale Difesa Rischio Chimico-Industriale ed Ecologico, C.N.R., Via Tiburtina 770, Rome, 00159, Italy b Department of Chemical Engineering, University of Rome "La Sapienza", Via Eudossiana 18, Rome, 00184, Italy The assessment of the hazard represented by the transportation of dangerous goods is the only reasonable basis for any policy of risk management and reduction. The use of such a quantitative approach, on the other hand, requires both the acquisition and the manipulation of a very large number of information: these steps are always time-consuming and seldom very accurate, since the need of limiting the calculation burden generally imposes the use of simplifying assumptions. This work presents a computer aided approach to transportation risk analysis coupling a risk assessment program with a Geographic Information System (GIS), providing accurate local information. The obtained benefits are more accurate risk estimates, a substantial cut of the time required to perform the analysis, a simplification of the data input step, and the possibility of displaying the results on the area map, together with other information, useful in the case of an emergency (location of fire brigades stations, hospitals, etc.). 1. INTRODUCTION The transportation of hazardous substances represents a significant risk source, which, in some cases, is comparable with that of fixed facilities [1]. The risk of such activities can be estimated by means of Transportation Risk Analysis (TRA), which is largely based on Quantified Risk Analysis (QRA) methodologies, developed for the chemical process industry [2]. However, the application of TRA to a practical case gives rise to a number of problems, primarily due to the fact that the risk source is moving: this means that most of the involved parameters change along the itinerary, thus obliging both to determine their values in a great number of progressive on route locations, and to repeat the calculations every time these values change. This point represents a great obstacle to a wider use of TRA: in most cases the problem is managed by dividing the route into rather large portions where the parameters are assumed to remain constant [3]. Of course, this approximate approach reduces both data acquisition and calculation burdens, but introduces additional uncertainties in the risk estimates. Moreover, any change in the itinerary requires both the acquisition of the data and the performance of the calculations for the new route, thus representing a work not much different from that of studying a completely new case: this circumstance does not encourage to examine alternative (and possibly less hazardous)
770 itineraries. As a matter of fact, TRA papers recently published report study cases relevant to well-identified products and routes [4], propose rigorous methodological approaches requiring a great number of information to be applied [5] or suggest simplified approaches to speed-up the analysis and to examine the overall problem for a whole country [6-8]. Geographic Information Systems (GIS) represent a relatively new tool with a great potential in a number of applications, including QRA, since they allow to manage databases related to territorial entities (plants, towns, roads, rivers, etc.). The application of GIS to area risk studies [9], risk analysis of pipelines [10] and of road transport of dangerous substances [11] has been recently suggested. However, commercial GIS do not cover the information needed for a TRA, and the problem is still that of data acquisition and databases preparation. This work proposes a computer aided approach to TRA aimed at improving both speed and accuracy of risk estimates. This result is obtained by coupling a risk assessment program and a GIS. The proposed approach allows to perform a TRA in a rather short time, using accurate local information, and getting other useful territorial information, such as the location of emergency services. 2. TRANSPORTATION RISK ANALYSIS A TRA properly concerns the transportation of dangerous goods: basing on this assumption, other phases, such as loading or unloading are not taken into account. At the same time, the hazards of concern are those deriving from the release of the transported substance into the environment, while those arising from the use of the transportation mean (such as the death of the driver or other persons involved in a road crash) are disregarded. Depending on the hazardous nature of the transported product (flammable and/or toxic), the amount and the physical state (gas, liquid, etc.) of the spill, the meteorological and environmental conditions, a number of different outcome cases (pool fire, toxic cloud, etc.) may follow the release. The risk can be evaluated by combining the frequencies of occurrence of each outcome case with the extension of the relevant impact zone, estimated by applying consequence analysis procedures [2]. In TRA both individual and societal risk measures are usual. The individual risk, IRx.yis defined as the probability that an individual will die in a given period of time by the consequences of the transportation hazard at a specific geographical location. For each portion of route, the individual risk IRx.y, can be expressed as [3]: n
IRx.y = T. A. Z R i i=i
m Si 9Z L i , j . Wj. Z Pi,j,k j=i k=l
(1)
being T the number of trips per year, A the local accident rate per kilometre, ~ the release probability for the i-th release size, Li.i the length of release location zone j, Wj the probability that wind blows in the direction of concern, P~i.k the probability of a fatality at location x,y given that accident outcome k occurs. The societal risk is usually given in the form of F-N curves, relating the cumulative frequency F of all the possible accidental events causing a number N of fatalities, vs. this latter. The values can be calculated from the following equations:
771
gg,i.k = T - A g - R i - L g
"Pi,k
Ng.i,k = CAi, k 9pD 8 . PFi, k
(2)
(3)
where CAi. k is the consequence area associated with incident outcome k, PD~ the population density for segment g, and PFi~ the probability of fatality. The results of these calculations are the frequency of incident outcome k for release size i on segment g (Fg.i.k), and the associated number of fatalities (Ng.i.0. The variables appearing in eqs.(1-3) depend on information related to the transport case (substance involved, physical state during transport, amount transferred per trip, number of trips, type of tanker, typical release scenarios, possible evolutions of the release scenarios, etc.), to the characteristics of the territory along the route (meteorological conditions, prevailing wind direction, type of environment, population density, etc.) and of the route itself (type of road, accident rate, etc.). Some of the above factors, such as weather conditions, accident rate, population, etc., are actually variable along the itinerary, and the extension of the zone at risk, as well as the number of individuals exposed to the risk, should be recalculated any time they change. Such a rigorous approach generally represents an unaffordable task; therefore, in current practice the route is divided into a manageable number of segments [3]. However, even in this case, a TRA is a time-consuming task, since all the relevant data have to be collected and organised, and a great number of calculations have to be performed to assess the consequences of all the outcome cases under each assumed environmental condition. 3. PROPOSED TRA M E T H O D O L O G Y All the variables involved in a TRA can be roughly divided into route-independent and route dependent ones. Route-independent parameters include data about the transportation case, about the substance, and about the accidental scenarios and their expected evolution: such information can be obtained from the literature, from the statistical analysis of historical data, or using proper techniques (such as fault-tree or event-tree analysis). Routedependent parameters, which are typical of a TRA, include data about the local accident rate, weather and environmental conditions, and population: such information can be obtained at a number of locations along the route or sufficiently close to it, and should be associated to their respective geographical location. Since this type of data can be effectively handled by means of GIS, the integration of a risk assessment program, giving the routeindependent data, with a specifically tailored GIS appears as a suitable mean for performing TRAs based on accurate territorial information. A GIS and a TRA program were then developed, separately tested, and combined to give the integrated GIS based approach to TRA. The GIS program MapRisk, based on ArcView software, was specifically tailored for TRA applications [11]. MapRisk coverage includes the whole Italian road network, divided into 1 km segments, the population of built-up areas [12] and the location of the meteorological stations [13]. The associated databases contain the accident rate relevant to
772 each 1-km portion of main roads and motorways [14-15] and the weather data (temperature, wind velocity, etc.) recorded at the meteorological stations [13]. The risk assessment program, TrHaz, performs the individual and societal risk calculations, basing on transport case data, product related information, and weather, accident rate and population density data for each portion of the route [16]. The product database contains information about the probability of the assumed release scenarios, the probability of the final outcome cases for each release scenario, and the expected impact zone for each outcome case. The first data are based on the analysis of historical records of accidents [3,17-18] and the last ones are derived from the results of consequence analysis, carried out independently using a commercial software [19]. By separating the consequence analysis step, which is very long, from the risk assessment one, the latter becomes much faster; however, using this procedure, the impact zone extension is estimated for some average weather condition, possibly different from the actual one. To overcome this inconvenience, the consequence analysis calculations were repeated for 6 different combinations of temperature and wind velocity, selected as representative of the most usual weather conditions in Italy during the year. The proper impact zone data are then used by the risk assessment program, depending on the route and the period(s) of the year chosen for the trip(s). The integration of GIS and risk assessment programs, schematically shown in Figure 1, gives rise to the overall tool, TrHazGis, performing the computer aided TRA.
Input I I
Product
I
Route
Product database l ]
~ ...... 1Probability of release scenarios "1 Probability of outcome cases Impact zone of outcome cases
Period of the year
GIS q~ Weather database Accident database
Risk Assessment "1
Individual risk
]~
§ __.~l Population database I
~.1
Societal risk
Figure 1. Schematic diagram of the proposed TRA approach.
[
773 The input phase include the selection of the product, of the route and of the season of the year. The route is automatically subdivided into 1 km portions, and the weather conditions, population density and accident rate values relevant to each segment are taken from the databases. The information about the weather condition of each segment is used to select the proper impact area for each release scenario and outcome case, which, combined with the respective probabilities and with the accident rate of the segment under exam, allows to estimate the individual risk as a function of the distance from the route. By integrating this information with that concerning the population density in the impact zones, and repeating the calculations for each segment of the route, the societal risk associated to the trip(s) is also evaluated. The whole procedure requires a few minutes to run on a Pentium II PC, and the results are directly available in graphical form. The proposed approach ensures that the TRA is carried out using accurate local information, which can be easily updated, thus increasing the reliability of the results. The obtained risk measures can be used to verify the compliance of dangerous substances transportation activities with existing regulations, in the case of the societal risk, or to identify hazardous spots along the route, in the case of the individual risk. Moreover, using the GIS, the risk deriving from this type of transport can be compared, and combined, with that deriving from other sources to give the overall risk in a certain area. Of course, the risk measures can be used also to compare the risk of different itineraries, and of different transportation means (a similar study for rail transport is being carried on). However, the information obtained using the proposed approach is not limited to the classical risk measures, since the GIS represents a useful support also for managing transportation emergencies. In fact, in case of an accident, it is possible to locate directly on the GIS the impact zones, taking into account the actual wind direction: this allows to specify the zone to be evacuated, the roads to be blocked, the emergency forces to be activated, etc. 3. CONCLUSIONS The proposed approach for a computer aided transportation risk analysis effectively combines route-independent and route-dependent parameters in order to obtain reliable risk measures suitable to verify their compliance to proper absolute thresholds or to compare different risks. The program TrHazGis is structured in order to perform rapidly and accurately the calculations: to this end, the time-consuming consequence analysis is carried out separately, for a number of different weather conditions, and the results are collected in a database. The short time needed to get the results and the huge local data bank provided enable to very effectively use TrHazGis for comparing the risk of different itineraries. This can greatly help the management of the hazard represented by the transportation of dangerous substances on a sound scientific basis (public authorities can prohibit or impose restrictions to the transit along a certain itinerary). The use of TrHazGis also allows to display on the GIS map iso-risk curves and impact zones, which can be compared with other data layers, such as population density, available routes, emergency services location, etc. This type of information is particularly valuable in performing risk studies of industrial areas, including chemical process industry, transportation, etc.
774 ACKNOWLEDGEMENTS The contribution of M.Conforti to part of the computational work and the financial support of the Italian National Research Council (C.N.R.) are gratefully acknowledged. REFERENCES 1. D. Egidi, F.P. Foraboschi, G. Spadoni and A. Amendola, Rel. Eng. and System Safety, (1995) 75. 2. CCPS, Guidelines for chemical process quantitative risk analysis, New York, AIChE, 1989. 3. CCPS, Guidelines for chemical transportation risk analysis, New York, AIChE, 1995. 4. R. Bubbico, G. Dore and B. Mazzarotta, J. Loss Prev. Proc. Ind., 11 (1998), 49. 5. S. Bonvicini, P. Leonelli and G. Spadoni, SRA 1998 Annual Conf. "Risk analysis: opening the process", Paris, 1998, vol.2, 717. 6. G. Tiemessen and J.P. van Zweeden,. Proc. 9-th Symp. on Loss prevention and safety promotion, Barcelona, 1997, 299. 7. S.A. Gadd, D.G. Lemming and T.N.K. Riley, Proc. 9-th Symp. on Loss prevention and safety promotion, Barcelona, 1997, 308. 8. R. Bubbico, S. Di Cave and B. Mazzarotta, 16th ESReDA Seminar on safety and reliability in transport, Oslo, 1999. 9. F. Bellezza, S. Contini, M. Binda and G. Spadoni, Safety and reliability, Eds. Lydersen, Hansen &Sandtorv. Rotterdam, Balkema vol. 1 (1998) 67. 10. G.Uguccioni, R.Fiore, M. Sinisi and U. Minelli, Safety and reliability, Eds. Schu~ller & Kafka, Rotterdam, Balkema, 199, 421. 11. R. Bubbico, S. Di Cave, and B. Mazzarotta, Chemical industry and environment HI, Ed. R.Zarzycki & Z.Malecki, Krakow, 1999, vol.2, 333. 12. ISTAT, 13 o Censimento generale della popolazione e delle abitazioni, Roma, 1992. 13. ISTAT, Statistiche meteorologiche anni 1984-1991, Ann. n.25. Roma, 1994. 14. ACI-ISTAT, Atti 52 ~ Conf. del traffico e della circolazione. Stresa, 1996. 15. ACI, Analisi dell'incidentalit~ stradale a livello nazionale e regionale, provinciale e nei comuni con oltre 250.000 abitanti. Roma, 1997. 16. R. Bubbico, S. Di Cave and B. Mazzarotta, SRA 1998 Annual Conf. "Risk analysis: opening the process", Paris, 1998, vol.2, 665. 17.Health & Safety Commission, Major hazard aspects of the transport of dangerous substances, London, HMSO, 1991. 18. OSH-ROM, HSELINE, C15DOL, MHIDAS, NIOSHTIC. London: Silver Platter, 1998. 19. DuPont Safer System, TRACE 8 User Guide. Westlake Village, 1998.
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rightsreserved.
775
Using PHA Results for Real Time Operator Support during ASM Sourabh Dash and Venkat Venkatasubramanian* Laboratory for Intelligent Process Systems, School of Chemical Engineering Purdue University, W. Lafayette, IN 47907, USA Process Hazards Analysis (PHA) and Abnormal Situation Management (ASM) are important activities aimed at plant safety and maintenance in the chemical process industries (CPI) today. While PHA is carried out @line, ASM monitors the plant on-line. However, the inherent objectives in both of them are similar- to identify hazards, try to avoid/mitigate them and plan for emergencies. PHA results are a valuable source of information from which ASM could potentially benefit. An integrated framework, combining these two methods is proposed here. In order to manage the large number of results in a systematic and organized manner, use of a hierarchical representation of the plant is described. An automated methodology based on topology and function for the same has been developed. The application of the framework is illustrated using an industrial case study. 1. INTRODUCTION Modern chemical plants are highly complex and integrated, processing large volumes of materials and operating at extremes of pressure and temperature. This makes them susceptible to various failures that can lead to hazardous situations such as the accident at Bhopal, India (Lees, 1996). Hence there has been an increased awareness in the academia and industry alike to develop methods that prevent and mitigate occurrence of such situations. Process Hazards Analysis (PHA) and Abnormal Situation Management (ASM) are two such methods that are used by industrial practitioners to improve the design, performance, and ensure the safety of a process. These methods are briefly discussed below: Abnormal Situation Management (ASM) An abnormal situation is any departure of a process from its acceptable range of operation. Abnormal situation management (ASM) involves dealing with these situations through timely detection, diagnosis and countermeasure planning during on-line operation. It is an important part of safe and optimal operation of chemical plants. An estimated $20B is lost annually by the petrochemical in the US alone due to insufficient ASM (Nimmo, 1995). An effective ASM methodology that properly addresses these issues can thus have significant economic and safety impact. There have been several techniques proposed to address the individual issues of ASM, especially fault detection and diagnosis (Venkatasubramanian et. al, 1995). Mylaraswamy and Venkatasubramanian (1997) describe a hybrid, distributed, multiple-expert based blackboard framework for fault diagnosis called Dkit which was shown to successfully diagnose different fault scenarios in the Amoco Fluidized Catalytic Cracking Unit (FCCU). Realizing the importance of ASM, Honeywell launched an industrial consortium (ASM Home Page, 1995) of major oil companies, software vendors and university-based research groups to develop the next generation process control system called Abnormal Event Guidance and Information System (AEGIS). The blackboard framework of Dkit has been utilized for the design and development of the AEGIS prototype.
Process Hazards Analysis (PHA) PHA deals with the systematic and proactive identification mitigation and assessment of potential process hazards, which could endanger the health and safety of humans and cause serious economic losses. This is carried out off-line. It is an important activity in Process Safety Management (PSM). The importance of PHA was recently underscored by OSHA *Author to whom correspondence should be addressed
776 PSM standard title 29 CFR 1910.119 (OSHA Home Page, 1994). A wide range of methods such as Checklist, What-If Analysis, Failure Modes and Effects Analysis (FMEA), Fault Tree Analysis and HAZOP (Hazards and Operability Analysis) are available for performing PHA (CCPS, 1985). HAZOP is the most widely used and recognized as a preferred PHA method by the chemical process industries. It is a difficult, labor- and knowledge-intensive activity that can benefit from automation in several ways. An automated HAZOP analysis system would reduce the time and effort involved in a HAZOP review, make the review more thorough and detailed, minimize or eliminate human errors, facilitate documentation for regulatory compliance and make the study results available on-line. A digraph model-based expert system called HAZOPExpert (Vaidhyanathan and Venkatasubramanian, 1995, 1996) for continuous processes has recently been reported to have successfully emulated the human experts' reasoning and identify all hazards on several industrial case studies. In this paper an integrated framework that combines the relative strengths of ASM and PHA and its application to an industrial case study is described 2. INTEGRATION OF ASM & P H A - MOTIVATION AND ISSUES Motivation: The motivation for integrating ASM and PHA mainly stems from the fact that the
inherent objectives of both is to deal with situations that are outside the normal behavior of the plant. They try to prevent, mitigate and plan for emergencies. Their complementary nature is shown in Figure 1. While ASM tries to reason from symptoms to causes, PHA is aimed at examining the consequences of the various deviations. The results from a PHA can be used to assist the diagnostic systems in their task. They represent information concerning the plant's behavior and safety characteristics that is already available. So, if the results of an available safety analysis can be used to supplement the diagnostic systems, there will be less need for the generation of separate process information sources for them, thereby making them more useful.
Figure 1: ASM and PHA PHA Results (Cause & Consequence information): The PHA results comprise of deviations, their causes, consequences and possible corrective actions that need to be taken in the event of an abnormal situation. As can be imagined, these results can provide crucial information during on-line monitoring of the process to ward off any impending danger by helping the operator act in advance. Since hazards analysis' results are required by law for most chemical plants and hence is available early in the design stage, an additional effort to manually construct this knowledge for ASM is unnecessary. Also HAZOP results often include operability issues and these can be used to track production and quality. The PHA results are also required by the OSHA regulations to be retained throughout the life of the plant. The results are required to be updated periodically and whenever any substantial modifications are made to the process that demand a review of the existing results. Currently in the process industry, the HAZOP results are entered in a spreadsheet by the team and these results are printed out for regulatory compliance and stored with the manuals. In the next section a hierarchical representation of the plant is developed in order to address these issues. 3. EFFICIENT ORGANIZATION & DISPLAY OF PHA RESULTS - HIERARCHICAL REPRESENTATION
A hierarchical representation of the plant would facilitate efficient storage of the PHA results as well as make fast and effective display of the same possible. This would allow the operator access the results at different levels in the hierarchy making the person more cognizant of the process
777 situation as a whole allowing him/her to make crucial decisions pertaining to countermeasure planning in the event of abnormal situations. One can look at the process of hierarchy construction as either abstraction - grouping units together or disaggregation- breaking down systems into its components (Stephanopoulos et. al, 1990). The former is a bottom-up approach while the latter a top-down. Here, a hierarchical scheme and an automated model-based bottom-up development strategy for the same is proposed. The framework builds the hierarchical representation consisting of data structures called logical units (representation of the plant at different levels) and relationships, which relate the logical units across levels. The inputs to the framework consist of the following: functional classification of equipment,
connectivity information from P&ID and configuration models (from a model library). F R A M E W O R K INPUTS
Functional Classification: The equipment are classified based on their function as shown in Figure 2. The UNIT-OPERATION-EQUIPMENT carry out physical and chemical transformation e.g., distillation column, reactor etc. The numerous receivers, tanks, and accumulators etc., which don't directly serve in the processing but aid the UNIT-OPERATION-EQUIPMENT accomplish their goal, are termed the AUXILIARY-EQUIPMENT. Together they are grouped as TRANSFORM-EQUIPMENT. The pipes, valves and pumps whose main purpose is to transport material are classified as TRANSPORT-EQUIPMENT. ;i@ii;k::iiiN::i:i 9 The sensors and various indicators that .................~........................ measure the variables in the process are .................................................................................................................... TRANSP()RT :i::R;~:Nsi:;6i(~:i MONITOR '....CON' .........[.ROI.: ............... classified as MONITOR-EQUIPMENT. The ................................................................. ...................~.................................................................................................controllers ......... are grouped as CONTROL~• ...................................................
...................................
~ ...............................................
i
,.[3.N !.i.!;L0!! [!}.R:A.i!i.[0 [NII
i .............................
~t .........................
i AUXILI,,\RY
EQUIPMENT.
...........................................................
Figure 2" Functional Classification Logical Units and Relationships: The logical units in the proposed hierarchy are: Equipment, Pipeline/Control-system, Input-Output-Unit go-unit), System/Sub-system and the Plant For example, the pipeline logical unit consists of TRANSPORT-EQUIPMENT and serves the function of transportation. The different levels in the hierarchy are related by means of relationships. For example, the relation is-the-io-unit-of-system relates a IO-UNIT to a SYSTEM. It is a many-to-many relation meaning that there could be more than one IO-UNIT related to more than one SYSTEM and vice-versa. The hierarchy is shown in Figure 3. L EVEL 5 !
"\
"~ i$-the-i~176
l
i
i
i
s ~ "~'~t,
/ ......
-~~,
LEVEL 4 ,,
L ......
HI IERAP, C H I C A L O E r - _ , O M P O S I T I O N B A S E D
Ol'-I F U N C:T'I O N & C: O N N E C T I V I I ' " (
Figure 3: Proposed Hierarchy & its Construction
Level 1. Equipment level: Consists of all equipment. This is the bottom-most level.
778
Level 2. Pipelines~Control-Systems level: Consists of pipelines (contiguous train of transformationequipment) and the control-systems (essentially the control-loops). Level 3. Input-output-unit level: The transformation-equipment and their input- and output-pipelines comprise this level. The control-systems are related to the io-units using is-the-cs-of-io-unit and to the systems/sub-systems using is-the-cs-of-system. Level 4. System/Sub-system level: The systems and the subsystems identified are included here. It is related to the above level by the-system-consists-of-io-units. The systems are related to the subsystems using the-system-consists-of-subsystem. Level 5. Plant level: The top level. There is only one plant. It is related to the above level through the relationship the-plant-consists-of-systems. Configuration Models: These models describe the manner in which UNIT-OPERATION-EQUIPMENT such as distillation column, reactor, absorber, stripper all achieve are configured/arranged with the AUXILIARY-EQUIPMENT. They are used to identify the SYSTEM and SUBSYSTEM logical units. For example, the configuration model of the distillation column would include the distillation column, with the reboiler at the bottom, the condenser and the reflux-drum at the top making the DISTILLATION-SYSTEM consist of the REBOILER-SUBSYSTEM and the CONDENSOR-SUBSYSTEM. The algorithm for the bottom-up construction is shown in Figure 3. At Level 1 the plant consists of all equipment. These are then grouped into the four classes of equipment discussed earlier. Using connectivity information from P&ID, TRANSPORT and MONITOR/CONTROL equipment are further grouped to form the pipelines and control-systems at Level 2. Level 3 consists of the UNITOPERATION equipment and the AUXILIARY equipment along with the connected pipelines and control-systems to form UNIT-OP-IO-UNITs and AUX-IO-UNITs. These IO-UNITs are then grouped using the configuration-models to form the SYSTEMs and SUB-SYSTEMs at Level 4. The various logical units are related by means of relationships. Level 5 consists of PLANT. The PHA results are stored by relating them to the different logical units resulting from this representation of the plant. 4. T H E I N T E G R A T E D F R A M E W O R K
In this section an integrated framework for ASM and PHA is presented. The proposed framework is shown in Figure 4. Broadly, the framework consists of: 9 A monitoring system to monitor the states of important process variables 9 A structured database of PHA results, which essentially is a safety model of the plant. 9 A system to retrieve and display the results Its off-line and on-line components are explained below in some detail.
Figure 4: Integrated Framework OFFLINE
PHA "state estimator": The plant is analyzed off-line for hazards during the PHA. This component contains the organized database of PHA results (causes and consequences), stored per the hierarchical representation developed earlier and the retrieval methods. The retrieval methods are search methods, which search the organized database to find the causes and consequences for the
779 detected deviations. To find the causes for example, they would search the database to find causes that can explain all the detected deviations. These are then sent to the DKit module. Similarly they also find the consequences of the deviations. Automated hierarchy construction module." This implements the representation developed in the last section. The PHA results are stored by utilizing the representation developed in the last section. ONLINE Monitoring & Detection module: The plant is monitored on-line for any kind of abnormal deviations in the sensors. The detected deviations are sent to the "PHA State Estimator" and the "Suite of Diagnostic Methods" described below. Suite of Diagnostic Methods (DKIT)." A suite of diagnostic methods is used since it is known that no single method measures up well on all diagnostic criteria. It has an open architecture and both process model-based and process history-based methods can be implemented here. The input to this module consists of the detected deviations and the causes from the PHA "state estimator". The potential list of causes can then be generated based on some conflict resolution scheme and ranked by confidence, which are then sent to the Results Manager. Results manager module." The user interacts with this module. The causes and consequences of the different deviations are posted here. This module also uses the hierarchical representation generated offline. This way, it gives the operator access results to different levels in the plant. Here the different logical units appear at the top using which the user can easily navigate through the plant to view the results. 5. CASE STUDY
...... 7,7~
7"~'7 ~
Figure
5: Sour Water Stripper Plant P&ID
. ......
In this section, the application of the integrated framework to a real-life industrial case study will be presented. The case study is a sour water stripping plant shown in Figure 5. The process treats a refinery sour water stream that is separated in a surge drum to remove slop oil from the sour water. The sour water is pumped into a storage tank where any carried over slop oil can be skimmed off. From the storage tank the sour water is through a heat exchanger to a steam stripper where ammonia and hydrogen sulfide are stripped from the water. Details of the case study can be found in Srinivasan et. al (1997).
PHA results from an automated system HAZOPExpert for this case study (Vaidhyanathan and Venkatasubramanian, 1995) are available. Here, they constitute the only knowledge base - PHA "state estimator" The results comprise 734 possible deviations resulting from 279 causes and 854 consequences. These results were organized according to the hierarchy generated using the automated scheme implemented in G2. The hierarchy of the full plant is shown in Figure 6. The bottom-most layer consists of 26 pipes, 5 flow control valves, 5 non-return valves, 5 pumps and 6 controllers.
780 A simulator for the plant was built in gPROMS (Barton and Pantelides, 1994). A state transition model is used to model the hybrid behavior exhibited by the surge drum, depending on the state it is in. Details of the assumptions and simplifications can be found in Srinivasan et. al (1997). The state of the plant is monitored using 5 NT LEVEL5 P~,TLAVER sensors" Surge-drum-interface-level, Surge/ I ~_ drum-side-level, SWStorage-flow-out, OverheadLEVEL4 SYSTEMand and Stripper-bottoms-level. SUBSYS1EMLAYER accumulator-level, The detected deviations are sent to the PHA "state estimator" as queries and the retrieval methods search for the causes that explain all the sW~'l'O NiT -o .14o.UNtT LEVEL LEVEL3 observed deviations. Since HAZOPExpert results IO-UNI1 -UNI1LAYER L are quite thorough and complete there a cause is ~ ! Nrl.' LEVEL; found that can explain the all the deviations in LEVEL2 PIPELINES/ 1PELINI ~ilLiiiii211 ~m,r.o~.. O LSVSTEMS the plant at a given time. Similarly consequences LAYER ___---....... are also retrieved. su~e.oRuH-1.sYsr~
i.............................. oooooooDooo
i:............................... oDoooooooooi
i:.............................. ooooooooooo
LEVEL1 EQUIPMENT
Figure 6: Sour Water Stripper Plant Hierarchy 6. CONCLUSIONS ASM and PHA are important activities in the CPI today. The motivation and issues involved in their integration were discussed. To manage the possibly large number of PHA results in a structured and systematic manner, an automated scheme for hierarchical representation of plants was presented. The integrated framework utilizing PHA results and the hierarchical representation was then proposed. Finally, the framework was illustrated on an industrial case study.
REFERENCES ASM Home Page, The abnormal situation management joint research and development consortium, (1995) http ://www.iac.honesavell.com/Pub/Ab SitMang/ Barton, P. I. And C. C. Pantelides, "Modeling of Combined Discrete/Continuous Processes", AIChE Journal, 40(6), 966-979, (1994) CCPS, Guidelines for Hazard Evaluation Procedures, New York (1985) Lees, F. P., Loss Prevention in Process Industries. (Vols. 1-3), London, Butterworths, (1995) Mylaraswamy, D., and V. Venkatasubramanian "A Hybrid Framework for Large Scale Process Fault Diagnosis", Comput. and Chem. Engng. 21, $935-$940, (1997) Nimmo. I., "Adequately address abnormal situation operations" Chem. Eng. Prog., 91(9), 36-45, (1995) OSHA Regulations on Process Safety Management (Standards - 29CFR), (1994) http ://www.osha-slc.gov/OshStd_data/1910_0119.htm! Srinivasan, R., V.D. Dimitriadis, N. Shah, and V. Venkatasubramanian, "Integrating KnowledgeBased and Mathematical Programming Approaches for Process Safety Verification", Comput and Chem Engg 21, $905-$910, (1997) Stephanopoulos, G., G. Henning, and H. Leone "MODEL.LA A Modeling Language for process engineering- I. The Formal Framework" Comput. and Chem. Engng. 14(8), 813-846, (1990) Vaidhyanathan, R. and V. Venkatasubramanian, "Digraph-Based Models for Automated HAZOP Analysis" Reliability Engineering and System Safety, 50, 33-49, (1995) Vaidhyanathan, R. and V. Venkatasubramanian, "HAZOPExpert: An Expert System for Automating HAZOP Analysis" Process Safety Progress, 15(2), 80-88, (1996) Venkatasubramanian, V., S. N. Kavuri, and R. Rengaswamy, "A Review of Process fault Diagnosis" CIPAC Tech. Report, Purdue University, (1995)
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
781
Leak Detection and Localisation in Pipes and Pipelines Gerhard Geiger a, Withold Gregoritza a, and Drago Matko b aUniversity of Applied Sciences Gelsenkirchen, Faculty of Electrical Engineering Neidenburger Str. 10, 45877 Gelsenkirchen, Germany bDrago Matko, University of Ljubljana, Faculty of Electrical Engineering Trzaska 25, 1000 Ljubljana, Slovenia This paper is concerned with model-based leak detection an localisation methods. The keycomponent is the pipeline-observer, basing on a mathematical description of the pipeline dynamics. Three models for the pipeline are investigated: the non-linear distributed parameters model, the linear distributed parameters model and the linear lumped parameters model. The non-linear distributed parameters model was simulated using the special program PIPESIM and provides the best results, however it involves the highest computational demand. With very small changes of the signals around a working point, the linear models also provide useful results however, with a greater change of working point conditions which may be caused by the leak, the results of linear models become biased. 1.
INTRODUCTION
Pipes and pipelines are used widely in the chemical and petrochemical industries for the transport of fluids (liquids, LPG, gases). Many of these fluids are in some sense dangerous. It is therefore often necessary to install leak-monitoring systems, especially due to legal regulations. As to the difference between pipes and pipelines, only the length parameter is considered; in this sense, pipelines are nothing other than long pipes. In this paper, leak monitoring comprises the detection and localisation of leaks. This paper is concerned with a model-based approach: with the use of a mathematical model description of a pipeline in the form of a pipeline observer, where there is a leak the leak flow and leak position can be calculated [ 1]. The aim of this paper is to compare different mathematical models with regard to their usage in the model-based leak-monitoring scheme. The basic model is a non-linear distributed parameter model obtained by applying the principle of mass conservation and Newton's second law of motion [2]. All other methods are obtained by linearisation and Laplace transformation leading to Multi-Input Multi-Output (MIMO) models [3].
2.
OBSERVER-BASED LEAK MONITORING
Observer-based leak monitoring requires a pipeline model in the form of a pipeline observer to compute the pipeline states assuming no leak. Further discussion will be focused on
782 r ...........................
i !
-0
.,--. . . . . . . . . . . . . . . . .
Flow (F) . . . . . . . . . . . . . . . . . . . . . . . . . . .
:
Pressure(P) . . . . . . . . . . . . . . . . . . :
i
N
I~iet .--...... Temperature(~ of ~tch....... : 0~Ua i ~ J
v
;
;--Temperature (T) of Grounot-=
+
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/
i
;
@@@
i
(~
TG,OTB,O Po
PI TBj TG,I
t!-
i
Pipeline-Observer
F~ ~ Y
x
outletmeasured and estimated fl~
at inlet and
x(k)-F~(k)-f)(k) y(k) = F o (k) - #o (k)
Instrument Error Analysis
+
a discrete-time d a t a processing s c h e m e using the discrete time k - k. To with sampling time TO. The difference between the
Leak Classification I Classification: - Leak yes/no - if yes: leak flow and localization
will be referred to as residuals. The leak flow can then be estimated using
#tea~ (k)= x ( k ) - y(k) , the leak position estimation is given by
Xleak( k ) -
where
- y(k)
x ( k ) - y(k)
. Lp
(2)
Lp denotes the length of the pipeline
[1]. Use mass flow q - M , volume flow 12 or velocity v for flow variable F . In order to eliminate noise on flow measurements, a moving average filter will be applied to the residuals. Fig 1" Observer-based Leak Monitoring.
3.
O B S E R V E R DESIGN The basic mathematical model of a pipeline is the non-linear distributed parameter model
A Op Oq 2 a Ot Ox 10q _ 2(q) 2 Op - - - - + pg sin o: + q = --A Ot 2DAZfi Ox
(3)
obtained by using the equations for continuity and momentum for a compressible, viscous, isentropic, homogenous and one-dimensional flow. p is the pressure, q the mass flow, A the cross-section of the pipeline, a the velocity of sound, ,~ the constant density of the homogenous fluid, c~ the pipeline inclination, /l the dimensionless friction coefficient and D the diameter of the pipeline. A general solution is not available; however, a transformation into four ordinary differential equations grouped into two pairs of equations using the characteristics method is possible [4]. The PIPESIM programme was used to solve the equations (3) numerically. PIPESIM can also be used to simulate the non-linear pipeline behaviour in the case of a leak.
783
3.1.
Linear pipeline model with distributed parameters
The non-linear equations (3) are linearised and written in a form using notations common to the analysis of electrical transmission lines [3]. Also, the gravity effect can be included within the working point so c~ = 0 is supposed. The transcendent transfer function of such a model is obtained by the Laplace transformation and corresponding initial and boundary conditions. Introducing the characteristic impedance Z K = a/(Ls + R ) / ( C s ) and n - ~[(Ls + R). Cs , the linearised model of the pipeline can be written as follows:
1 QL
1
1_
. 1 1 Z x sinh(nLp )
_
1
=-211.Eol
where L - 1/A , R -(/],(-q)-q)/(A2pD),
PL
G22 -- - G l l
l__~coth(nLp) ZK
(4)
and C - A / a 2 are the inductivity, resistance and ca-
pacitance per unit length, respectively. ~ is the flow at the working point.
3.2. Linear pipeline models with lumped parameters The pipeline as a lumped parameter system according to (4) can be presented with secondorder transfer functions in the form ~
bll,2 $2 -t- bll,lS + b11,0
e_ST.,,,
Gl(S)=
.,
all,2 $2 -nt- all,iS nt- 1
G21(s)=
e-St..2,
b212 $2 J- b211S '1- b210 '
'
a21,2 $2 + az1,~s + i
(5)
'
where Td, o is the dead time. The transcendent transfer functions are approximated by a rational transfer function with dead time however only for a class of well damped pipelines. The parameter bu,0 , i.e. the static gain of the transfer function is obtained from the first term of the Taylor Series expansion of the transcendent function. The term aij,2
-
1/(Oo is determined
from the eigen-frequency o)0 of the pipeline, which can be interpreted as follows: the shock wave originating at one end of the pipeline returns after reflection at the other end of the pipeline with the opposite phase. The half-period of the oscillations is consequently equal to the time needed by the shock wave to travel along the pipeline and back. This gives the radial eigen-frequency o)0
=
~/(2~/~.Lp
) Next, the high frequency gain is approximated from the
transcendent function on the assumption of a well damped pipeline C . ~ .
( R L p ) / 2 >>1 and
known dead time is applied. In this way, four coefficients of the transfer function (5) are determined. The remaining two are obtained using a Pade approximation of the transcendent transfer function [3]. Since the high frequency gain approximation is valid only under certain conditions, the derived models are only valid for one class of pipelines. The parameter a11,2 = a21,2 = 1/w 2 = 4/7"/.22 L p L C is the same for G~I and 021 , the remaining parameters for
(5) are as follows:
78,4 Cross t r a n s f e r function G21 (s), G21 (s). Since this transfer function connects quantities at different ends of the pipeline, the dead time for (5) is known - it is the time needed for the shock wave to travel along the pipeline, so Td,21 -
1/LpR, so the coefficient
b21,0 -
Lp ~/LC. The
1/LpR. The high frequency gain is determined by
1
;i_+mla21 ( j ( . o ) l --- ; i m
=2
I R -~-ja,'Cj('Osinh(gp L V(N + jo)L)ja~ )
yielding b21,2 - 8//7/"2 CL2p- ~ e
~e_,Y~- ~ VL
r7
K 1 .
20
L3 g 3 c 2 f / . 2 _ 5.
P
2
(6)
- c~(RLp)'2 . The remaining coefficients are determined by a
Pade approximation of G21 (s), and with K =-48L2pR2LC~/Le
b21,1 ----
static gain of G21 (s) is
LpRLC~ 2 + L2pR2CIr2~ILc + 6 .
_ ~_~
~z. 2 + 24LpRLC we get
Ir2L~ILC
R~a(g. L - g . LpR.~-~ + L2pRiC) K - 6---6 7 . L4pg 4 c 2 ] z . 2 -[- 3 "LpR2LCIIs 2
a21,1 --
(7)
L3pR 3C~2"~-~-6" Lp RIc2L4LC + 6" L21z'2
g:rg2(6,t-6. tpg~/tC -]-L2pR2C)
T r a n s f e r function Gll (s), Gll (s). The dead time of the treated transfer function is zero (Td,l~ - 0 ) since it connects a change of the flow at one end of the pipeline if the pressure changes at the same end. The static gain of G11(s) is
1/LpR, so the
coefficient b11,0 =
1/LpR.
The high frequency gain is determined by
limial,(j~o)l=liml/ jo)c
coth(Lp4(R + j(_oL)ja~ =
(8)
yielding bll,2 = 4/7/ 2 CL2p~LC. The remaining coefficients bll,1 and all,1 are determined by a Pade approximation of Gll (s)"
bll,1 =
2 2 LpC(96"L-96"LpR'~/-CL +16.~iL-3"~ 2 LpR C 8. Tg2(3. L- L2pR2C) 3 3 C~/LC__ 1/24 "12 9L2pR2LC__ LpR
all,1 --
gf~2(3.L-L2pg2C)
. L4pR4C2~2
+3 . L2~ 2
(9)
785 4.
A P P L I C A T I O N TO A REAL P I P E L I N E
The models (3), (4) and (5) were verified using data measured on a real pipeline with the following data: length of the pipeline Lp=9854m, diameter D=0.2065m, relative roughness (the most uncertain parameter of the model, estimated on the basis of the static drop of pressure in the pipeline) kc=0.0957mm, inclination c~ = -0.1948 ~ and the following fluid data: density p = 832kg/m 3, kinematical viscosity v = 7.10 -7 mZ/s and velocity of sound a=1139m/s. The stationary fluid velocity prior to the leak occurrence was 2.45m/s. A 2%-leak rate (1.36kg/s corresponding to 5.9m3/h) was generated at t=300s at 56.4% of the pipeline length where the outrunning fluid was filled into a tank truck. In Fig. 2, the time courses of the residuals x and y are shown while Fig. 3 and Fig. 4 depict the estimated leak rate and leak location.
inlet o f t h e pl ] 0,05
-~
............. I.... :1 .
,.,
,.i
o.os
~!ilill
9
i.
i
-0.05
, I'
Outlet o i l
pipeline
i,~,i,~-(j ':
I
.............. 'l"', ~i I,~ i
0
200
inlet of the
'h'"i,t,
"
.
.
.... , " .
II
I
inlet o:f t h e pipeline
I
,.~,, ~ . ~ 'I
i
............. ~:.l....,. i....i ...... .~..... 1..!,
o.o~
"
'":
" ' :!
! ' ! " "
b')
.
400
tie 600
200
400
t[s]
600
400
200
t[s]
600
t[s]
Fig. 2: Residuals of the leaking pipeline calculated by: PIPESIM model (left), linear distributed parameter model (middle) and linear lumped parameter model (fight)
7
.
6
~- . . . . . - . . . . . . . . . . .
7
7
6 ............ =--............ . .9. . . . . . . . . '.....
. . . . . . . . . --~ ............,'i......... ~.........................
i i.......
~
;
. . ,--, .
:
~,-~
i . . . . . . . . . . . . . . . . . . . . . . .
........................
200
[ ....
real 400 t[s]
J 600
:...~. --..~.~....~. ............,i........./~: .........................
~ ..........
1 ........... 0 L~-~J
6 .................................................. -.-. ~.~-.~.~-~ i- . . . . . - . . . . . . . . . . .
,
...........................
i[ .........................
1
1 ...........
0
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200
400 t[s]
600
200
I
real 400
I 600
t[s]
Fig. 3: Estimated leak rate: PIPESIM model (left), linear distributed parameter model (middle) and linear lumped parameter model (fight)
786 0.7 0.6
0.7
0.7
0.6
06
0.5
0.5
............
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1
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........... !............ i ....................... i estimated i real
0.3 200
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600
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...........
0.3 200
0.4 ............i
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....
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estimated real
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600
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Fig. 4: Estimated leak location: PIPESIM model (left), linear distributed parameter model (middle) and linear lumped parameter model (fight) The residuals were filtered according to Section 2 with a moving average filter of time length 60s before using them for leak localisation. No significant differences can be established. All methods exhibit a small bias for leak rate. This is probably due to approximate evaluation of the real leak rate, which was done manually. The linear models have a significantly larger bias when estimating the leak location. This bias is the consequence of the shift of the working point due to increased flow caused by the leak. REFERENCES 1. Geiger (1998). Application of a Model-Based Method for Leak Detection and Localisation. GMA-Kongress Meg- und Automatisierungstechnik 18./19. Juni 1998, Ludwigsburg. VDI-Berichte Nr. 1397. 2. Streeter, Wylie, Bedford (1998): Fluid Mechanics. 9th edition, McGraw-Hill. 3. Matko; Geiger; Gregoritza (2000): Verification of various pipeline models. 3rd MATHMOD-Symposium, Vienna (Austria), February 2-4, 2000. 4. Wylie, Streeter (1993): Fluid Transients in Systems. Prentice-Hall.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
787
Industrial Applications o f Intelligent Systems for O p e r a t i n g P r o c e d u r e Synthesis and Hazards Analysis for Batch Process Plants Jinsong Zhao, Shankar Viswanathan and Venkat Venkatasubramanian * Laboratory for Intelligent Process Systems, School of Chemical Engineering, Purdue University, West Lafayette, IN 47907 Operating Procedure Synthesis (OPS) and Hazard and Operability analysis (HAZOP) are two important areas in batch process development. They are time-consuming and knowledge-intensive, and could benefit from automation. Recently, two knowledge-based systems for automating OPS and HAZOP, called iTOPS and BHE respectively, were developed and integrated. The integrated system has the capability to perform consistent HAZOP analysis based on the information in the generated operating procedures. In this paper, the integrated system is illustrated using one large-scale pharmaceutical industrial case study. 1. I N T R O D U C T I O N The increasing trend towards the production of higher-value-added products by chemical or pharmaceutical industries has stimulated considerable interest in batch processes. In the current environment of intense market competition, batch process industries stand to benefit from faster and safer process development. One day of delay to the market may cost millions in potential profits since today's blockbuster drugs may make $500 million to $1 billion a year [ 1]. In the process of batch process development, two important areas in batch process development that take considerable amount of time and effort are operating procedure synthesis (OPS) and process hazards analysis (PHA) as they are often manually performed. Thus there exists substantial motivation to develop computer-based approaches for OPS and PHA for batch processes. In this paper, we discuss the results of fielding several industrial applications of such automated intelligent systems for OPS and PHA. 2. I N T E L L I G E N T S Y S T E M S F O R OPS A N D H A Z O P
OPS is systematic synthesis of a sequence of elementary tasks an operator needs to manage a batch process safely and optimally. PHA is the proactive identification, evaluation, and mitigation of hazards. CCPS defines hazard as "an inherent physical or chemical characteristic that has the potential for causing harm to people, property or the environment" [2]. The importance of OPS and PHA is underlined by OSHA PSM Standard 29 CFT 1910 which was enacted in 1992. This standard regulates that major chemical plants should perform PHA on a regular basis when a new process is launched or any change happens in an existing process. It also requires that at least every five years after the completion of the initial PHA, PHA shall be updated and revalidated to assure that the process hazard analysis is consistent with the current process. It is also regulated by this standard that written operating procedures that provide clear instructions for *Author to whomall correspondence shouldbe addressed
788 safely conducting activities should be developed and implemented for chemical processes covered by the standard. PHA is a time-consuming activity. It is estimated that the complete PHA of a typical process could take 1-8 weeks for a PHA team. Typically, there are tens to hundreds of processes per year to review in a chemical or pharmaceutical company, it is easy to imagine how hard the PHA team would work. Moreover, it is considerably difficult for the team to keep the process hazard analysis consistent and systematic during so long and tiresome a period of time for PHA. An intelligent system which helps in automating the entire PHA study would reduce the time, effort and money involved in a review, make the review more thorough and detailed with human errors minimized. OPS is also a time-consuming activity. It is often manually performed according to the experiences of process engineers. The experience-based method is also short of consistency in process development because different engineers can generate different versions of operating procedures. Once the procedure is generated, PHA has to be performed for this process to identify the potential hazards. Corrections to the procedure have to be made to prevent the potential hazards found out in PHA. For example, according to the results of PHA, safeguards might be added to the process. Values of process variables might also be changed if they were set too high or too low. In addition, human errors in the operating procedures are inevitable so that there are usually some corrections of this kind of mistake. Whenever there is a correction necessary, the whole procedure has to be designed again and the diagram of the procedure has to be drawn again. Recently, Viswanathan et. al. [3] presented an intelligent tool for operating procedure synthesis (iTOPS) by using grafchart-based methods. The input to iTOPS included process materials, process equipment and high level process description called Block Process Sequence Diagram (PSD) indicating the sequence of the main tasks. Then process sequence diagram (PSD) and master operation records (MOR) were automatically generated. MOR is a series of detailed operation instructions. As HAZOP is the most widely used PHA method, an automated HAZOP analysis expert s y s t e m Batch HAZOPExpert (BHE) for batch processes was developed [4-5]. Batch recipe was represented by two-tier Petri-Nets. One was Recipe Petri Net (RPN) representing the sequence of main tasks such as reaction, separation and etc. Associated with each main task, there was a lower level Petri Net called Task Petri Net (TPN) containing a series of subtasks. Associated with each subtask, there was a digraph-based qualitative model that captures the cause-effect relationships of the variables in the subtask. The subtask digraph models are process-generic, which means that they can be used in different processes. Since PSD contains the process specific information such as process materials, equipment, operating conditions and so on that is required by BHE for performing HAZOP analysis, Sriniwasan and Zhao et. al. [6] integrated iTOPS and BHE together. In the integrated system, once the OPS is done, i. e. after the PSD and MOR are generated, the PSD-based representation of batch processes is translated into the Petri Net-based representation required by BHE. Meanwhile, during the translation, the digraph models are automatically associated with the corresponding subtasks. Then BHE can perform HAZOP analysis based on the process information derived from OPS. To get process-specific knowledge such as the hazard critical properties of the process materials, material and equipment databases are connected with the integrated system. Material interaction database is used to capture hazards caused by possible side reactions. Block PSD can be determined based on the chemist's process description, iTOPS
789 generates PSD and MOR based on the Block PSD. Through the OPS-PHA interface, block PSD and PSD are converted into the Petri Nets., and therefore, process specific information flows from PSD into Petri Nets. Then BHE can automatically analyze all the potential hazards resulting from abnormal deviations to the process variables, and the final HAZOP analysis report is output to a Word file. 3. I N D U S T R I A L A P P L I C A T I O N S OF ITOPS AND
BHE
The industrial applications are based on the intelligent systems iTOPS and BHE mentioned above. This paper focuses on the applications of these two intelligent systems in pharmaceutical industry. In the following, one large-scale pharmaceutical industrial case study is addressed. A B C
C + E +F waste
y
I
~k
el F ~~ Reaction
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E
~
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Vacuum Distillation
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~ ~] F1 "]Filtratioi
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~ H waste
I
+ G Disposal
H F2 Filtration
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~
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R6 Vacuum Distillation
[
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+
waste
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Extraction ]
,--
p
q, waste Fig. 1 Process flow diagram
This industrial case study compromises of OPS and HAZOP analysis of a large-scale pharmaceutical process. According to the chemist's process description, to produce the final product P, thirteen tasks including two main exothermic reactions, one neutralization, six vacuum distillations, two filtrations and two extractions were performed. The first reaction is:
790 A+B
(1)
c+D ) O + G
Then the intermediate product 0 was transformed to the final product in the second main reaction: O+J
(2)
I )P+L
Separations were used to separate out the solvents, catalysts and by-products involved in this process. Totally, twenty eight process materials and nine pieces of equipment such as reactors and filters were involved in the process. Fig. 1 shows the process flow diagram of the process.
Operating Procedure Synthesis by iTOPS To perform OPS, iTOPS provides friendly user interface/workspace to specify the following process-specific information: process materials with necessary physical properties such as density, molecular weight and so on; process equipment with capacity information; process chemistry (reactions and separations) and the high level process description - Block PSD. The workspace for the property specification of a process material is illustrated in Fig. 2. Once the process specific information is input, OPS can be done in the order of 2-3 minutes by iTOPS. According to the OPS results, 148 operations were generated by iTOPS in order to complete the thirteen tasks. These operations include twenty three different kinds of operations such as charge, heat, cool, hold, distillation, filtration, extraction and so on. Due to the length limitation of the paper, the whole picture of the PSD generated by iTOPS can not be shown in this paper.
Fig. 2 Workspace for material property specification
HAZOP Analysis by BHE BHE converts the Block PSD and PSD into Petri Nets-based representation of the product recipe. In HAZOP, hazard-critical properties of process materials and equipment are necessary. The hazard-critical properties of a process material include quantitative properties such as boiling point, flash point, melting point and decomposition point as well as qualitative properties such as flammability nature, corrosive nature, toxic nature
791 and so on. The properties of process equipment needed by HAZOP include the design temperature and design pressure. Most of the quantitative information can automatically derived from the database if the material or equipment is available in the database. However, the qualitative information and the quantitative information of the materials and equipment not present in the database should be specified by users. BHE instructs users to specify the information it needs. Instructions BHE gives are then listed on a spreadsheet. For example, if the decomposition temperature of material A is not available in the database, the instruction such as "Please specify the decomposition temperature of A" will be given by BHE. Clicking on the instruction can activate a property dialog of the material where users can specify the decomposition temperature. Similarly, dialog-based interfaces are designed to help users complete all other instructions BHE gives. Table 1 Node-1 HAZOP Results Reported by the Team and BHE Deviations Safety Hazards from the Team Safety Hazards from BHE Low None 1. Incomplete reaction 2. Highly hazardous reactant not Temperature consumed Vaporization of volatile materials High none Temperature Low Agitation Too low an agitation may cause 1. Incomplete reaction an incomplete reaction within 2. Highly hazardous reactant not the specified time consumed 3. Non-uniform concentration due to poor mixing 4. Poor temperature distribution High Agitation None Level close to the maximum volume Short Time Too short a reaction will result 1. Incomplete reaction in incomplete reaction 2. Highly hazardous reactant not consumed High Level None Close to the maximum volume High None Operator exposure to hazardous Concentration material B when sampling As PSD only indicates the sequence of operations, it is hard to accurately identify the position of a reaction. BHE also gives an instruction to ask users to insert necessary reaction subtasks into the Petri Nets of the corresponding reaction tasks. After all the instructions given by BHE are completed by users, HAZOP can be automatically performed by BHE in about 15 minutes for this case study. Totally 1148 possible high/low/zero deviations to the process variables were analyzed by BHE, and 68 potential safety hazards were reported. All of the twelve safety hazards reported by the HAZOP team were captured by BHE. Table 1 compares the safety hazards of Task-1 reported by the team and BHE. From the table, it can be found that some important safety hazards were neglected by the team. For example, the reactant B is a highly hazardous material. To control the product quality, sampling is required in the operating procedure. Therefore, BHE indicates a
792 potential operator exposure to hazardous material B during sampling. However, the team was not able to flag this out. In equivalent situation, BHE can reproduce similar results. For example, BHE reported that high Temperature could cause emission of volatile materials from condenser in all of the six vacuum distillation tasks while the human team only flagged this hazard in Task-2. Up to now, sixty industrial processes have been generated by using iTOPS in one of our industrial partners since January 1998. Twelve pharmaceutical processes have been tested with BHE, of which eight processes were tested with the integrated system. Similar observations were obtained. Due to the length limitation of the paper, more case study results can not be shown here. If interested, please contact authors. 4. CONCLUSIONS Compared with the OPS and HAZOP analysis done by engineers without the aid of such automation tools, up to 50% savings in time and effort have been seen in all of the industrial applications. BHE generated more consistent HAZOP results in the process variable deviation analysis than the team did. Documentation of PSD and MOR becomes much easier. BHE considered many more potential hazardous scenarios thus performing a more comprehensive analysis. Faster and safer process development can be achieved by using the intelligent tools- iTOPS and BHE. REFERENCES
1. Basu P. K., "Pharmaceutical Process Development is Different!", Chem. Eng. Progress, 98(9) 75-82 (1998). 2. Center for Chemical Process Safety (CCPS), Guidelines for hazards evaluation procedures, 2nd edition with worked examples. American Institute of Chemical Engineers, New York, 1992. 3. Viswanathan S., Johnsson C., Srinivasan R., Venkatasubramanian V., and Arzen K., Automating operating procedure synthesis for batch processes - I&II Implementation and application, Computers and Chemical Engineering, 22(11) 1673-1696 (1998). 4. Srinivasan R., Venkatasubramanian V., Automating HAZOP analysis of batch chemical plants: Part II Algorithms and Application, Computers and Chemical Engineering, 22 (9) 1357-1370 (1998). 5. J. Zhao, S. Viswanathan, R. Srinivasan and V. Venkatasubramanian, Automated process hazard analysis of batch chemical plants, AIChE Annual Meeting, Miami, Florida, Nov. 1998. 6. S. Viswanathan, J. Zhao and V. Venkatasubramanian, Integrating operating procedure synthesis and hazards analysis automation tools for batch processes, Computers & Chem. Eng., 23 $747-750, (1999).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
793
Model-based safety verification under uncertainty H. Huang, C.S. Adjiman* and N. Shah Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BY, United Kingdom. This paper deals with a novel, quantitative approach to process safety verification under uncertainty in either model parameters or process inputs and disturbances. The work is based on the concept of a region-transition model (Adjiman, 1999). 1. INTRODUCTION Process plants tend to be ever more complex and operated closer to constraints, with greater degrees of heat- and mass-integration. While this results in more efficient process economics, it also exposes the plant workers and even the public to potentially more dangerous situations (Lees, 1996). This has led to an increased emphasis on safety analysis studies in both industry and academia. Generally, safety analysis should answer the following questions: i. Hazard identification. What can go wrong? ii. Risk assessment. What are the consequences and effects? iii. Risk mitigation. What are the appropriate methods to render the risk acceptable? Many people believe that hazard identification is the most important step in safety analysis on the grounds that "a hazard identified is a hazard controlled", because a reputable company is unlikely to expose its business to the financial consequences which could result from failing to eliminate or control a major hazard. A number of approaches have been proposed for hazard identification. Most of these methods are based on qualitative techniques (Torney and Pitblado, 1996). More rigorous model-based qualitative approaches have also been developed and implemented as toolkits supporting HAZOP teams (Venkatasubramanian and Vaidhyanathan, 1994). These approaches, although thorough and computationally efficient, sometimes suffer from ambiguity because of the qualitative nature of the reasoning, and as a result tend to be conservative in that they identify some hazards that are unlikely or impossible in practice. A few quantitative, model-based approaches (Dimitriadis et al., 1997; Adjiman, 1999) have been devised in recent years. A combined qualitative-quantitative approach which exploits the strengths of each has also been proposed (Srinivasan et al., 1998). The quantitative approach of Dimitriadis et al. (1997) is based on the modelling of the process system and its interaction with the environment and control system through a state transition network which recognises the hybrid nature of such systems. At any time, the
*Author to whom all correspondence should be addressed (email: [email protected])
794 system is one of a number of discrete states. Each state is characterised by a set of continuous describing variables, a set of (usually differential-algebraic) equations that determine the dynamic behaviour of the system when in that state and a set of transitions to other states. The transitions depend on logical conditions based on the continuous variables being satisfied. The model is used to perform safety verification through an optimisation approach. Regions of the operating space are defined as unsafe, and other regions are identified as possible initial conditions. The optimisation problem attempts to find a set of disturbances that drives the process from an initial condition to an unsafe condition as quickly as possible within a finite horizon. The approach requires convex models and was applied mainly to linear systems, as it is computationally expensive. It also identifies a single unsafe behaviour. This paper describes a procedure that aims to extend the previous work in quantitative techniques, and does so in three ways: a) it is easy to consider nonlinear processes; b) a whole range of unsafe behaviours (rather than one very specific operating policy) are identified; and c) uncertainty in model parameters is easily treated. The next section describes the modelling approach taken here. This is followed by the safety verification algorithm and an example. Finally, some conclusions are drawn. 2. T H E R E G I O N T R A N S I T I O N M O D E L The state-transition network (STN) based model described leads to realistic, numerical evaluations of hazards. However, its applicability has been limited by the numerical methods available for the model analysis, which have tended to require linearity. Recently a new region-transition model (RTM) (Adjiman, 1999) has been proposed. Like the STN, (Dimitriadis et al., 1997), this model is based on the concept of hybrid system transition networks. In addition, the interval representation of uncertain information such as model parameters and operating ranges is introduced. In order to make conservative estimates of system behaviour, interval bounds are assigned to these parameters. As a result, the state variables that depend on these parameters are defined by interval bounds instead of scalars. One use of this model is to determine which regions, as opposed to individual points, of the initial operating space can guarantee safe operation. This model can also deal with nonlinear systems in a straightforward fashion.
2.1 Model assumptions The following assumptions are made: Determinism. While determinism is commonly required of all equations describing a hybrid system, this assumption underestimates the complexity of chemical processes and is usually justified by the necessity to obtain a straightforward model. As our model is proposed to deal with complexity, this assumption is relaxed and only the practical requirement that all inputs should be deterministic is imposed. Time-invariance. The equations describing a state, the transition relations and state initializations are assumed to be time-invariant, so that the memory of the process is limited to its latest state. Despite this limitation, time-invariant models can account for changes in the parameters as long as the rate of change is constant. Instantaneous transitions. In general hybrid systems, the variable set may change upon a transition and transitions may be non-smooth This generality is preserved in our model, and the only assumption regarding transitions is that they are instantaneous.
795
2.2 Modelling approach In process optimisation, we generally aim to find the single point in the variable space (the optimum) which results in the best value for an objective function. However, in many other problem classes such as safety analysis we are actually interested in a range of values of certain key parameters or variables rather than a single point, especially to distinguish between safe and unsafe operating regimes. A means of capturing this requirement is the concept of a region. A simple region r is defined by a state s and a hyper-rectangle bounded by the describing state variables [x[, x~] and external inputs [uL, uu], and denoted by (s, [xLs, xvs] • [uL, uu]), where subscripts L and U denote lower and upper bounds respectively. A region is a union of simple regions. The union of hyper-rectangles corresponding to the same state s is denoted by Xs. If the system might be in any of the M states in SM and in any hyperectangle Xs, the corresponding region is denoted by Iq - UsssM(S, XS). This concept is illustrated in figure 1. In the figure, circles represent states; rectangles represent simple regions; solid arrows represent transitions and the dotted arrow represents how a region evolves within the same state. In this system there are three possible states denoted as 1, 2 and 3. Suppose that at time t, the system is located in the region Iq of state 1. The region Iq is a union of three simple regions rl, r2 and rs. The figure shows that, under the same dynamic evolution equations of state 1, there is a transition from region rl to region R: of state 2, a transition from region rs to region R3 in state 3 while 1"2 remains in state 1 but evolves to a different simple region, r4, at the next time instant, t+l. This means the system is in three states at t+l. This is a distinguishing feature from the other hybrid system models (e.g. Dimitriadis et al., 1997) where the system is uniquely in one of the states at any time.
("
t~
u
i ! /i: U
Fig. 1: RTM example
Fig. 2: Safety verification
Hence, the dynamic evolution of the discrete and continuous components of the hybrid system are such that the system may be in multiple states and/or regions at certain points in time. In order to describe the fact that we are interested in ranges of values of individual variables rather than point values we utilise the formalism of interval arithmetic.
2.3 Interval arithmetic (Moore, 1966) An interval, X = [aL, au], containing a real variable x is characterised by two scalars aL and av such that aL -< x _
796 inclusion of functions. Consider a real-valued function f(x). The interval function F(X) is an inclusion isotone off(x) if: x e X implies that f(x) e F(X), and in general Y _c X implies that F(Y) c F(X). Several methods can be employed for function inclusion, including natural inclusion and problem-specific inclusion. The natural inclusion of a function f(x) is obtained by replacing each occurrence of the variable x with an interval including it, X, and replacing real arithmetic operations with interval arithmetic operations. This method is simple, but may result in very large intervals for the inclusion.
2.4 Uncertain model parameters Model parameters such as reaction rates have usually been gathered experimentally or estimated from correlations. Any uncertainty in model data must be taken into account in order to provide a conservative result in safety studies. Although we do not know the exact value of a parameter, we usually know with confidence that a certain range can bound the value of the parameter. This bounded range can be easily expressed in intervals in our proposed model. For instance, consider a first order reaction A ~ B, rA = -kACA, where rA is the reaction rate; CA is concentration of A and kA is the reaction kinetic constant. If, when taking experimental error and catalyst decay into account, the kinetic constant is known to vary between 1.2 and 2 s -1, the reaction rate can be expressed as ra = -[1.2, 2]CA. Thus, the uncertainty of the kinetic constant results in a range of possible reaction rates. In safety studies the bounds of the rates are usually our concern. 3. S A F E T Y V E R I F I C A T I O N P R O C E D U R E In the region transition model, a whole region rather than one point in the region is taken into account. As a result of this method, in a dynamic system, different parts of the region can lead to different states. From a safety verification perspective, a initial region (initial conditions) has two destinations: safe or unsafe. Therefore, safety verification based on this model is to identify which parts of the initial region can ensure safe operation throughout a given time horizon, and which parts lead to potential unsafe behaviours during this period. This idea is illustrated in figure 2. In the figure, the subregion of initial conditions rl leads to unsafe behaviour while the subregion r2 remains safe. The curved borders between the safe and unsafe initial regions indicate the nonlinearity of the model. Based on the RTM, a safety verification algorithm (Adjiman, 1999) has recently been described. In this algorithm, the input is the region R0 comprising the initial condition space Ri, controls Ru and disturbances Ro. R, denotes the union of unsafe regions. The algorithm computes an under-approximation RL of R,, which contains only initial states that lead to the unsafe space O, and an over-approximation R0 which includes the initial states that lead to undecided and unsafe behaviour. At first, the under-approximation is empty and the overapproximation is the entire initial region. The under-approximation is built through the addition of unsafe regions and the over-approximation is improved by removing safe regions. The two approximations provide information about which initial regions may lead to unsafe operations. Based on this algorithm, here, we introduce a new algorithm. Instead of using underapproximation and over-approximation, we create three sets of regions, Runsafe, Rsafe, Rundecided to store separately those initial regions which could lead (by the end of the evaluation horizon) to unsafe, safe or undecided operation respectively. The new algorithm consists of an outer
797 loop and inner loop. The outer loop generates progressively smaller input regions by branching on the variable intervals Flunaeekletl and returning the final results. The choice of branching variable is made in fashion that aims to extract maximum information about the behaviour of the two new input regions. The inner loop identifies the reachable states of a given input region by the end of time horizon. Ultimately, the search procedure terminates, having evaluated the whole of the input space in terms of its ultimate safety within a given tolerance. To illustrate the algorithm, consider the simple tank example of Dimitriadis et al. (1997). In figure 3, the initial volume of material in the tank (Io) is unknown, and is treated as the initial region. At first time step (tl), at the top, part of Io evolves out of Vmax, while on the bottom part of Io evolves to less than Vmin. Thus, these two parts of Io are placed in the unsafe initial region list and the remaining parts are evaluated further. As this computing process proceeds to the end of time horizon (t/-/), the initial region Io is split in to three sections: 12 ensures the safe operation of the tank while 11 and 13 lead the system to the danger of overflow or drainage separately. 4 AN E X A M P L E
4.1 No uncertainty in parameters A batch reactor is fitted with a cooling jacket and is used to undertake the exothermic gasphase reaction A --~ B. The reaction rate is first-order with respect to A. The aim is to achieve a certain minimum conversion X* of A by a specified time tma~. During the operation of the reactor, it may be in one of two states. $1 is the safe state denoting normal operation. $2 is an unsafe state, and arises if during operation the temperature exceeds a threshold temperature Tmax which is associated with the onset of thermal runaway. If the system remains in state $1 throughout operation, the minimum conversion condition is evaluated at tmax. If X < X*, then the process is said to have ended up in an undesirable state $4, otherwise it ends up in a safe state $3. 2
-
-"T //lax
to
-___
"_A_
t1
t2
Fig. 3: Tank example
tH
Lr Fig. 4: STN for reactor
The state-transition network for the reactor is depicted in figure 4. The system is modelled using material and energy balances, an ideal gas equation of state and a constant rate of cooling as long as temperature crossover does not occur. The RTM-based safety verification procedure is applied using a problem-specific function inclusion, considering the initial charge and physical dimensions as constant, but treating the initial temperature, To, and the rate of cooling, Q, as input variables for safety verification. The results are depicted in figure 5. It is seen that too high an initial temperature will always result in unsafe operation, regardless of
798 the rate of cooling. An initial temperature that is too low, or adequate but with too low a rate of cooling, will result in undesirable operation in that the conversion target is not reached. The dark regions at the boundaries reflect small undecided regions associated with the tolerance of the algorithm. 550
550
(9
(9
UNSAFE
500
,~
UNSAFE
500
!._
(9 o.
E (9
450
450
m .
"E
e,, 400
I-
400
UNDESIRABLE 350 - 1500
-_.1300
- 1100
-900
UNDESIRABLE -700
Q rate of heat removal
Fig. 5" Reactor results
-500
350 -1500
- 1300
- 1100
-900
-700
Q rate of heat removal
-500
Fig. 6" Effect of uncertainty
4.2 Uncertainty in parameters It is often the case that some model parameters are uncertain. In our case, we now assume that the rate constant is not known exactly, but may be described by an interval with bounds + 2% of the nominal value. Our method can automatically deal with it as it handles intervals naturally. In this case, the results are as illustrated in figure 6. It can be seen that the overall trend is the same, but the undecided regions are larger. 5. CONCLUSIONS A novel procedure for safety verification has been described. It is based on the RTM approach. In contrast to previous methods, it is able to distinguish between entire regions of the operating space, and deals naturally with nonlinearity and uncertainty. The approach has been described with reference to a small example. Further research is required in the area of efficient computation of the outcomes associated with particular input regions and on efficient branching strategies for undecided regions. REFERENCES Adjiman, C.S., "Safety verification in chemical plants: a new quantitative approach", Computers chem. Engng., 23, $581 (1999). Dimitriadis, V.D., N. Shah and C.C. Pantelides, "Modelling and safety verification of discrete-continuous processing systems", AIChE J., 43, 1041 (1997). Lees, F.P., Loss prevention in the process industries 2 na Ed., Butterworths, London (1996). Moore, R.E., Interval Analysis, Prentice Hall, Englewood Cliffs, NJ (1966). Srinivasan, R., V.D. Dimitriadis, N. Shah and V. Venkatasubramanian, "Safety verification using a hybrid knowledge based mathematical programming framework", AIChE J., 44, 361 (1998). Tomey R. and R. Pitblado, Risk assessment in the process industries, ICheme (1996). Venkatasubramanian, V. and R. Vaidhyanathan, "A knowledge-based framework for automating HAZOP analysis",AIChE J., 40, 496 (1994).
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
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Computerized Screening of Chemicals for Energy Release Hazards B. Keith Harrison a "EGLB 244Department of Chemical Engineering, EGLB 244, University of South Alabama, Mobile, AL 36695, U.S.A. The computer program CHETAH is described. The program is unique in that it predicts the energy release hazards of pure chemicals or mixtures. CHETAH has a 25 year history of successful application and continues to be refined by ASTM Committee E27.07. Examples of CHETAH's use are given. The program also has a sizable database of thermochemical properties and makes predictions of thermochemical properties based on molecular structure. The accuracy of CHETAH for reactive hazard prediction is assessed against a test database.
1. I N T R O D U C T I O N Since 1974 some form of the computer program named CHETAH has been provided by the American Society of Testing and Materials (ASTM) for predicting the energy release hazards of pure chemicals and mixtures (CHETAH, 1998). This program is provided as a not for profit public service by a volunteer members of ASTM from industry, government, and academia. It is believed to be the only generally available computer program that makes predictions of which chemicals or chemical mixtures might be explosive hazards. The program does this by applying a series of four thermodynamic and structure based correlations and using the results of the calculations to provide an overall indicator of the potential hazard. CHETAH, which stands for the Chemical Thermodynamic and Energy Release Computer Program, is used routinely by a number of chemical and pharmaceutical companies as a part of their hazard evaluation process for the synthesis of new chemicals and for reactor designs. It is anticipated that such computer based assessment methods will see increased use in the years ahead as an economical way to promote increased safety and loss prevention in design and development of new chemical processes. The program has undergone continual refinement through the years by ASTM Committee E27-07. The most recent release of the program is Version 7.2, released in 1998. There is a long history in the scientific literature of cases in which it has been used successfully. Over 60 references to the use of CHETAH have been noted in the literature. Typical recent citations include Grewer et al. (1999) and Huang et al. (1996).
800 2. USE OF C H E T A H F O R P R E D I C T I O N OF R E A C T I V E H A Z A R D S
CHETAH is used in the Chemical and Pharmaceutical industry to predict the potential for deflagration or detonation of a pure chemical or a mixture of chemicals. The emphasis is on the evaluation of chemicals or mixtures of chemicals before synthesis or manufacturing to prevent accidents. In addition CHETAH can be used to predict the flammability of chemicals and to predict the enthalpy of combustion and combustion products. The combustion calculations can be accomplished for chemicals composed of a very wide range of elements and is unique to CHETAH. CHETAH can also be used to predict the enthalpy of a user specified chemical reaction or the equilibrium constant for a reaction, whether the reaction involves a hazardous situation or not. An example of CHETAH being used is provided by Cardillo and Nebuloni (1991) in which over two thousand mixtures relative to waste streams from vapor degreasing, dry cleaning, and solvent extraction were screened. These authors were able to use CHETAH to define the proportions of certain waste chemicals that when combined will yield or not yield a reactive hazard. Predictions were made as to the amount of an inert compound required to maintain certain mixtures as non-reactive hazards. The authors point out that an experimental effort surveying all of these mixtures would be prohibitively expensive and time consuming. Another example of the use of CHETAH for reactive hazard screening is given in Figure 1, a triangular diagram showing compositions of 3 chemicals and the explosive limits predicted by CHETAH in contrast to known incidents. Here mixtures of the three chemicals, perchloric acid, acetic anhydride, and water are considered. Mixtures of these chemicals are sometimes employed in industry as electro-polishing solutions. All compositions combining these three chemicals can be represented on a triangular diagram. CHETAH was employed by the author to predict reactive hazardous combinations of the three chemicals. The results of this effort are shown on the figure as a border between mixtures predicted to be reactive hazards and those that were not predicted to be hazards. Also indicated on Figure 1 is a region of unsafe mixtures as given by Nester and Vander Voort (1992) citing earlier work by Medard and Sartorius (1950). A screening of mixtures of the three indicated chemicals by CHETAH has correctly pointed out the possibility of a reactive hazard and is in reasonable agreement in terms of regions of unsafe compositions. It should be noted that although predictions by CHETAH do not exactly agree with the literature indicated reactive region, Nester and Vander Voort point out the locations of the unsafe regions is ambiguous. Figure 1 also shows compositions corresponding to two known reactive incidents, one of which involved the loss of seventeen lives (Herr, 1947). CHETAH correctly predicts a danger of explosion for both cases.
801 Perchloric Acid
A 8o/ V I ~ /\iX/\ -,o 7
'~
-~ Known Accidents 9
Detonation Limit shown by Nester and Vander Voort
/XA-AAA
",o'
'
M\
Water
Anhydride
Fig. 1 Explosive Electropolishing Solution Compositions
3. O T H E R RELATED USES OF CHETAH CHETAH is also used to predict thermochemical properties of chemicals including ideal gas enthalpy of formation, ideal gas Gibb's free energy of formation, ideal gas entropy, and ideal gas heat capacity. This is accomplished for organic and organometalic compounds using the well established Benson's group contribution technique (Benson, 1976). The database of Benson groups that is used within CHETAH has been added to by the ASTM committee through the years and is to believed to be the largest in existence. The program has been designed to be flexible allowing the user to apply it even to cases for which the desired Benson groups are not available. Techniques appropriate for this extension are discussed in the users manual for the current version and are not published elsewhere. Also believed unique to CHETAH is the ability to predict the thermochemical properties of a wide range of inorganic species composed of any appropriate combination of 117 different ions. CHETAH also includes a backup heat capacity prediction method that was developed within the ASTM committee that applies to any compound with no exceptions. The net result is that CHETAH is very useful for the prediction of thermochemical properties for a very wide range of compounds. These predictions are useful in making calculations to assess possible reactive hazards as well as in many chemical process process and design calculations.
802 4. A C C U R A C Y HAZARDS
OF
CHETAH
FOR
THE
PREDICTION
OF
REACTIVE
No comprehensive reporting of the predictive success of program has ever been accomplished beyond a tabulation of the performance of the program in the correlation of the original data set used for training the program. The original dataset was indeed correlated quite well as reported in the original users manual (Seaton et al., 1974) by the hazard assessment methods used in the program, but correlation is no guarantee of predictive success for new cases not in the original database. Certainly through the years a lot of anecdotal evidence accumulated concerning the accuracy and usefulness of the program. Complicating matters, in recent years changes related to the predictive methods used in the program as well as to the program interface have been made, based on the analysis of experts. The ASTM Committee E27-07 felt that an evaluation of the actual predictive accuracy of the program needed to be accomplished. Over several years with the cooperation of several chemical and pharmaceutical companies, a database of the impact sensitivity of 754 chemicals and chemical mixtures has been assembled. This database of actual experiments on a wide variety of pure chemicals and mixtures is essential to development of technology in this important application area. About 178 of these datapoints were used in the original correlation work associated with the design of the computer program and not suitable for assessment of the predictive ability of CHETAH. The other 576 datapoints were however available for testing the performance of the program and also for exploring the success of proposed new predictive methods. These 576 test datapoints were used to evaluate each of the four hazard assessment correlations used within CHETAH as well as the composite overall energy release hazard indicator calculated in CHETAH. Each of the four methods used within CHETAH have strengths relative to certain compound classes. The overall composite reactive hazard score calculated by the program is the result of an expert system weighting of the four methods. The analysis reveals that CHETAH gives the correct overall prediction relative to reactive hazards in approximately 70 % of the cases as shown in Figure 2. For a conservative evaluation, the CHETAH hazard assessment correlation based on the maximum heat of decomposition of a compound or mixture should be used. This predictive method correctly predicts a shock sensitive composition to be shock sensitive in 89 % of the test cases.
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Figure 2. Accuracy of CHETAH Results
The study of the performance of CHETAH is being used to further refine the program. An analysis of the results for peroxides is leading to the development of new reactive hazard predictive technology for this class of compounds. Also a proposed predictive method based on adiabatic temperature rise for decomposition reactions has been evaluated. Results indicated the proposed method has some validity, but the proposed method showed no improvement to the overall prediction furnished by the existing CHETAH program.
5. C O N C L U S I O N The ASTM computer program, CHETAH, is a useful tool for prediction of reactive hazards of chemicals. Such predictions are useful for avoidance of accidents before the synthesis of chemicals, in existing process operations, and in new process designs. CHETAH has a long history of application for these purposes and is now shown to be accurate in its predictions in approximately 70 % of 574 test cases. CHETAH is shown to be useful as a conservative tool when the maximum energy release correlation is used, preferring to err on the side of predicting a hazard when there is none. The results of this study are being used to further refine the predictive methods used in this computer program. CHETAH was never intended to replace physical testing of materials and should only be used as part of an overall scheme involving both physical testing and other predictive tools.
ACKNOWLEGEMENTS William H. Seaton and Fuat Ling are acknowledged for their contributions in accomplishing and processing the CHETAH calculations on the test database.
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REFERENCES Benson, S.W., Thermochemical Kinetics, Methods for the Estimation of Thermochemical Data and Rate Parameters, 2nd ed., Wiley, 1976 Cardillo, P. and Nebuloni, M., 'Reactivity Limits of Aluminum and Halohydrocarbon Mixtures: Determination by the ASTM CHETAH Program,' J. Loss Prev. Process Ind., 5(2), 81-88, 1991. CHETAH TM , Version 7.2, The ASTM Computer Program for Chemical Thermodynamic and Energy Release Evaluation (NIST Special Database 16), ASTM Subcommittee E27.07, ASTM, West Conshohocken, PA, 1998. Grewer, T., Frurip, D.J., and Harrison, B.K., 'Prediction of Thermal Hazards of Chemical Reactions,' J. Loss Prev. Process Ind., 12, 391-398, 1999. Herr, F.A., 'Los Angeles Plating-Plant Explosion, 'Metal Finishing, 45(3), 72-73, 107, 1947. Huang, C., Harrison, B.K., Madura, J., and Dolfing, J., 'Gibbs Free Energies of Formation of PCDDS: Evaluation of Estimation Methods and Application for Predictinng Dehalogenation Pathways,' Environ. Toxicology and Chem., 15(6), 824-836, 1996. Medard, L. and Sartorius, R. 'Explosive Properties of Perchloric Acid-Acetic Anhydride Mixture,' Mem. Poudres, 32, 179-196, 1950. Nester, R.C. and Vander Voort, G.F., 'Safety in the Metallographic Labortory,' ASTM Standardization News, 34-39, May, 1992. Seaton, W.H., Freedman, E., and Treweek, D.N., CHETAH The ASTM Chemical Thermodynamic and Energy Release Potential Evaluation Program, DS51, Philadelphia, PA, 1974.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
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A HYBRID MODULAR HIERARCHICAL APPROACH FOR FAULT DIAGNOSIS IN COMPLEX TRANSIENT PROCESSES N.J. SCENNA *# l, B. DROZDOWICZ # + 2, S.J. BENZ* 3, and E.J. LAMAS #4 *Grupo de Investigacitn Aplicada a la lngenieria Quimica, UTN, E. Zeballos 1341, 2000 Rosario, Argentina +Laboratorio de lnteligencia Artificial Fac. lngenieria. Univ. Nac. Entre Rios OINGAR-CONICET, Avellaneda 3657, 3000 Santa Fe, Argentina. nscenna@arcride,edu.ar*;bdrozdo@arcride,edu.are;[email protected];elamas@arcride,edu.ar4 Abstract: A diagnostic problem is characterized by a set of observations that must be explained. Dynamic discontinuous high non-linear models characterize the diagnosis problem for batch processes. A hybrid multi-modeling approach and a modular architecture based on neural networks is applied to perform the diagnosis task. A Fault Diagnosis System for batch distillation columns is presented. 1. INTRODUCTION Batch processes are used in the manufacture of pharmaceutical products, certain polymers, fine chemicals, etc. The time dependence of variables and parameters characterizes batch operations. Thus, the "normal state" is a set of desired temporal trajectories that can be labeled as normal evolutions. A diagnostic problem is characterized by a set of observations that must be explained. When the observed behavior is different from the expected one, it is said that the system has a faulty state. Thus, given the set of symptoms (temporal observations) we must deduce (diagnose) which fault/(s) can explain the abnormal behavior (observed symptoms). Therefore, modelling the diagnosis problem requires to deal not only with the discontinuous non-linear models inherent to batch distillation, but also with dynamic evolutions and the need of supervising initial conditions. There are very few reported works about Model Based Fault Diagnosis Systems (MBFDSs) or empirical systems for batch distillation columns. 2. PROBLEM SOLVING METHODOLOGY. The system theory tells us that complex problems or tasks (like fault diagnosis) can be analyzed and decomposed into a smaller set of genetic strategies. This is the basis over which almost all the existing tools for fault diagnosis in chemical processes are being built. From both the expertise and the theoretical points of view a hierarchical modular architecture is very convenient for complex processes. In control theory, for example, when attempting to satisfy the requirements of the non-linear control system design, it is often difficult to find continuous control laws useful in all the relevant regions of a plant parameter space. If we know how the dynamic changes according to the operating conditions, it is possible to implement a piecewise controller using different control laws according to the plant state. This principle is known as "gain scheduling" (Shamma and Athans, 1996) (Jacobs and Jordan, 1993). Essentially, the entire plant evolution (dynamics) is divided into several operating zones covering the full operating range. Each zone is characterized by a time-invariant (linear) approximation of the plant. Between adjacent or time successive operating zones, gains are interpolated or scheduled. These authors found that suitable designed modular architectures can learn how to perform nonlinear control tasks using a piecewise control strategy, avoiding adverse effects. For example, when neural networks (NNs) are used for process modeling, a modular architecture avoids problems such as crosstalk, excessive time for the training stage, etc. Moreover, they suggested that this strategy is not limited to learning a piecewise feed-forward control law, but can be useful for a variety of other control problems, such as identification, state reconstruction, etc. We suggest that this approach is also a good choice for our problem. In this way, "local" (spatial or temporal) models can be used for process modeling in a sort of multimodeling cooperative architecture. A limitation in this approach is that accurate models for high non-linear plants (global or local) are very hard to formulate and model parameters are very difficult to determine. Indeed, we need a "supervision system" switching different valid models as functions of the time, the state of the process, etc. (this naturally implies a hierarchical strategy).
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2.1 A Hybrid Modular Hierarchical Approach. Using a modular architecture, we can divide the temporal evolution of the process in a set of atemporal invariant modules (models). This approach can be found in (Tarifa and Scenna, 1999), who applied it to a batch reactor using a set of linked time invariant qualitative models. They generated one MBFD prototype for a sulfolene batch reactor. Each qualitative partition of the process was called Pseudo Continuous Block (PCB). Different temporal parts of the batch process evolution are qualitatively assimilated to continuous processes. Another example of modular design approach is the hierarchical NN system proposed by (Maurovouniotis and Chang, 1992). (Drozdowicz et aL, 1999) used this approach for the implementation of a fault diagnosis system for a batch reactor. If compared with the signed digraph based model proposed by (Tarifa and Scenna, 1999), this is faster and has the possibility to overcome the lack of adequate process models. However, the MBFDS (based on SDGs) is more efficient to identify situations not predefined, taking into account its deep knowledge model based characteristics. 3. AN EXAMPLE. A FAULT DIAGNOSIS SYSTEM FOR A BATCH DISTILLATION COLUMN. The Fault Diagnosis System will be implemented using a set of Neural Networks, that is, a modular system, identifying each Neural Network with a portion of the temporal evolution of the process. Besides, the control unit includes an Expert System (ES) conforming a hybrid hierarchical system, which uses the outputs of the NNs in addition to process variables for the identification of fault causes (Fig. 1). I Ir..i
Process
vl i KI
t ~ !i i
B,. Vent t:
,.it=
Expert System
[ Generic Rules ...i Rules I ]Rules I .......
i ecg A I
I PCB B
I PCB N
Figure 1: System Architecture Figure 2: Flowsheetof the batch distillation(n-hexanen-heotane orocess). A set of Self-Organizing Neural Networks is used for the fault detection process. The self-organizing model belongs to the class of vector coding algorithms, because the model provides a topological mapping that optimally places a fixed number of vectors into a higher dimensional input space. For the training process, the Self-Organizing Mapping (SOM) algorithm is used. The fault evolutions are obtained from a process simulator. Each training set must contain the principal measured variables of the process, for each time t. In a SOM structure, the neurons are placed at the nodes of a usually one- or two-dimensional lattice. The neurons become selectively tuned to various input patterns (vectors) or classes of input patterns in the course of a competitive learning process. In competitive learning, the output neurons of the network compete among themselves to be activated or fired, resulting that only one output neuron or a reduced group o f neurons is on at any time. The location of the neurons so tuned (i.e. the winning neurons) tends to become ordered with respect to each other in such a way that, a meaningful coordinate system for different input features is created over the lattice. The SOM algorithm is therefore characterized by the formation of a topographic map of the input patterns, in which the spatial locations (i.e. coordinates) of the neurons in the lattice correspond to intrinsic features of the input patterns.
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This spatial movement of the activated neuron zones is related to the temporal and dynamic evolutions of the input patterns. Then, it is possible to analyze the dynamic evolution of a high dimensional domain by means of the analysis of the SOM output trajectory on the bidimensional neural network structure. 3.1 Process Description A batch distillation of an equimolar mixture of n-hexane and n-heptane is analyzed. This batch operation consists of a 30 stage tower with a total condenser and the reboiler. The operation of the batch distillation unit comprises the following procedure. In order to begin the batch distillation, distillation heat must be provided to the reboiler. A target set point of 5E+06 kJ/hr in 10 minutes is specified on the Duty Control. Once the product specification of 99.5 % n-hexane is reached, the distillate valve can be opened to remove product from the batch column. The Distillate Mole Fraction of nhexane is continuously monitored. Once its value begins to drop, the product flowrate must be stopped by ramping the distillate valve opening to 0 kg/h in 2 minutes. When the Distillate valve is closed, the reboiler heat can be stopped. A ramp target set point of 0 kJ/hr with a duration of 5 minutes is provided. After heat application to the reboiler has ceased, the reboiler and the condenser can be emptied by opening both the distillate and bottom valves. The evolution of the batch process is simulated with the process simulator HYSYS (see HYSYS Manual, 1996). A scheme of the flowsheet is presented in Fig. 2. 3.2 The "normal" operation. The "normal" dynamic operation of the batch distillation unit comprises the procedure before mentioned. Different runs varying the initial conditions with the process simulator in order to have a group of normal trajectories have been made. This group of trajectories defines a normality band for each variable at each time (table 1). We assume that a normal batch evolution may be partitioned into four Pseudo Continuous Blocks (PCBs). The postulated PCBs corresponding to this process are described in table 2.
,Trajectode Name , Initial C b ~ p i ! ( C 6 ) ' i~lnifial Normal 0.5 Normal Comp - 1 0 % 0 .4 5 Normal Comp + 1 0 % 0 .5 5 Normal Reb. Level + 1 0 % 0.5 Normal Reb.Leve1-10% 0.5
Valid from the beginning to the achievement of 50% level in the condenser From the end of PCB A to the achievement of Xhex= 0.995 (Temperature T = 342 K). From the end of PCB B to the moment in which the distillate valve is closed ( Xhex begins to decrease)
R~i!iL:e~l 80% 80% 80% 88% 72%
From the end of PCB C to the moment in which the distillate and bottoms pumps are activated.
Table 1: Normal evolutions.
Table 2: PCB List Figure 3 and Figure 4 show the normal level evolutions and the temperature evolutions of the reboiler and the condenser, respectively. Both figures point out the PCB A and PCB B border limits for the normal operation. 365370 .........................- - . . . . . . . . . . . . . . . . . . . . i
Tem Reboiler perature Vessel
360
Condenser Vessel
g
355
~.
350
Temperature End
A
. . . . . . . End PCB B
345 340 335
PCB
~ ~ v
100 90 80 70
Cond. Liquid Percent Level Reboiler Liquid Percent Level End PCB A
60
so 40
....... End PCB B
30 20 10
0
2000 Time (sec.)
4000
Figure 3: Temperatureevolution. Different PCBs
0
0
,
,
1000
2000
3000
4000
Time (sec.)
Figure 4: Level evolutions. Different PCBs
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3.3 The Fault Set. The data of the faults evolutions to be considered for the NN training process are obtained using the dynamic process simulator HYSYS. Each training set contains the principal measured variables of the process, for each time. Various types of failures are simulated at different stages of the process, like variations in the feed initial composition, different reboiler faults, temperature and level sensor faults and reflux valve faults. In PCB A the main faults are associated with the reboiler, initial compositions and sensor failures. Figure 5 shows an evolutions set for reboiler duty, based on faulty and normal operations. Faulty states are due to a higher or a lower magnitude of the heat duty, occurred at different times during the ramp period.
Figure 5: Reboiler Duties for Faulty and Normal Operations.
Figure 6: Reflux flows for reflux valve faults (the maximum opening capacity can not be reached) with different fault amplitudes.
In PCB B, faults in the reflux valve are added to the before mentioned fault set. Figure 6 shows the reflux flows across the different PCBs when a reflux valve fault is present. Is interesting to note the change in the PCBs limits produced by the faults. In these example the system implemented is capable to make the diagnostic before the change in the PCB limit. In other cases, these kinds of situations must to be addressed by the ES. 3.4 The Fault Detector Module. SOM Outputs. The variables are conveniently filtered. Based on the trajectories considered as normal, a normality band was defined. The differences with respect to the variable values in faulty evolutions were calculated upon this band. The signs of these differences were used afterwards for the network training and evaluation. So, each variable in the training patterns has a (+1) if the variable value is above the normal range, a (-1) if it is below and a (0) if it does not get out of the band. Figure 7 shows the network output in the presence of different faults produced at the beginning of the PCB A (not simultaneous faults). The discrimination power of the trained network could be observed. Also the system has shown a good response wen a unknown fault was present (fig. 7) identifying the abnormal situation without overlapping with the trained faults. To evaluate the NN identification capacity for the same fault at different times (within the PCB A) several cases are shown in Figure 8 (low Qr). All these cases were identified on the SOM NN lattice by the same group of wining nodes.
Figure 7: PCB A. Different faults at the same time
Figure 8: PCB A the same fault (Lower Qr) at different times.
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Figures 9 and 10 show some of the results obtained for faults simulated in the PCB B. In figure 9, faults are shown in their possible maximum open capacity in the reflux valve. In figure 10 the discrimination power of the SOM for the considered fault set is shown.
Figure 9: PCB B The same fault (Reflux Valve) with different fault amplitudes at the Same Times (PCB B)
Figure10: Qr high and low together with the faults in the level and temp. sensors in the reboiler (PCB B).
3.5 System Performance. The upper hierarchical level is an ES developed to receive the outputs generated by the SOMs and to identify the current state of the process (see Fig. 1). The conclusions obtained by the ES must be shown to the user in a friendly format. The ES must contain the necessary knowledge to identify the principal characteristics of the process variables, gain changes, the corresponding PCBs and the time at which the change occurs. So, the functions of the ES are: a) To define the SOM which must be activated, according to the current time and process evolution. b) To interpret the SOM outputs in order to identify the causes of the process malfunctions. c) To advice the operator when an abnormal situation appears, informing the causes which give origin to the detected fault. In those limited situations where a list of potential faults should be considered, the ES generates a ranking of possible fault causes with the associated probabilities. This information must be presented to the operator as friendly as possible, in order to avoid misunderstandings. The networks outputs (best matching units) are used by the ES for the final diagnosis. Then, when the network identifies an unique fault, the ES analyses the information generated by the NN in order to validate the detected fault while possible untrue symptoms generated, for instance by instrumentation errors, are discarded. In those cases, where the network is not capable of detecting with high precision an unique fault due to the overlapping of the actual output with respect to the different regions defined during the network training phase, the ES determines the most probable fault, by means of analyzing the process characteristic conditions. The Knowledge Base (KB) includes a set of rules IF-THEN, that verifies the behavior of determined characteristic variables which are associated with each fault. We choose a best matching unit vs t representation for the faults evolution to show its time dependence in an explicit way. This kind off representation allows to observe how faults that occurs at different moments activate the same group of nodes associated to each fault class. On the other hand also has been verified that in this example none of the trajectories shown in figures 7 and 10 produce an overlapping in the bidimensional lattice of the SOM NN. An important task in the design of the upper hierarchical level is the decision about which variables (measured or estimated) is convenient to be used at this level, and how to use this information in order to improve the performance of the overall system. For example checking the deviations in key variables or calculating the residuals from the process model. As a result of these approach it is possible to reduce the neural network size using an estimated variable (derivatives, process parameters, etc.) while the original values can be used by the ES to improve the diagnostic. For example if the fault Qr high occurs the ES can use information about the controller settings, and measured values in the auxiliary heating stream to refine
810
the diagnostic. In these cases three possible causes associated could be considered: high heater flow, high temperature and wrong configuration in the controller parameters. The ES must include the knowledge required to identify the right cause. In this way the KB could be structured in different specific classes of IF- THEN rules (Fig. 1). Finally, the same fault could produce different consequences, according to the PCB where the fault occurs. The corresponding fault patterns are identified by the ES as "different patterns". But even if these "different patterns" are received by the ES, it should be prepared to identify the same fault, considering the PCB activated and the process variables values involved in the fault. However, the recommendations generated to the operator, could be different according to the current fault consequences.
4. CONCLUSIONS A Fault Diagnostic System for batch distillation columns is presented in this work. Also the problem presentation and some guidelines for the system implementation are developed. To achieve both a general scope and a satisfactory efficiency, a hybrid hierarchical system must be implemented. The decomposition of hierarchies, which handle different modules, comes from the temporal evolution of the process. The handling of the high non-linearity of the model and the different natures of the represented knowledge impose a multiple-modeling cooperative approach. This defines the need of an ES working as a control level that organizes all the system architecture and the real time reasoning process (procedural knowledge). This architecture proved to be very efficient even for a complex transient process like the one here analyzed. Acknowledgement The authors gratefully acknowledge the financial support of the Agencia Nacional de Promoci6n Cientifica y Tecnolrgica; CONICET and the Universidad Tecnol6gica Nacional, which made this work possible. Also, they thanked to the students V. Cirulli, J. Francesconi and M. Mufioz, who made the most of the process simulations. REFERENCES Dae-Hee Hwang, Ahn Tae-Jin, T. J. Chai and Han C., "Real Time Monitoring for a process with Multiple Operations Modes", Workshop: On-Line Fault Detection and Supervision in the Chemical Process Industry, 4-5 June, Lyon, France, IFP-IFAC (1998). Drozdowicz, B., G. Simrn and N. Scenna, "Fault Diagnosis in Batch Reactors Using SOM Neural Nets", sent to IEEE Transactions On System Man and Cybernetics to be printed (1999). HYSYS Manual, 1996. Hyprotech Ltd. Jacobs R. and M. Jordan, Learning Procedure Control Strategies in a Modular Neural Network Architecture, IEEE Transactions On System Man and Cybernetics.,23, 2, March/April (1993). Maurovouniotis M. L. and S. Chang, Hierarchical Neural Networks for Process Monitoring, Comp. Chem. Engng., 16, 4, 347 (1992). Pascal J.C, and D. Andreu, "An Hybrid Reference Model for On-Line Detection in Discontinuous Systems", Workshop: On-Line Fault Detection and Supervision in the Chemical Process Industry, 4-5 June, Lyon, France, IFP-IFAC (1998). Rippin D.W.T., "Batch Process Systems Engineering: A Retrospective and Prospective Review", Comp. and Chem. Engng., 17, sl-sl3 (1993). Shamma J. and M. Athans, Analysis of Gain Scheduled Control for Nonlinear Plants, IEEE Trans. Automat. Contr., 35, 898-907 (1990). Tarifa E. and N. Scenna, "Some Aspects about Fault Diagnosis in Transient Processes", sent to I.E.C to be printed (1999). Tarifa E., and N. Scenna, "A Fault Diagnosis Prototype for a Bioreactor for Bioinsecticide Production", Reliability Engineering & System Safety, 48, 27-45 (I995).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
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D y n a m i c S i m u l a t i o n o f the B e h a v i o u r o f Pressure R e l i e f S y s t e m s Juha-Pekka Pokki *, Juhani Aittamaa and Markku Hurme Helsinki University of Technology, Laboratory of Chemical Engineering and Plant Design, P.O.Box 6100, FIN-02015 HUT, Finland [email protected], [email protected], [email protected] * author to whom correspondence should be addressed Abstract Pressure relief of near critical processes working at two phase region is simulated dynamically. The design procedure for pressure relief devices is demonstrated by an example. The example shows that the required relieving area of the safety valve is dependent on time and scenario. This is the basis for the selection of the type and the size of the safety valve. When the process relieves through the selected valve the outflow and the pressure of the protected vessel varies dynamically after the start of the relief. This phenomenon can not be discovered by the static approach that is normally used in the design of pressure relieving devices. In addition, the sizing equations are difficult to apply in near critical region and they contain values of coefficients that are sometimes difficult to obtain. The sensitivity may be so large that different size of the relieving device is selected. Usage of two common models, AP! 520 and DIERS, in the region of two phases is discussed. Introduction Often the pressure reliving devices are designed based on the assumption of static behaviour of the system but almost every case is dynamic. The static approach may be an appropriate method for processes where the emergency outflow does not change its phase. During the relief the fluid relieved may be in several states, liquid, vapour-liquid mixture and gas state. This gives challenges for the design of safety systems that must be capable to handle various process conditions. The reasons for the emergency relief include fire, reduction of cooling, power failure, blockage in some process equipment or piping, malfunctioning of some control equipment, feed at abnormal rate, abnormal weather conditions, etc. The consequence of these can be the increase in pressure of some process equipment. The equipment is designed to tolerate certain maximum pressure which may not be exceeded. The pressure higher than maximum may cause the equipment to rupture which must be prevented with pressure relieving devices. These operate independently of the control system. The pressure relieving device opens a particular route for the material to escape to a safe location. Principles In the previous paper, (Pokki et al., 1999), the dynamic simulator was presented. It simulates the abnormal process conditions of a chemical process operating even near the mixture critical point. This simulator allows also the calculation of polymer flow in design mode. In this paper a reactor containing fluid, polymer and catalyst is studied. The fluid in this work means not
812 only normal hydrocarbons but also quantum gases like hydrogen and nitrogen in liquid, vapour, gaseous, vapour-liquid or supercritical state. The reaction rate of each monomer depends on temperature and on concentration of monomer and catalyst. Reactors are usually protected with pressure relieving devices at the top of the vessel. The emergency outflow of this process contains always fluid but it can also contain polymer and catalyst particles. The outflow contains only fluid if the polymer settles down in the reactor or fluid, polymer and catalyst if the polymer and catalyst do not settle down once the abnormal event starts. This simulator operates in two modes, in design and in rating mode. The design mode is based on the specification that to keep the pressure of the reactor at the set pressure of the pressure relieving device a certain amount of mass per time must be removed from the reactor. In the rating mode the pressure relieving device is specified and temperature, pressure and many other thermodynamic properties of the reactor are calculated. This mode requires the specification of the pressure relieving device. It includes its discharge area, valve coefficients and the characteristic curve, i.e. the opening of the valve as a function of pressure difference. The fluid flowing through the relieving device may split into two phases. In the previous paper this was not studied because the mass flow was the primary interest. This paper presents the developments made in the simulator. However, the simulator still assumes that the pipe before the pressure relieving device is short enough and the pipe after the pressure relieving device is wide enough that the flow is not affected by the conditions before and after the pressure relieving device.
Valve models API 520, (American Petroleum Institute, March 1993), is a widely used recommended practice in petroleum industry. The equations are divided into vapour or gas equations and liquid equation. The selection between critical or subcritical vapour or gas flow is based on critical pressure ratio that is calculated from.
Per~P1 = [2/(k + l)] kIck-l)
(1)
where ratio of heat capacities k = cp/Cv and pcf is the critical flow throat pressure and pl is the upstream relieving pressure. As pcf > Pb the vapour or gas flow is critical the API 520 equations (2)-(4) are used and as pcf < Pb the vapour or gas flow is subcritical the API 520 equations (5)-(7) are used, where Pb is back pressure. As the relieved fluid is liquid the API 520 equations (9)-(12) are used. This requires that the fluid remains liquid as it flows through the valve. When critical conditions exists some questions arise. The heat capacity Cp increases near the critical point and the value of k can be much larger than 2 which is the biggest value mentioned in API 520. Also in the near critical conditions the value of compressibility deviates much from unity. Two phase flow is computed as a combination of these equations. This procedure is not clearly specified in API 520. The fluid is flashed either to critical flow pressure or back pressure whichever is greater. Then the area for vapour is computed with vapour or gas equations and area for liquid is computed with liquid equation. The area of the valve is equal or greater than the sum of vapour and liquid areas. According to Leung and Nazario (1990)
813 the API 520 method may lead to undersizing of pressure relieving devices. They modified the API 520 liquid equation by using the pressure drop from upstream pressure to critical flow pressure, not the pressure drop from upstream pressure to downstream pressure. This modification gave better results in their work. The DIERS method, (Leung, 1996), for two phase flow is more clearly defined than API 520. It is said to apply for systems where thermodynamic reduced temperature is less than 0.9 and reduced pressure less than 0.5 and above those conditions it underestimates the mass flux and thus overpredicts the area. The method is developed primarily for one-component flashing fluids and limited success for multicomponent mixtures exhibiting minor vapour phase composition change during venting. The selection of model is a complicated task because the models available are limited to certain conditions but the process may enter various conditions during emergency outflow. The calculation procedure of DIERS determines first the parameter omega, 0), and then the critical pressure ratio tic = p c / p o is iterated from
1]2c+( 0)2 - 20)~1 - 13c)2 + 20)21n(13~)+ 20)20-13 c)=0
(2)
In this work the explicit expression for critical pressure ratio is obtained from Nazario and Leung (1992), and used as an initial guess for iteration. r/c = 0.5 5 + 0.217(ln(_o)- 0.046(ln(.0) 2 + O.O04(ln(.o)3
(3)
The equation above gives a good initial guess as co > 0.4, but if 0 < co < 0.4 r/c = 0.3 is a good initial quess. As critical pressure ratio is solved the mass flux G is calculated from
G ) 1 - - I - 2 / 0 )~.l n / ~\ P/ +o (J 0 ) - l I l - / P ) ) ) / 0P 0) / ~
( p o / l ~ 0 /2
/
--1 +1
(4)
where v 0 is specific volume at inlet conditions. If pc < Pb the flow is not choked and p is set equal to Pb. If pc > Pb the flow is choked and mass flux G is computed from
Gc/(Po /Vo ),,2 =13c/oi,2
(s)
The area of the orifice is then A = W / K G , where W is the required mass flow and K is the valve discharge coefficient. Basic equations The basic equations presented in the previous paper, (Pokki et al., 1999), are not repeated here. To summarise the model it was assumed that the reactor is a vessel that has a defined volume. The shape of the reactor was not included. If the reactor is high the pressure in the bottom is higher due to hydrostatic pressure and there may be also temperature gradients. The heat capacity of the wall of the vessel and the effect of solved polymer in the fluid are
814 ignored. In the previous paper the speed of sound was used as the speed of flowing fluid. The validity of this assumption decreases as the fluid becomes dense. That is why the kinetic energy term is removed. Example This example consists of a reactor, volume 30 m 3, two storage tanks, both 70 m 3 and a buffer tank, 120 m 3, see Figure 6. The reactor is in a runaway condition, the buffer tank is not affected by fire but the storage tanks are exposed to fire. The content of the reactor is mainly propylene and small amounts of near boiling hydrocarbons. The weight fraction of polymer to total mass of monomer and polymer is 0.15. It is assumed that the reactor is already at the two phase region as the runaway phenomena begins. The reaction rate inside the reactor is temperature dependent and follows the bell shape curve. The contents of the storage tanks are in two phase region. The storage tank number 1 is nearly full of liquid and the relief is two phase flow but the storage tank number 2 is half full of liquid and the fluid relieved is vapour only. The purpose of the buffer tank is to smoothen the flow to the flare and partly store the vented material. The design procedure starts with the definition of the user-given worst case scenario. Reasons for a runaway reaction are for example equipment malfunction, power failure, human error and fire. The runaway reaction is usually much more dangerous situation than the fire only. The equation of state used is Peng-Robinson. (Peng and Robinson, 1976) At first the system is simulated in design mode to find the first estimate for the area of pressure relieving device assuming that polymer does not flow out. The set pressure of safety valve, SV, of reactor is 3.5 MPa(a), the set pressure of SV of vapour-liquid filled tanks 1 and 2 are 1.5 MPa(a). The flow from buffer tank to flare starts as pressure increases over 0.2 MPa(a) and the flow area is set to 10332 I I l m 2 t o keep its pressure below 1 MPa(a). The reactor reaches the set pressure soon and the required area can be seen in Figure 1. The area requirement for vapour-liquid filled storage tanks 1 & 2 can be seen in Figure 2. The larger requirement of area in storage tank 1 than tank 2 is due to two phase flow and stronger influence of fire.
3000
4000 AP1520
3000 2000
-L
1000
<
0
DIERS
500 time/s 1000
Fig. 1 Requirement for the instantaneous area, SV of reactor.
ZOO0 1000 E E0 <
tank1,API1 5 ~ tank2, AP1520 ~k"~
~""~
..... t
__~___I tankl'DIERSt
4 500
time/s
1000
Fig. 2 Requirement for the instantaneous area, SV of storage tank 1 and 2.
Now the first estimates for areas of SV are known assuming the polymer stays inside the reactor. In two phase region the boiling may mix the polymer. If the polymer flows out, its
815 mass flow is essential to know. Next the same scenario is simulated in design mode when the hydrocarbons and polymer flow out. Because the difference in density between hydrocarbon and gas is big the flow of polymer is assumed to end as the content of the reactor enters from two phase region to dense gas region. The polymer flow is in the beginning of the relief 9 kg/s and decreases almost linearly to 2 kg/s as the relief stops. This gives the basis for the design of area of SV for reactor. The area taken by the polymer must be taken into account when also polymer flows out compared to only hydrocarbons flow out. The next step is to simulate the system in rating mode. The area of SV of reactors and of storage tanks 1 and 2 are 2800 mm 2, according to API 526, (American Petroleum Industry, June 1995). These areas are selected based on the results of design mode shown in Figures 1 and 2. In this case the modified API 520 and DIERS gave almost identical areas for two phase flow. This should not be understood as a general rule but in this example the discharge coefficient of DIERS is set to 0.75. This is an indication that the results are sensitive to the valve models used and their coefficients. Figures 3 and 4 show the temperature and pressure trends of the systems. Now the valve models used are DIERS for two phase and API 520 for gas flow. Polymer is assumed to stay in the reactor. The two phase relief form storage tank 1 behaves very smoothly but the vapour relief from tank 2 oscillates. A longer simulation time than 1200 s is required to find out if the pressure starts increase in tank 1.
450500 [1........r.....e......a......~..................qi.............................. 4
...........................................
reaot~ I
2.5 350 300
I ~
tank1........ i
250
2] tank1
0.5 ~
200
0
0
500
timels
1000
Fig. 3 Temperature in rating mode.
I ~ l
0
500
..........................
time/s
1000
Fig. 4 Pressure in rating mode.
Storage tank 2 is also exposed to fire but its heat input is only 50 % of that of tank 1 because the liquid amount is smaller. In this example the safety valve is selected the same size because the volume and purpose of the storage tanks are identical in this example. The runaway of the reactor causes the highest peak in the pressure of the buffer tank. The storage tank 2 causes the smaller peaks. The streams are shown in Figure 5. The storage tanks behave much more slowly than the reactor. As the vessels are exposed to fire the surface temperature of the wall of vessel becomes very high. This causes a risk for rupture of vessel material.
816 45 40 35 30 25 20 15 10 ~5 ~o E 0
/
~
/,
reactor / buffer
'
tank 1 /
I
buffer
[---'--.......t a n k - ~ ~
]'qll\lll 500
I! time/s
I]
I
1000
Fig. 5 Streams in the system in rating mode.
Fig. 6 Schematic figure of the example
Sometimes the capacity of the flare may be the limiting factor in plant design. The dynamic simulation helps in adjusting the volume of buffer tank and flow to flare. In general the increasing the volume of buffer tank lowers the peak in pressure and shifts it later. Decreasing the size of the SV to flare decreases the flow but increases the peak pressure and makes the peak take place earlier. These general ideas are much easier to study by dynamic simulations. Conclusion It is shown that the results of the simulation are sensitive to the relief valve models used. The models suffer also experimental validation at near critical conditions. Also the selection of the scenario is very important in finding out the most severe conditions the system may enter. The near critical conditions are a challenge for the simulation and the modelling. The consideration of the dynamic nature of the system is crucial in design of pressure relieving devices. The dynamics reveals the behaviour of buffer tanks that affect the back pressure of pressure reliving devices. Acknowledgements Financial support from Neste Research Foundation to (J-PP) is gratefully acknowledged. References American Petroleum Institute, (June 1995), API Standard 526, 4 th ed., API, Washington, DC. American Petroleum Institute, (March 1993), API Recommended Practice 520, Part I - Sizing and Selection, 6th ed., API, Washington, DC. Leung, J.C., (1996), Chem. Eng. Progress 92 No 12, 28-50. Leung, J.C. and Nazario, F.N. (1990), J. Loss. Prey. Process Ind., 3,253-260. Nazario, F.N. and Leung, J.C., (1992), J. Loss. Prey. Process Ind., 5,263-269. Peng, D.-Y. and Robinson, D.B., (1976), Ind. Eng. Chem., Fundam. 15, 59-64. Pokki, J.-P., Aittamaa, J., Keskinen, K.I. and Hurme, M., (1999), Comp. chem. Engng. Supplement, 399-402
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
817
From Environmental Care to Sustainability the role of CAPE tools and methods Jan Venselaar 1 Akzo Nobel Engineering Consultancy, Safety & Environmental Engineering, P.O. Box 5136, 6802 EC Arnhem, The Netherlands Present technological practice will not suffice to address environmental issues on the long run. Radical changes in processes and products and in operating business are required to attain real sustainability. Chemical industry will have to aim for drastic reduction of resource use and switch to renewable resources. Chemical engineering must focus on the key enabling technologies and develop the methodologies and tools to implement these. CAPE tools are essential in this respect. New and optimised tools are needed to support sustainable solutions. Experts in CAPE and sustainable development will have to combine forces on this. 1 INTRODUCTION
A reader survey by Chemical Engineering Progress on the views and expectations for the new millennium showed that the two main issues seen as pivotal for the chemical engineering profession are the tempestuous development in use of computers/PC's, process automation and simulation tools and the increasing influence of environment related factors (Mascone 1999). Environment care has brought about many changes in technology and operations. Supporting that computer aided tools and methods have been adapted and newly developed. The last decade sustainable development has become a new paradigm world-wide, for politics, economy and technology. That again asks for new approaches, new technologies and new computer aided tools to meet that. This paper introduces the development from environmental care to sustainability, and summarises the consequences sustainable chemical engineering has for CAPE. 2 SUSTAINABLE DEVELOPMENT
Sustainable development was set on the agenda by the Brundtland Commission. Its definition: "meeting the needs of the present without compromising the ability of future generations to meet their own needs" is acknowledged world-wide. In practice it is translated to better use of the available resources, better distribution of prosperity world-wide and taking into account already present environmental problems and a future growing world-population. Taking into account an increase of world-population with a factor 2 in the future, a justified claim for increased prosperity globally (thus an increased consumption with at least a factor 3 till 4) and the fact that environmental problems exists already, it is concluded that to reach real sustainable development in the future a substantial improvement in the efficiency with which we use resources is needed, at least with a factor 20. That concerns use of raw materials, energy, space, preventing emissions 1 presentlychairman of the EFCEWorking Party on Environmental Protection
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and loss of ecological values such as biodiversity and improving quality of live. To do that, totally new processes, products and ways to organise our economic activities are needed: a "system change". Just 'simply' improving the present technology and ways of doing business may suffice to reduce the environmental impact on the short term. However with a growing population, growing consumption and a growing economy, that effect will quite likely be off-set by that growth. We need to do better and drastically so. Real sustainable development requires a shift from optimisation and better management, to new more 'eco-efficient' processes and products and ultimately to new system approaches to supply the services we need. Figure 1 illustrates this.
O
sustainable options 20
... 10
"~
ment
5
optimisation !
2000
I
2010
2050
Fig. 1. Attainable improvement in the three development steps It should be clear that technology alone cannot alone lead to sustainability. An integral approach including technological, cultural and socio-economic factors is asked for. 3 FROM ENVIRONMENTAL CARE TO SUSTAINABILITY
The three phases in taking care of the environment: optimisation, process and product improvement and eventually sustainable options, have each a different approach with their own technology requirements and need for specific tools. Besides, although in the end sustainable options must dominate, the other approaches are essential and will stay so. Each real solution will be a mixture of the three and the tools have to cover that. Optimisation mainly involves the 'classic' environmental technology (commonly 'endof-pipe' and remediation) and measures aiming at so-called 'good-house-keeping', formalised in environmental management systems. This is being implemented in most industries and concentrates on the own production activities. It is for the present the major approach to handle environmental issues. Dependent on the earlier situation a reasonable improvement, with a factor 2 or 3 seems attainable, but not more. Structural process and product improvement takes it a step further aiming at optimising existing processes and introducing new efficient processes. For products, a broader scope of environmental effects is taken into account, for the whole lifecycle,
819
including the effects of resource exploitation and actual use. The aim is prevention instead of "cleaning after the fact". Much research and development takes place and it will form the main basis for environmental improvement in the coming decade(s). Implementation however is still sluggish for the moment because of technological, financial and business constraints. Key approaches are pollution prevention, recycling, process integration, process intensification, higher conversion, efficient separation technologies and in particular for product development: total lifecycle management. Another emerging development is 'industrial ecology'. In due course all this must evolve into 'sustainable chemical process technology'. That will take time. Elements of such an approach are use of renewable resources, as biomass and solar energy, 'dematerialization', strict total lifecycle product and mass balance management, product-service systems, focussing on other ways to fulfil customer-needs. Elements from the two other approaches will be essential. This phase will take decades to be reached. The research programs to that aim are for the moment focussed on defining how such a future could look like, which technology might be promising or essential and which key technologies have to be developed as first steps towards it. (DCO 1999, Venselaar 1999) The first two approaches are the main options for the short and the medium term. Effective sustainable solutions imply in many cases drastic changes and can be reached only stepwise. Existing processes and installations will continue to be used in the coming decades, nevertheless substantial environmental improvement and reductions in resource use is required. That is a challenge but offers also opportunities to investigate routes to more structural improvement. That stimulates evolution where, for the short term, revolution is not always economically viable. 3 TOOLS AND METHODS FOR SUSTAINABLE DEVELOPMENT
Many CAPE tools 2 are used for improving environmental performance and to aid environmental management. More are being developed, being modified and as yet not envisaged tools are certainly needed in the future. They differ in application and form. Applications are modelling, assessing and predicting, improving performance or support design. It are physical and statistical models calculating 'exact' results (insofar the input is exact), general assessment protocols which support inventory and evaluation, often offering only qualitative results. Besides there are many procedures and methodologies, checklists, and decision schemes framed into a software program, sometimes incorporating expert systems. There is much interest in the field because environmental issues are accepted as critical design factors and sustainability has become a new design paradigm. A division based on applications, in view also of the different approaches towards sustainability appears useful. One should keep in mind however that such division is not rigid. Such categories are: 1. determining effects of specific environmental issues, such as concentrations after dispersion, physical effects, direct health; 2 The concept "tool" used here covers the whole range of means for inventory, assessment, optimising, simulation, calculation, selection, guiding process control and whatever is possible, based on more or less formal theories, methods and procedureswhich are translated into computer programs or are 'framed' in software to make them better usable (such as databases and spreadsheets).
820
2. inventory and assessing total environmental impact or 'unsustainability' for processes, product chains cycles, activities etc.; 3. modelling and improving specific environmental activities, technology, equipment 4. modelling and optimising whole processes, installations, group of installations, industrial system etc with emphasis on environmental performance; 5. design paradigms and methodologies, selection of (more) sustainable options, processes, resources, components; 6. support for care systems, environmental management, product chain management. Many tools are just 'normal' CAPE tools used with an environmental purpose or simple adaptations of standard tools. There is also a growing body of specialised tools developed specifically with environmental aims in mind. The review hereafter is only a concise one and from an environmental angle. It underlines the specific uses, the variety 3 and draws attention to interesting new options specifically useful for aiding sustainable technology development and design. 3.1 Effects of specific environmental issues
These are the 'basic' tools for 'visualising' quantitatively the effects of emissions, pollution, etc. They are mainly based on physical and chemical models sometimes combined with statistics for climate and such, focusing on one specific environmental aspect. They are used to check if requirements set by laws, regulations and permits are met or to compare different measures or design options: - dispersion of pollutants in air, for specific conditions or yearly averages; - spreading of pollutants in soils sometimes including chemical conversion; - real time tracking of gas-clouds in case of incidents for warning purposes; - fate of substances in the atmosphere, eg ozone depletion. It does not lead directly to improvement options but helps to assess measures for their result. The precision of the tools is continuously improved, by inclusion of better models for the influence of surroundings, taking into account adsorption and decomposition of compounds and predicting actual effects on people and ecology. 3.2 Total environmental impact and 'unsustainability'
This category comprises tools for Environmental Impact Inventory and Assessment and for Environmental Performance Indicators (EPI's). Well-known are the Product Life Cycle Assessment tools (LCA). They are essential to monitor environmental care and sustainable development and get as such much attention, also in publicity and politics, because 'yard sticks' are in demand. They provide the possibility to assess, compare, define the major issues for environmental impacts. Agglomerated numerical results are used sometimes to rate the process, activity or product on an environmental or sustainability scale. It forms the basis for 'green-labels' for products and business. The tools provide systematic inventories of environmental impacts and resource use during the whole lifecycle of installations and products. They can include a translationstep to get a specific value in terms of the chosen 'yard stick'. Results can become 'corrected' or weighed with a factor to account for (political) relevance. The differences between the various methods and tools are large. The most obvious one is the choice of yard stick. Some examples are: 3 This paper can only give a schematic overview of relevant CAPE tools. Therefore no references are given but see Cano-Ruiz & McRae (1998) and Pistikopoulos (1999) for some more information.
821
-
energy: Energy Use Accounting ; money: Environmental Cost Accounting, Life Cycle Costing; with variations in methods defining 'costs' eg based on willingness to pay, investment or damages; - mass, area used: materials intensity per service, total area used, the Ecological Footprint; the Rucksack method (Wuppertal Institute) - more abstract measures in relation to set targets, distance to target method (DTT), weighted impact (political, social, 'scientific'), Environmental Burden. A distinct 'one and only' environmental impact parameter does not exist. Nevertheless each has its useful applications. A not unimportant goal is creating awareness and improving understanding, of industry, society and politicians. That influences such choice too. The tools differ furthermore in the way they handle data, selection and translation of impacts, system boundaries and correction an weighing factors. Main developments are aimed at improving the models on these points, trying to get better insight, better description and making links with other tools, eg for designing.
3.3 Specific environmental activities, technology, equipment
This category comprises to a large extent just the standard tools for processes and equipment, applied to environmental processes and equipment. Special is that much of the conditions are not so normal compared to 'ordinary' process conditions. Concentrations are low and often quite variable in time, flows and conditions change easily, and the requirements are quite severe, final concentrations to be reached extremely low sometimes. So the tools are to be adapted to that circumstances.
3.4 Total process, installation and activity modelling and optimising
Very specific tools exist to improve environmental performance such as: - energy consumption (HeatPinch, Exergy Analysis, network based methods); - water consumption (WaterPinch, and many other); - mass balance efficiency, mass exchange networks; - waste-reduction: a nearly uncountable number of Pollution Prevention schemes. This category comprises many of the 'normal' chemical engineering tools (such as those from AspenTech). Essential is that environmental parameters play a critical role. In the framework of SUSTECH a program called CAPRI is set up to develop more sophisticated process engineering design and control tools which explicitly take environmental and sustainability parameters into account too. 'Translation' and inclusion of new design methodologies: such as Concurrent Engineering, Process Integration, Process Intensification, Process Synthesis into the various tools needs much attention. Such tools should be developed too for 'industrial ecology', for networks of industries, optimising the combined performance through utility sharing, clever use of byproducts.
3.5 System change approaches Sustainable development needs radical changes, new approaches and systematic search and evaluation over a wide area. That requires special tools: - new design approaches, stimulating other process-routes, resources, technology; - selection tools and intelligent information tools eg to find technologies available and generic process synthesis approaches proven useful; - modelling and simulation tools which take into account more factors as costs, resource availability, economic development, etc. The Forrester models used for
822
the Club of Rome fall in this category. For practical use such models should be aimed on much smaller systems, an industry branch, a product chain. Here the potential of combining various tools becomes the most obvious. 3.6 Care systems aimed at environmental performance Main tools here are administrative, making inventories, keeping records of activities, emissions, regulations, storage. They can be linked to the actual process information systems, to have a more or less real time overview of the situation and warn when deviations occur. Simulation tools for the production system as a whole would be interesting to assess strength and weaknesses of the organisation and operation. 4 SOME GENERAL REMARKS AND CONCLUSIONS
Trends anticipated and desired are, from my view: - a general tendency for more sophistication and realistic models; - ongoing "computerisation" of selection, information and decision tools; - more and more linking and "crossbreeding" of all categories of tools to make them more versatile and to strengthen their usefulness as aid in real sustainable development. An example is the LCA tool which principles are used quite broadly. Nevertheless the growing influence of computer tools in development and design is not unconditionally positive. A remark made in the earlier mentioned reader survey noted: "PCs will .... speed our work, but they also take away some of our ability to make common-sense decisions". And whatever way you look at it, "the main creative, most versatile tool for engineering is still the human brain". Computer aided models and tools must stay supporting and advising tools. The engineer should exercise the final responsibility. That necessitates too that those tools are transparent so the user can to a sufficient extent trace and interpret its results. At the same time CAPE tools should be so user-friendly, that all engineers can use them. Only so they will be used to their full potential and really contribute to better environmental performance and sustainable development. Lastly, no CAPE tool should exist without environmental constraints and the drive towards sustainability as a crucial parameters in their set-up and performance. REFERENCES
Cano-Ruiz & J.A., McRae, G.J. (1998) Environmentally Conscious Chemical Process Design. Annual Review of Energy and the Environment, 1998 (23) 499-536 DCO (1999), Sustainable Technological Development in Chemistry, Improving the Quality of Life through Chemistry and Agriculture, DCO report (Netherlands' Foundation for Development of Sustainable Chemistry), Wageningen NL, 1999 Mascone, C.F. (1999), Engineering the Next Millennium Chem. Eng. Progress 1999, October, 102-12 Pistikopoulos, E.N. (ed) (1999) Design and Operation of Sustainable and Environmentally Benign Processes, Special Issue Computers & Chemical Engineering, December 1, 1999 Venselaar, J. (1999), Need and Opportunity for Sustainable Process Engineering, towards 'our common future', Proceedings ECCE2, Montpellier 1999
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
823
Accounting for sustainability requirements in process design M.P.C. Weijnen, P.M. Herder and H.D. Goel Delft University of Technology, Department of Technology, Policy and Management, Industry, Energy and Environment Group P.O. Box 5015, 2600 GA Delft, The Netherlands
Abstract The incorporation of sustainability requirements in process design calls for a new process engineering paradigm, and for a new knowledge management strategy that effectively supports the development of life cycle process models in the chemical process industry. 1. INTRODUCTION Sustainable development criteria play an increasingly important role in chemical industry decision making, i.e., from strategic, high level business decisions, down to process and plant design decisions. The urge for sustainable development of the business is concerned with the need to safeguard the long term continuity of the business: 9 ensure a stable and competitive business-economic performance, 9 protect the quality of the natural environment and its resources, 9 ensure acceptance of the business by customers and society at large. Business-economic motives have driven the development of the chemical industry, and economic criteria govern the decision making in all business functions of chemical companies. At the process engineering level, appropriate methods and tools are available to enable process engineers to evaluate the economic impacts of their decisions, whether it be in process design or in manufacturing operations. The business-economic dimension of sustainable development is fully internalized in all decision making processes, and embodied in a variety of assessment tools. This stage has not yet been reached for the environmental protection dimension of sustainable development. Although this challenge was recognized and accepted by the industry since the 1970'ies, its embodiment in methods and tools to support strategic and operational level decision making is still lacking. The enormous improvement in environmental performance that the chemical industry nonetheless achieved in the past decades thrived on established process engineering paradigms and practices. Structural approaches to environmental protection are still in their infancy. As the chemical industry already finds itself facing another dimension of sustainable development, concerned with the "license to operate", a drastic re-alignment of business and innovation strategies is called for. In this paper we will explore how the chemical industry might develop a truly integrative, three-dimensional approach to the challenges of sustainable development, what new paradigms are emerging and how these may be embodied in new methods and decision support tools. We will focus on the design process, as this is the creative process where innovations are embodied in new plants.
824 2. INNOVATION SHAPING PARADIGMS A paradigm is defined by Wei (1996) as: ".... the whole constellation o f things that defines one scientific discipline or profession and sets it apart from other disciplines... " Paradigms shape the way we look at the world around us, and the paradigms of chemical process engineering shape the process innovation strategies in the chemical industry. Similarly, business engineering paradigms shape the business development strategies of chemical companies and their organization.
2.1. Paradigms of process engineering The evolution of chemical process engineering starts with a pre-paradigm period, before 1915, when chemical engineering was building on empirical insights from mechanical engineering and chemistry. In 1915, Arthur D. Little introduced the concept of unit operations as "steps common to most industrial processes, such as heat transfer, distillation, fluid flow, filtration, crushing, grinding, and crystallization", thus establishing the first paradigm that characterized chemical process engineering as a discipline. The empirical unit operations approach was not extended with a new paradigm until the 1960's when transport phenomena were recognized as a basic principle. This development nourished fundamental research and mathematical modeling, a development that was strongly encouraged by the surge in computing power becoming available for chemical engineers. The impact of this paradigm, that has effectively turned chemical engineering from an art into a science, is still evident from the progressive development towards higher levels of detail in the focus of R&D: from general transport phenomena, through computational fluid dynamics and to molecular modeling as present day sources of process and product innovations. Process integration is suggested by Bogle and Perris (1998) to be the new paradigm of process engineering. The widespread adoption of heat integration since the conception of pinch technology (Linnhoff, 1982), and the promising developments of process integration towards mass exchange networks support their suggestion. Remarkable developments that can also be categorized as process integration, albeit in a much broader sense than in its original interpretation for heat exchange networks, are the integration of unit operations into hybrid systems, and process intensification. 2.2. Paradigms of business engineering As the chemical industry developed and the scale of manufacturing operations was expanded, also the volume of waste streams expanded to a point that their processing into byproducts became profitable. The business engineering paradigm of optimizing raw material efficiency through product diversification has largely shaped the complex present day petroleum refineries and the (petro)chemical industry. Until two decades ago, these complex process systems were managed by complex hierarchically structured organizations. Innovations were primarily technology driven, and markets for new products were created subsequently. Since then, the globalization of the economy, the recognition of different dynamics in the markets for different petroleum products and chemicals, and hence, the need for market driven innovation have driven the chemical process industry to a drastic restructuring. In the present day business environment, shareholder value is on top of the priority list, urging companies towards lean and mean business-driven organizations. The focus on core business paradigm created the present day situation in which only a limited nmrtber of world-wide players or strategic alliances operate in each base chemical and polymer market. At the company level, the complex hierarchical organizations have been
825 replaced by independent business units, and within these units each plant is operated as an independent profit center. Natural consequences of the business re-engineering paradigms are a focus on core competencies, a dwindling interest in general research, cross-cutting technologies and engineering skills, and a limitation of process integration efforts to an intraplant scale. Inter-plant (site wide) process integration is virtually limited to intra-company initiatives, and is only becoming scarcer as the different plants in complex production sites are now owned and operated by different companies. 2.3. New challenges- new paradigms? The chemical industry's response to the environmental challenge has so far been firmly rooted in the established paradigms of the process engineering profession. A variety of new unit operations was developed and added to existing plants, either to separate environmentally harmful components or to convert these into harmless substances. The fundamental insights acquired since the transport phenomena paradigm have helped to produce very sophisticated environmental technologies such as membrane separation and selective catalytic conversion technologies, and also to improve the selectivity and efficiency of separation and conversion operations in the primary process. The impact of the process integration paradigm is evident e.g., from the enormous energy efficiency improvements achieved in many companies. In spite of these achievements, however, it is felt that the industry's approach to environmental protection is more ad-hoc than structural. Although the paradigm of a structural, sourceoriented approach to environmental protection has gained acceptance, it is not yet embodied in process design engineering practice. As long as suitable methods and tools are lacking, opportunities to add value to the process and create a competitive edge will be missed. 3. THE DESIGN PROCESS
A good design process starts with a correct formulation of the design problem, specifying system boundaries, design constraints, performance criteria for the design, and the design space. In the conceptual design phase, process alternatives are generated, synthesized, optimized and evaluated on the basis of the specified performance indicators. The need to consider sustainability requirements as early as in the conceptual design phase, is emphasized by the fact that decisions made at this stage typically determine 80-85% of the overall process costs (Westerberg et al., 1997). 3.1. System boundaries The sustainability challenge forces the process designer to adopt a new perspective to the system of 'plant to be designed'. In comparison with the established practice of designing a plant as a stand-alone system, the system boundaries are significantly widened in both the dimensions of time and distance: The designer needs to take a life span perspective of the plant and its products, and he needs to take all possible interactions with the plant surroundings into account (Villermaux, 1996). The environmental perspective does not only relate to the natural environment, but may include neighboring plants that may have an interest in utility sharing, exchange of by-products, etc., thus reducing the overall environmental impact through external process integration. The life span perspective implies a cradle to grave assessment of the design, and implies that the plant must be designed for responsiveness to changes in the business environment (e.g., market, legislation) during its life span.
826 3.2. Constraints and performance criteria
Thus far environmental considerations are incorporated in conceptual design by treating them as constraints to the design problem, designated by environmental regulations and (foreseeable changes in) legislation. A structural approach to environmental protection, however, requires that environmental issues are systematically addressed as design objectives, to which purpose suitable environmental performance criteria must be defined. The environmental dimension of sustainable development entails more than meeting emission limits: it is about maximizing the efficiency of material, energy and water use, about the use of inherently benign substances and production methods, and about pollution prevention throughout the life span of the plant, including its demolition. A major hurdle in applying sustainable development performance criteria is their illdefined and qualitative nature. Even more so than for the environmental dimension, this problem is encountered in dealing with the social dimension of sustainable development. Clear definitions and indicators, both quantitative and qualitative, are needed to support the implementation of these criteria in process design. As shown by Herder (1998), project commissioner, design engineers and other experts involved in a process design need to arrive at a shared definition of design objectives in the design problem formulation stage, to be explicited in a comprehensive Basis of Design, and at a shared agreement on the hierarchy of design objectives, in order to avoid expensive re-work in later stages of the design. 3.3. Methods and tools
In industrial practice, the search process for viable process alternatives within the confines of the design space largely relies on heuristics, and the performance indicators on the basis of which the selection is currently made, are mainly economic indicators (e.g., return on investment). Besides the well-established methods to assess the economic viability of process alternatives, a limited number of methods and tools is available to support an evaluation of the ecological impact of plants and products. Especially environmental life cycle assessment (LCA) methods, originally developed for discrete products and specific materials, are gaining interest for process evaluations. Such evaluations address a wide range of emissions and their environmental effects, but cannot handle non-quantifiable environmental effects (e.g., those concerned with persistent chemicals). Other problems involved are concerned with, e.g., the ranking of alternatives, requiring an ambiguous aggregate score for each alternative. In analogy with the environmental life span approach, also economic design evaluations are increasingly treated in a life span perspective, aimed at minimizing the so-called total cost of ownership or TCO (Ishii, 1997). However, the integration of environmental and economic objectives in the design of sustainable, green or clean processes is still in its infancy. Basically, the present framework of methods and tools is not able to handle design requirements that cannot be converted into costs, and requirements that are of a nonquantitative nature. Major hurdles to be taken for an integrative sustainability performance evaluation of technology and business altematives, including the social dimension of the sustainability issue, are concemed with data uncertainty and ambiguity (quality of information) and the lack of systematic and objective assessment methods (quality of information processing). The quality of information and information processing challenges in a design engineering context are, in fact, problems of knowledge management.
827 4. KNOWLEDGE MANAGEMENT More than data and information, knowledge is a crucial asset, also considered as the fourth production factor. The process engineering knowledge and experience embodied in the design, construction and operation of the existing process installations are recognized by the industry as a critical success factor in the competition for the future markets. Especially design engineering is a knowledge management challenge in itself, as this activity not only draws on many sources of explicit knowledge (data bases, process models, previous designs, etc.) but mainly relies on the implicit knowledge and experience of the experts involved. In spite of the extensive documentation stored on previous designs, it is estimated that only 20% of the knowledge acquired through previous designs is actually captured and reused (Westerberg et al., 1997).
4.1. Knowledge management in conceptual process design As many design projects are cancelled along the way, most companies are hesitant to involve a large design team in the early stages of the design. Hence, the conceptual design is made by one experienced process design engineer, or a small number of designers. It is up to the designer to consult other experts in this crucial stage of the design. Quite often, however, he will find himself under tremendous time pressure and is thus not encouraged to seek information and opinions from other experts. As he relies on his experience, he is prone to making many of his design decisions implicitly, either not even realizing that he is doing so or simply not recognizing the need to explicitly document the 'why' of many of his decisions.
4.2. Knowledge management challenges in business driven organizations In the new business organization of many companies in the process industry, the business functions of research and development, design engineering and manufacturing support have been reduced in size and redistributed over the new business units in such a way as to deliver tailor made services to their business unit. Excellent conditions have thus been created for knowledge sharing between process development, design and operation within the business units. The downside of this change is a deterioration of conditions for knowledge sharing across business units. In the present day situation of harsh international competition between lean and mean business driven organizations, attention is now focused on innovative approaches to knowledge management to ensure that lessons learned from previous projects are captured and shared between different experts, business units and business functions.
4.3. Knowledge management strategies for sustainable development In the strategies employed by the industry to overcome the barriers to knowledge sharing, two fundamentally different approaches can be distinguished: the actor oriented approach and the systems oriented approach. In the actor oriented approach the individual professionals are fully recognized as the carriers of crucial, largely implicit, knowledge. Information and knowledge systems can support, but never replace, the professionals as the generators and carriers of the company's knowledge assets. The systems oriented approach seeks to retrieve the professionals' implicit knowledge and make it explicit, so that it can be stored independently from the professionals that created the knowledge, to be retrieved and reused when wanted. In the design engineering practice of the process industry and engineering contractors both strategies are more or less successfully employed. Within the design process, the parallel or concurrent approach can be seen as an actor-oriented approach to improving
828 the sharing of knowledge between disciplines and between phases of the process life cycle. Between business units, knowledge sharing is achieved through skill groups, workshops etc. An effective response to the sustainable development challenge can only be found in a balanced combination of the actor and system oriented strategies. With respect to the knowledge contents of the design process, many sources of explicit knowledge are used already, and these will only increase with the need to retrieve e.g., operational performance data and environmental impact data for process life cycle evaluations. As information collection strategies and data quality assurance are standardized, the possibilities for explicit knowledge storage, retrieval and processing will also be improved. However, even though the role of explicit knowledge systems is expected to grow, these will never be able to replace the actor oriented strategy. On the one side, there is the fact that experts are not eager to make their expert knowledge explicit and that personal communications are their preferred way of sharing knowledge. On the other side, the many ill-defined and non-quantifiable criteria that figure in process design, can only be dealt with if project commissioner, designers and other experts involved arrive at a shared understanding and interpretation of these design criteria. 5. CONCLUDING REMARKS The search for a structural, source-oriented approach to environmental protection can be seen as a logical step following the paradigm of process integration. The models and tools to effectuate this approach arc in an early stage of development. The life cycle approach is being adopted in "sustainable" process design with respect to both the economic and the environmental dimension, through TCO and environmental LCA, respectively. A promising development is the integration of both dimensions in so-called life cycle process models (Bolton and Pcrris, 1999). Such models can ensure the quality of the knowledge, and the quality of the knowledge processing throughout the life span of the plant. At this point in time life cycle models are not yet available, and knowledge sharing relies largely on communications between experts. The actor oriented approach to knowledge sharing will by definition remain crucial in dealing with ill-defined and qualitative design criteria. If the industry is to deal effectively with the social dimension of sustainable development, this strategy might be further developed to include external stakeholders in the design problem formulation and conceptual design stage. REFERENCES
Bogle, D., Perris, T., CAPE and Its Roles and Benefits, Summerschool in Innovative Process Development, GSCE/Kemira Research Foundation, Savonlinna, Finland, August 3-5, 1998. Bolton, L., Perris, T., A Vision of Future Industrial Needs and Capabilities, CEFIC/SUSTECH, PMSC, Version 1.0, July 1999. Herder, P.M., Weijnen, M.P.C., Quality Criteria for Process Design in the Design Process - Industrial Case Studies and an Expert Panel, Computers chem. Engng, Vol. 22, Suppl., pp. $513-$520, 1998. Ishii, N., Fuchino, T., Muraki, M., Life Cycle Oriented Process Synthesis at Conceptual Planning Phase, Computers chem. Engng., Vol. 21, Suppl., pp. $953-$958, 1997. Linnhoff, B., Hindmarsh, E., Understanding Process Integration: The Pinch Design Method for Heat Exchanger Networks, Pergamon Press Ltd., Oxford, England, 1982. Villermaux, J., New Horizons in Chemical Engineering, Proc. The 5th World Congress of Chemical Engineering, July 14-18, San Diego, U.S.A., 1996. Wei, J., A Century of Changing Paradigms in Chemical Engineering, ChemTech May, pp.16-18, 1996. Westerberg, A.W., Subrahmanian, E., et al, Designing the Process Design Process, Computers chem. Engng., Vol. 21, Suppl. pp.S1-S9, 1997.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScience B.V. All rights reserved.
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An Intelligent System for Identifying Waste Minimization Opportunities in Chemical Processes I. Halim and R. Srinivasan* Laboratory for Intelligence Applications in Chemical Engineering, Department of Chemical and Environmental Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 Pollution prevention is one of the major issues facing the chemical industry worldwide. Increasing environmental awareness and regulations have put pressure on the chemical industry for implementing waste minimization at the source rather than relying on end-of-pipe treatment. Conducting a waste minimization review is time-consuming, expensive and labor- and knowledge-intensive. An automated system that performs waste minimization analysis would reduce the time and effort required for a thorough review and thus is attractive. In this paper, we propose a knowledge-based system, called ENVOPExpert, that can detect and diagnose waste generation in any chemical process, and identify process-specific waste minimization options. ENVOPExpert has been tested on an industrial hydrocarbon separation process. We also present ENVOPExpert's results for the case study and compare it with waste minimization options suggested by a team of experts. 1. I N T R O D U C T I O N Billions of tons of hazardous wastes are generated annually from the chemical industry worldwide. Apart from creating potential hazards, these wastes also represent losses of valuable materials and energy from the production units. Traditionally, control of this pollution relies heavily on waste treatment process added at the end of the production line. Such "end-of-pipe" treatment does not actually eliminate waste but simply transfers it from one medium (air, water or land) to another. Increasing public awareness of the impact of industrial pollution on both human health and the environment has shifted the paradigm of pollution prevention from end-ofpipe treatment to waste minimization at source. Waste minimization is defined as any technique, process or activity which avoids, eliminates or reduces a waste at its source, or allows reuse or recycling of the waste for benign purposes (Crittenden and Kolaczkowski, 1995). When implemented, the benefits of waste minimization are obvious: 9 Waste minimization offers economic benefits through cost saving in waste treatment and disposal, reducing raw material, energy and utility usage, and increasing process productivity. 9 Waste minimization improves the health and safety of the employees by reducing the risks associated with handling hazardous materials.
Author to whom correspondence should be addressed (email: [email protected])
830 9 Waste minimization reduces company liabilities by environmental regulations. 9 Waste minimization improves company's public image.
reducing risks
of breaching
A number of different methods for waste minimization have been previously reported in literature. These can be broadly classified into quantitative and qualitative approaches. In the quantitative approach, pinch analysis or numerical optimization is generally applied to search for potential energy savings and emission reduction. The qualitative approach to waste minimization includes methods such as hierarchical systematic procedure, onion diagram and Environmental Optimization (ENVOP) technique. ENVOP technique is a waste minimization procedure that follows the approach of Hazard and Operability (HAZOP) analysis in process safety (Isalski, 1995). Like HAZOP analysis, during an ENVOP study, each line and unit is evaluated systematically by combining process variables (such as pressure, temperature, volume, etc.) with qualitative deviation guidewords (such as more, less, etc.) to identify and analyze waste minimization options. Table 1 lists some common process variables and deviation guidewords used in the ENVOP study. Table 1. Process variables and deviation guidewords used in ENVOP study Kejword Flow Temperature Pressure Level Composition Equipment
Deviation No, More, Less, Recycle/bypass More, Less More, Less More, Less Change, Add, Remove, Phases More/Larger, Less/Smaller, Alternatives
Waste minimization is a team-based effort requiring significant skill, expertise and experienced team members. A thorough waste minimization procedure is therefore time consuming, expensive and knowledge- and labor-intensive. This has caused a major technical barrier for implementing waste minimization program within the industry. The application of Artificial Intelligence (AI) techniques particularly expert systems to automate waste minimization procedure is therefore attractive. Despite the importance of automating waste minimization procedure, there has only been limited work on-going in this area. Huang and Fan (1995) developed a hybrid intelligent system to solve waste minimization quantitatively by establishing an optimum mass or heat exchanger network based on the degree of controllability. Luo and Huang (1997) developed an intelligent decision support for waste minimization that is applicable only to the electroplating process. In this article, we present a knowledge-based system called ENVOPExpert, that can automatically identify, diagnose and analyze potential waste minimization options in any chemical process. The task of the ENVOPExpert can be stated as follows: Given a flow diagram
and process chemistry of a chemical process plant, the goal is to identify opportunities to minimize process waste generated in that plant. In this article, we present the basic framework of ENVOPExpert with its application to a case study from literature. The organization of this paper is as follows: in the next section, the waste minimization methodology implemented by ENVOPExpert is described. In Section 3, the performance of ENVOPExpert on an industrial case study is illustrated and the results are compared with the experts' results.
831
2. ENVOPExpert: A K N O W L E D G E - B A S E D WASTE M I N I M I Z A T I O N SYSTEM In a chemical process, the overall transformation of raw materials and energy into desired products is usually accompanied by the generation of waste (See Figure 1). In the broadest sense, waste is any material or energy input into a process that is not incorporated into the desired final product (Jacobs, 1991). The origins of each material component in the waste stream can be traced to one or more of the following: (1) unrecovered raw materials (2) unrecovered products (3) useful by-products (4) useless by-products (5) impurities in raw materials (6) spent process materials. Therefore, the problem of waste minimization is equivalent to identifying every occurrence of these in the chemical plant under study and eliminating it. ENVOPExpert implements such waste minimization using a two-step approach: (1) Waste detection and diagnosis, and (2) Waste minimization options generation and analysis.
Process boundary Raw materials
[
Products PROCESS PLANT
Energy
Waste
I Figure 1. Process plant layout
The first task of ENVOPExpert is to identify the source of each material component that makes up the waste stream. For this purpose, a process-graph (P-graph) (Friedler et al, 1994) is used to represent the material flow within the process. In a P-graph, a material stream is represented by a circle, an operating unit by a horizontal bar and connections between the material streams and operating units by directed arcs. Figure 2 shows a P-graph representation of the input-output material streams shown in Figure 1. In ENVOPExpert, all input and output material streams across the process boundary are classified into four classes: raw material, product, utility and waste streams, and the purpose of each material used in the process is categorized as useful or non-useful. In order to identify each source of waste within the process, a qualitative material balance of each component in the waste stream is established using a P-graph starting from that waste stream back to the raw material stream. This is done by simulating the process qualitatively (including propagation of materials in the process) to determine all the materials and waste components present at different parts of the unit process. Once the P-graph of each waste component is established, the next step is to identify each of the raw material stream and the unit operation that contains and generates the non-useful material (waste) and each of the separation unit that separates the useful and non-useful materials ineffectively.
Raw material ~
Product
Waste Figure 2. A P-graph representation of process plant material stream layout After all unit sources have been diagnosed, to generate waste minimization options,
ENVOPExpert performs a heuristic analysis using functional models of Modarres (1996) and cause-and-effect digraphs to find options for minimizing the waste generation at those sources.
ENVOPExpert uses the four fundamental functional modeling concepts: goal, function, structure
832 and behavior to build systematic structures to achieve waste minimization objective. The overall waste minimization goal can be achieved only if each structure of the process plant (feed stream, reactor, separator, etc) performs their waste minimization function and each process variable in the unit (such as flow rate, pressure, temperature etc.) are regulated so as to attain the unit's intended function. To represent cause-and-effect interactions between each process variable, qualitative causal models are embedded into the knowledge system of ENVOPExpert. The various common waste minimization options generated from ENVOPExpert are as follows: 9 Optimize feed conditions by reduction of impurities and minimization of excessive materials used in the process 9 Optimize reactor conditions by increasing raw materials conversion and minimizing waste by-products formation 9 Optimize separation system to separate the useful components from the useless ones 9 Recycle or recover-recycle of valuable components in the waste stream
ENVOPExpert is implemented in an object-oriented framework using Gensym's G2 expert system shell. ENVOPExpert system consists of three main elements: (1) knowledge representation framework, (2) inference engine, and (3) integrated graphical user interface. The knowledge representation framework is separated into two distinct categories - process-specific knowledge about the plant under study and the underlying chemical and physical phenomena and process-general waste minimization knowledge captured in the functional and cause-and-effect models and generic methods and rules for identifying the source of waste and the minimization options. Through graphical user interface, the user inputs all the information about the process plant in terms of process flowsheet, materials properties, status of input-output material streams and process chemistry. The process-general knowledge on the other hand, remains the same for every chemical plant. The inference engine consists of rules and methods that integrate the process general and process specific knowledge to identify waste minimization alternatives. The output of ENVOPExpert is a set of diagnosis results that identify the waste generation source and suggestions to the user on possible waste minimization alternatives for the plant. 3. ENVOPExpert ANALYSIS CASE STUDY We have tested ENVOPExpert by performing waste minimization analysis on an industrial case study. This case study involves a hydrocarbon separation process, which was first described by Isalski (1995). The waste minimization analysis of this process had been performed by a team of experts and the results are available for comparison with ENVOPExpert's analysis. Figure 3 shows the flowsheet of the process. A vapor containing a mixture of hydrocarbons (C1 to C5) is the feed to the separation plant. The mixture is initially compressed to a higher pressure followed by a condensation using cooling water inside a heat exchanger. The resulting vaporliquid mixture is passed to a flash separator, where the bottom liquid is used as product and the collected vapor at the top of the separator is sent to a flare system as waste stream. The P-graph representation of the qualitative mass balance of the process and the functional model shown in Figure 4 and Figure 5 respectively are first constructed automatically by ENVOPExpert as described above. Waste minimization analysis based on this P-graph and functional model reveals that the waste stream is generated due to low condensation yield of vapor to liquid hydrocarbons in the heat exchanger and excessive vapor fed to the process (material source). The next step of ENVOPExpert is to find options for minimizing wastes. Based on the results from the previous step, the minimization algorithm focuses on the heat
833 exchanger unit of the waste stream using causal models. The comparison between the options generated from ENVOPExpert knowledge-based and team expert's result is shown in Table 2. As seen from the table, ENVOPExpert is able to successfully identify the sources of waste and the basic waste minimization solutions.
Figure 3. Flowsheet of hydrocarbon separation process
Figure 4. P-graph representation of material path of inlet-outlet stream Material source
Utilitysource C)---q
Energy ransfer
0
Materialsink-1
I Separate
Transport
~
O
Materialsink-2
Utility sink Figure 5. Flow modeling structure representation of separation process 4. CONCLUSIONS Waste minimization is one of the most important issues facing the chemical industry today. Performing waste minimization analysis is however labor and knowledge intensive and would gain by automation. In this paper, we proposed a knowledge-based system called ENVOPExpert that automates waste minimization analysis for any chemical process plant. The system comprises of process-general and process-specific knowledge. The process-specific knowledge comprises of user supplied plant information including the flowsheet, materials, stream status and reaction chemistry. The process-general knowledge consists of heuristic rules and methods,
834 which diagnose the sources of wastes using P-graphs and recommend waste minimization alternative using functional models and cause-and-effect digraphs. We have tested ENVOPExpert on a simple hydrocarbon separation case study. The comparison between the waste minimization options generated by ENVOPExpert and by a team of experts shows that our framework is able to accurately identify basic waste minimization solutions. Currently ENVOPExpert models are being extended and tested on other more complex case studies. Table 2. Comparison of waste minimization team's results and ENVOPExpert analysis of hydrocarbon separation process Source Feed stream Compressor
Heat exchanger
Separator Waste stream
Waste minimization team's results Less hydrocarbon feed to the plant Larger compressor power
More cooling water flow rate Lower temperature of cooling water Use other coolant (glycol) Larger heat transfer area Add second cooler after heat exchanger Recycling waste stream Use heavier hydrocarbon to absorb waste vapor Provide vapor recovery system after separator
ENV OPExpert analysis Prevent excessive hydrocarbon feed Decrease temperature of hydrocarbon feed Increase compressor power Decrease temperature rise inside compressor Improve compressor design Increase flow rate of cooling water Decrease temperature of cooling water Improve heat exchanger (shell and tube) design Use alternative cooling agent Improve separator design Direct recycling or recovery-recycling of vapor waste stream
REFERENCES: Crittenden, B. and Kolaczkowski, S. Waste Minimization: A Practical Guide, Institution of Chemical Engineers, Rugby, Warwickshire (1995). Friedler, F., Varga, J.B. and Fan, L.T. Algorithmic Approach to the Integration of Total Flowsheet Synthesis and Waste Minimization. In Pollution Prevention via Process and Product Modifications, 86-97, ed. M.M. E1-Halwagi and D.P. Petrides. American Institute of Chemical Engineers, New York (1994). Huang, Y.L. and Fan, F.T. Intelligent Process Design and Control for In-Plant Waste Minimization. In Waste Minimization through Process Design, Chap. 13, ed. A.P. Rossiter. McGraw-Hill, New York (1995). Isalski, W.H. ENVOP for waste minimization. Environmental Protection Bulletin, 34,16-21 (1995). Jacobs, R.A. Waste Minimization-Part 2: Design Your Process for Waste Minimization. Chemical Engineering Progress, 87(6), 55-59 (1991). Luo, K.Q. and Huang, Y.L. Intelligent Decision Support for Waste Minimization in Electroplating Plants. Engineering Applications of Artificial Intelligence, 10, 321-333 (1997). Modarres, M. Functional Modeling for Integration of Human-Software-Hardware in Complex Physical Systems. In Intelligent Systems and Soft Computing for Nuclear Science and Industry: Proceedings of the Second International FLINS Workshop, 189-204, ed. Ruan Da. World Scientific, New Jersey (1996).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
835
A C A P E tool for evaluation of adsorber-rcactor systems for treatment of exhausts from mobile sources J. Jirfita, M. Kubi6ek b and M. Marek a aDepartment of Chemical Engineering, bDepartment of Mathematics Center for Nonlinear Dynamics of Chemical and Biological Systems Prague Institute of Chemical Technology, Technickfi 5, 166 28 Praha 6, Czech Republic 1. I N T R O D U C T I O N
More than 50% of overall unbumed hydrocarbon pollutants from mobile sources arise during the cold start of engines. HC-traps combined with reactors form the basis of more than twenty proposed solutions patented in the last three years. Most often are the corresponding flowsheets based on delaying emissions of hydrocarbons by adsorption until the catalyst reaches the light-off and then the trapped hydrocarbons are released and oxidized. The flowsheets often combine several adsorbent or catalyst beds, with possible arrangements including by-passing, secondary air injection, electrical heating, etc. Four examples schematically representing selected patented configurations [1-4] are shown in Fig. 1. Exhaust gas conditions (temperature, flowrate) and composition depend on engine type. All inlet conditions to abatement system (temperature, flow-rate, composition) vary with time significantly and very rapidly. The flowsheet and way of operation can be adaptively redesigned on the basis of actual conditions as it is also proposed in some patents. Simulation and adaptive predictive control of dynamic operation of combined systems is thus a proper field for CAPE tools. As far as we know, no specialized software has been described in open literature for such class of problems. Powerful CPUs become standard parts of car equipment. The built-in computers offer possibility of advanced control using various semi-empirical rules combined with more complex mathematical models. The software presented here enables not only to perform dynamic simulations but also to generate dynamic data for development and tuning of control algorithms. Typical problem formulation is represented by dynamic simulation of interconnected systems of monolithic catalytic reactors (with possible heat exchange) and adsorbers, where inlet and/or boundary conditions are time dependent. Switching among different flowsheets (either according to fixed scheme or adaptive) has to be also considered. Both reactor and adsorber units are described by dynamic 1-D models considering the axial heat conduction, hence a set of partial differential equations results. Models of reactors were described in detail in [5,6], adsorber model in [6].
836 Reactor model: 9
9 aT*
O2T* (1)
P % - - ~ - ~'z az ~ aT aT a(T_T. o c ~ - g - - - v p c ~ -gz - k . ( z ) -~
)
(2) 9ad
~c~ _- _ O(vc~) _ kc(z)_~(c~ _ cs at az
(3)
e
-~
_
a
k c ( z ) ~ _ ~ ( q - cs
"
2 vugtj
.j=~
(4)
Here * denotes the surface of the solid, T - temperature, c - concentration. Adsorber model: ~t - -kv---~z + ck ~
p ~c~,~ +--
-
p.,.%,.,. --0-7 -
Here ~-~and qs denote the respectively.
e
c)'l:
(5)
Ot - k(qs - qk)
Z,:
actual
(6)
k
and equilibrium concentration of adsorbed
species
2. SOFTWARE
The software devoloped in our laboratory enables dynamic simulations of complex nonstationary reactor-adsorber systems described by nonlinear PDE's (1)-(7). To achieve the highest performance a stand-alone executable software routine is built for each configuration and set of chemical species. The whole integration cycle is controlled by Master program, cf. Fig. 2. The Master program and necessary routines are written in FORTRAN 77. Master program may be generated automatically or written directly by the user for each configuration. User has to supply kinetic equations and data (for reactors) and adsorption equilibrium formulae (for adsorbers). Several standard forms of equations are pre-programmed in the package. Modules such as reactor and adsorber models and utilities are pre-compiled. Master program, kinetics and adsorption modules are compiled; then all modules are linked to the executable program. The software is used for dynamic simulations, i.e. studies of the effects of system parameters, comparison of different arrangements or optimization of operating parameters for systems operated in a nonstationary way. Although these simulations can bring valuable results themselves, much higher benefit can result when these routines are used for generation of data and testing adaptive control algorithms. This has been demonstrated for a nonstationary operated system of thermally coupled monoliths for NOx reduction, where a combined algorithm for predictive control (heuristic rules, artificial neural net predictions and model based predictions) was used [7]. 3. COMPARISON OF SELECTED CONFIGURATIONS Let us demonstrate the software capabilities on the evaluation of several basic configurations of adsorber(s)-reactor(s) systems. The inlet conditions correspond to the standard European driving cycle of gasoline engine [8], the time-course of inlet temperature is shown in Fig. 3.
837 Three main reactions taking place in the reactors were considered in this case: C O nu 0 2 -----)C 0 2 ,
C3H6-k- 0 2
"-'--)CO2 -1- H 2 0 ,
C3H8 -t- 0 2 ~
C 0 2 -Jr-H 2 0 .
CO is the non-adsorbing component, C3H6and C3H8 are adsorbed. Propene and propane are quite often used to simulate the real hydrocarbon mixture both in experiments and modelling, although the real gas composition is quite different. C3H6 represents an easy-to-oxidize hydrocarbon, C3H8 a difficult-to-oxidize hydrocarbon. The total length of the reactor and adsorber (both have the same diameter) in all configurations is 0.11 m. The residence time (at 298 K) is approx. 0.15 s. Typical inlet gas composition used in all simulations is CO 1%, C3H6 320 ppm, C3H8 160 ppm and 02 concentration has been varied. Starting temperature of the reactor and adsorber is 25~ and the adsorber is regenerated. Compared configurations (cf. Fig. 1) include: (a) Single monolith (will be used as a reference configuration), cf. Fig. 1a. (b) Configuration reactor-adsorber-reactor [ 1], cf. Fig. lb. (c) Configuration reactor-adsorber-reactor with adsorber bypass, where the switching from initial (adsorption) configuration to desorption configuration is performed [2], cf. Fig. 1c. (d) Configuration based on the sequence of adsorbent and catalyst beds [3], cf. Fig. l d. The sum of bed lengths does not depend on the number of beds. (e) Configuration with flow-reversal, here is the adsorber located between two catalyst beds and the direction of the flow periodically alternates [4], cf. Fig. 1e. We will also examine the influence of different levels of catalyst activity for C3H6 oxidation (which represents an easy-to-oxidize hydrocarbon). a
-I
IInitialization
b
Start
C
-fq--I, I--fqAdsorption phase (cold start)
d
Desorption phase (after lightoff)
e ..................
Switch flowsheet?
Make one integration step for all units in chosen flowsheet sequentially
(a) Switching time schedules or (b) tests for adaptive switching Time-dependent parameters (inlet, heating, cooling, by-pass or split ratios, etc.) (a) Time schedules (b) Tests for adaptive changes
Output record
"1 ~ ..................
Fig. 1" Selected patented configurations
End
]
Fig. 2: Block scheme of the Master program
838 4. SIMULATION RESULTS
4.1. Single monolith (Fig. la): In this case the light-off for a single monolith occurs approx, about 150-200 s after the start, cf. Fig. 3. Higher catalyst activity for C3H6 oxidation may bring only limited improvement here, cf. Table 1. In the following comparisons we will use an "improvementfactor" defined as the ratio between the emission from the single monolith and that from the studied arrangement. Values of the factor higher than 1 then indicate higher conversions than in the single monolith. Table 1: Total conversions for single monolith (over the entire driving cycle). Catalyst activity for CO C3H6 C3H8 C3H6 oxidation low 84% 85% 82% intermediate 84% 87% 83% high 85% 88% 83%
4.2. Configuration (b), of. Fig. lb: No improvement is achieved in this case, on the contrary the hydrocarbon conversion is worse. The first catalyst is heated by the exhaust gas, but the second catalyst is still cold, when temperature desorption occurs. Only additional heating applied to the second catalyst can accelerate the start-up of the second catalyst. Adaptive predictive control may be used to estimate the start-up of the heating and conditions for minimizing electrical power supplied for heating the second reactor.
4.3. Configuration (c), cf. Fig. l c: This configuration represents another approach to the reactor-adsorber-reactor sequence. The adsorber should hold hydrocarbons untill the first reactor reaches a light-off; then the adsorber is partially bypassed and the hot gas is used to preheat and start-up the second reactor. During this operation the adsorbed HCs are slowly desorbed and then oxidized in the second reactor. For the conditions considered (kinetics, adsorption equilibrium, inlet conditions) the adsorber must be cooled during the adsorption period to achieve satisfactory conversions. And although the temperature policy can be successful, there can be further problems, as it is demonstrated on the two results presented in the Table 2. The conversions differ quite significantly and the only difference in simulation has been the oxygen level considered (1% and 2%, respectively). The difference follows from the fact that when desorption peaks of hydrocarbons occured, the oxygen level dropped to zero for a certain time and hydrocarbons were emitted in spite of favourable temperature conditions in the second reactor. Even in the case, where 02 level was kept at 2%, the low oxygen concentration during desorption was the limiting factor and the presented results thus do not represent maximum attainable values. The tasks for predictive control are obvious - to estimate the amount of additional air injected, the start of air injection (or corresponding air/fuel ratio modulation), the time of switching and also to tune the by-pass ratio.
839 4.4. Configuration (d), cf. Fig. l d: Bed splitting delays HC emission, but the improvements of conversion are still very low even if the catalyst is quite active, cf. Fig. 4. Desorption caused that the emission peaks with concentration higher than the inlet one arise. The difference between single bed (N--l) and dual bed (N-2) is quite significant; for N-2 and N=3 the emission peaks are only slightly shifted. Again, only electrical heating can accelerate the start-up of the reactors. Similar rules as for case (b) may be applied here. 4.5. Flow-reversal (e), eL Fig. l e: This arrangement improves C3H6 conversion, but CO conversion is decreased (cf. Table 2). HC conversion improvements require short switching periods (10-20 s). CO conversion is decreased due to delayed start-up of both reactors. Although for considered conditions the flow-reversal configuration did not give satisfactory results, it is possible that adaptively controlled switching period may lead to significant improvements for different conditions. Flow-reversal operation is especially advantageous in cases where reactor bed temperatures are high enough and inlet temperature decreases. Then the periodic switching of flow direction prevents the blow-out of the reaction zone. For example, comparison of a single monolith and a single monolith with flow-reversal for the case where all conditions are the same except that the monolith is preheated (initial temperature is e.g. 250 ~ instead of 25 ~ shows that the flow-reversal operation (switching period 20 s) achieves nearly complete conversion of propene in the first 200 s, whereas for the single monolith the reaction zone is blown-out before the higher inlet temperatures ensure high conversions.
2000 300
~,~
N=~IilN=3
15oo
........
200
:,.
f!/~
600
400
~
200 1 - low activity 2 - intermediate activity h activity
~
,~ o
1000
~
500
N-1
!iii
100 / 3i~1 ./ !.i!,~ 0
g
300
600 time [s]
900
1200
Fig. 3: Time-course of inlet temperature for European driving cycle and outlet C3H6 concentration from single monolith for different catalyst activity
.................. 0
iiiii............................................ i....................... inl ,,. 100
200 300 400 time [s] C3H6 concentration from
Fig. 4: Outlet configuration (d), cf. Fig. l d. N adsorbent and N catalyst beds are in sequence. Catalyst activity- high.
840 Table 2.: Improvement factor (catalyst activity - high) Configuration (a) - single monolith (reference) (b) - reactor-adsorber-reactor (c) - reactor-adsorber-reactor with adaptive switching (the time of switching 200s, cooled adsorber, ratio bypass/total flow = 0.9) (d) - sequence of beds
(e)- flow-reversal (switching period 10s)
02: 1% 02: 2%
N= 1 N=2 N=3
CO 1 1 1 1.06
0.89 0.90 0.91 0.59
Component C3H6 C3H8 1 1 0.95 0.93 1.67 0.93 25 3.8 0.70 0.73 0.73 1.97
0.84 0.86 0.87 0.54
5. CONCLUSIONS
Developed software was tested on a number of patented configurations of catalytic afterburners for mobile sources. An example of such comparison is presented in Table 2. The possibility of easy set up of the executable program from available subroutines enables to efficiently compare various combinations of afterburners operating in fixed or adaptive configurations and to generate dynamic data necessary for testing and tuning different control algorithms. REFERENCES
1. Toyota Patent US5315824 2. General Motors Patent US5492679 3. Bayerische Motoren, Volkswagenwerk, Daimler Benz and Porsche Patent EP0866218 4. Matros Technologies Patent US5768888 5. M. Kubi6ek, P. Pinkas, J. Jir~t, D. Snita, M. Marek, Computers and Chem. Engng, 21, $757-$762, (1997) 6. J. Jirfit, F. St6pfinek, M. Kubi6ek, M. Marek, "Operation of reactor-adsorber systems for minimization of exhaust gases emissions" in Reaction Engineering for Pollution Prevention, Elsevier, (2000) 7. F. St6p/mek, J. Jirfit, M. Kubi6ek, M. Marek, Computers and Chem. Engng, 23, $317-$320, (1999) 8. G.C. Koltsakis, P.A. Konstantinidis, A.M. Stamatelos, Appl. Catal. B, 12, 161-191, (1997) A c k n o w l e d g e m e n t : This work was partially supported by Grant No. VS96073, Czech Ministry of Education and by Grant No. 104/99/1408, Czech Grant Agency.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
841
Quick identification of the wastewater biological treatment process by using shortcut techniques and previous plant operation data G. Maria a* C. Constantinescu b and P. Ozil c
a) Chemical & Biochemical Reaction Eng., University 'Politehnica', P.O. 15-253 Bucharest, Romania b) Department of Chemical Engineering, University 'Politehniea' Iasi, Romania c) Lab d'Electrochimie et de Physico-chimie des Materiaux et des Interfaces, Institut National Polytechnique de Grenoble, 38402 Saint-Martin-D'Heres, France For a biological wastewater treatment process significant benefits in safety and optimal monitoring can be achieved if a sufficiently accurate and reliable process model is available. Because of the bioprocess and plant complexity, reduced mechanistic models are preferred, the physical meaning of the parameters offering the possibility to interpret the estimate quality and to store the results in databanks. Periodic model parameter and structure updating is usually performed ('tendency modelling') with an effective procedure in order to compensate the plant, data and model mismatch. Recently, Maria & Rippin [ 1] proposed an effective shortcut estimator MIP which increase the solution quality and reliability by means of the estimation problem decomposition and transfer-of-information from previous similar studied processes. A novel route is investigated in this paper to quickly identify the kinetic characteristics of a wastewater biological treatment (WWT) process when new substrates are present. By using the available recorded collection of the plant previous transient operation, similar evolutions (concentration trajectories) are used by MIP to generate the new model approximate parameters corresponding to the current substrates and data. This route is exemplified for an industrial WWT plant by adopting a simple Monod kinetics and a perfect mixed biological reactor with activated sludge recycle. 1. INTRODUCTION The biological treatment process is one of the very important steps in removing a large number of pollutants from wastewater. For optimal process monitoring and simulation, mechanistic models are preferred, offering the possibility to better evaluate the estimate quality and to store the results in databanks. To overcome the lack of information over a wide range of operating conditions and influent quality, the ordinary differential (ODE) model, representing the dynamic evolution of the process variables, is of complexity depending on the amount of available information and on the utilisation scope. For real-time WWT plant monitoring this is realised by means of a compromise among qualitative and quantitative process knowledge, the dominant fast and slow modes of process dynamics, the macroscopic and microscopic state of the process, and the non-biological and biological elements of the state vector [2,3,14]. If the ODE model structure was identified, the next critical step is their parameter estimation by minimising the residual differences between data and model predictions in terms of output variables, with several techniques: indirect methods (objective function iterative minimisation with repeatedly model evaluation, NLS), or direct methods (based on model transformation and approximate problem solution in one step [4]). Because in the biological WWT there is an excess of degrees of freedom in adjustable parameters than the observed and manipulated variables, adequate modelling can lead to multiple solutions, even if a reduced model structure is tested. The model discrimination has to be coupled with model structure improvement (under certain physical meaning parameters and process constraints), and eventually model reduction (via observation lumping, parameter sensitivity analysis, principal component regression, ridge selection/trace analysis [5-8]), sometimes by means of an optimization rule [9,10]. If model predictions are rather weak, then the
842 parameters and even the model structure must be updated in various variable operating regions [ 11,12]. The WWT plants usually consist in a series of aerated basins, continuously operated under transient or quasi-steady-state conditions which have to be adapted to the pollutant and biomass characteristics. The key point in the process optimisation is the knowledge of the biokinetics. Extended models require a laborious lab-scale experimental strategy and a substantial computational effort to identify the parameters [ 13], even if this information may not always be immediately applied to the plant scale due to the variability of materials and procedures, fluctuations in quality and quantity of feed wastewater streams, sudden increase in substrate concentration or some inhibitory substances, deterioration of the sludge characteristics, or few observed species. The use of simple Monod kinetics for the microorganism growth can be satisfactory if the models are on-line adjusted according to the new information from the process sampled with an appropriate frequency. Classical model updating is performed via state-parameter recursive estimators applied to a given model structure and prior parameter and variance estimates (extended Kalman Filters, EKF [ 15]). Rigorous NLS successive regression is time consuming, and it is applied only for portion of data to generate prior information for subsequent EKF estimators. However, the EKF are very sensitive to the chosen model structure, data noise level, model linearizations, tuning factors, and prior information [16]. Various variants were developed in order to increase the solution reliability, for instance by introducing NLS steps with a certain frequency [17]. Structural changes in the model can be detected by combining chemometric and estimation techniques [7]. Shortcut estimation techniques replace the regression rule with an one-step solution of an overdetermined algebraic set obtained after the discretization (DP) or integral transformation (IP) of the ODE model. Maria & Rippin [1] proposed an improved shortcut estimator (MIP) by developing transfer-of-information rules from previous similar studied processes. As proved, the MIP is superior to the classical shortcut (DP, IP) or recursive (EKF) estimators even for poor-conditioned cases [1,18,19]. The scope of this paper is to quickly identify the kinetic characteristics of a biological WWT process by using the available collection of the plant previous dynamic operation and the novel MIP shortcut estimator for transferring this information. Similar plant evolutions, completely characterised, are used to identify the current bioprocess characteristics even if the pollutants are different. With the increased availability of portions of data in various estimation stages, on-line MIP can highlight possible changes in the model structure and parameters, avoiding solutions with no physical meaning and convergence problems of a subsequent NLS [20]. The estimation problem decomposition via MIP is exemplified for an industrial WWT plant with sludge recirculation and a Monod kinetic model. 2. BIOLOGICAL WASTEWATER TREATMENT PLANT MODEL In order to exemplify the WWT biokinetics shortcut estimation, a classical activated sludge treatment is approached. A Monod kinetics and an ideal continuous stirred tank reactor (CSTR) coupled with a settler for recycling the sludge are considered [21,22]. The plant schema (Fig. 1) basically involves the aeration and mixing of the influent in the presence of a flocculated suspension of micro-organisms which are supported on particulate organic matter. After several hours of residence time, the effluent is passed to a sedimentation tank where the flocculated solids are separated from the treated liquid. All operations can be considered isothermal for certain time intervals. A reduction of BOD organics, ammonia, nitrites, nitrates, and other substrates is achieved under certain optimal operating conditions. Part of the settled activated sludge is recycled to the aeration tank in order to maintain a quasi-constant sludge concentration. If the biomass is enough aerated and no resistance exists on the diffusion of substrate to the flocons, the isothermal CSTR bioreactor and ideal settler can be represented under dynamic operating conditions by a differential balance set of equations: vdSj _Q dt (Sj,in - S j ) + r s j " SJIt= 0 = S j, o ; dXj _q Xj, r dt -V
(where X j , S j
Q+q xj V =
+
rx J
. y j , r = y J Q+-------~q . y j I t = o = X j , Qw + q '
o,
(1)
biomass and substrate species (index J) concentrations; V = reactor volume; Q, Qw, q=
input, waste sludge, and recycled volumetric flow rates; r S , rX = substrate consumption and biomass
843
production rates; t - time). The main adopted model hypotheses are: I) CSTR bio-reactor sufficient aerated to ensure the necessary dissolved oxygen for the bioprocess; ii) isothermal conditions; constant pH and biomass characteristics for an analysed time period; iii) enough nutrients for biomass development; iv) constant flow-rates and liquid volume; v) the same substrate concentrations in the clarified water and biomass recycle; vi) negligible biomass in the clarified water; vii) inert material not interacting with the biomass. The biomass growth in the reactor is assumed to follow a Monod kinetics: ~x 1 ~sx rX - bX; rS . . . . , (2) K+S YK+S with a first-order death rate [21 ], and parameters usually in the range of: HE(O.03-5) (1~day); K E(O-300) (g Substrate/m3), b~(0.05-2) (1/day), Y~(0.2-1) (gVSS/g Substrate) [22,23].
x,,s
Q, S/n
"~[
X
(Q-Qw), S (Q+q), S, X ~
REACTOR
p,
Qw, S, Xr
J, Fig. 1. Biological WWT aerator and the attached settler unit.
3. MIP SHORTCUT ESTIMATOR Maria & Rippin [1,18] proposed a shortcut estimator (MIP) of the ODE model parameters with the following advantages: simplicity, rapidity, reliability for poor-conditioned cases, any convergence problem, any tuning factor required or model linearisations, possibility of using prior information from databanks. Starting from the observed concentration data vectors, the MIP principle is to transform the ODE set into an algebraic one by performing integral transformations, but also considering in the same manner the prior information about an analogous process (to whom kinetic data and parameters are known). The data are scaled in a common time domain by means of a scaling factor q~>0. The similarity analysis is applied to the pair of similar species, for instance by identifying portions of common reduced time domain where the current/historic process rate ratio is quasi-constant. In this interval, the integral rate ratio of the two simultaneously considered processes is decomposed, by estimating first the dominant reaction term parameters and neglecting the others. The rule is repeated in several time subintervals, and avoiding poor-conditioning by using the same relative parameter ratios as in the previous process. The MIP is effective in quickly on-line checking of a model structure when changes in species observability and parameter significance occurs, being superior to the classical EKF. A combination of MIP and NLS followed by an advanced estimate sensitivity / principal component analysis can be applied for portions of data [20], thus overcoming local solutions with no physical meaning. 4. EXAMPLES OF THE WWT PLANT DYNAMIC EVOLUTIONS The WWT biological process dynamics is characterised by a wide range of time constants, nonlinearities, imprecision and some irreproducibility of data, substantial stability punctuated by abrupt failures, and a sensitive, readily adaptable community of micro-organisms. Frequent fluctuations in pollutant concentrations, flow-rates, biomass characteristics, operation and mixing conditions make difficult a systematic system identification. However, some portions of the dynamic state-variable evolution, numerical filtered [24] and regularised by using smoothing spline functions [25], can be recorded during several days (weeks) and used for the process identification. Reduced models may suffice the short-term needs of system control-optimisation, if periodic parameter updating are performed to overcome the lack of identifiability due to the very high process intrinsic complexity. For instance, by considering the kinetic parameter set [l.t,K,b,Y] = [1 (1/day), 300 (g/m3), 0.1 (1/day), 0.4 (g/g)], the plant volume V- 20588 (m3), input Q= 150000 (m3/day), initial S o - 0 (g/m 3) and Xo= 2000
844
(g/m3), various WWT plant dynamic evolutions can be simulated with the approximate model (1-2), corresponding to step variations of [Qw, Sin, q/Q] = [500 (m3/day), 350 (g/m3), 0.33] for cases 1 & 3, and [Qw, Sin, q/Q]= [700 (m3/day), 300 (g/m3), 0.33] for cases 2 & 4 (Figure 2, dash lines). 5. E S T I M A T I O N OF THE KINETIC P A R A M E T E R S BY U S I N G
MIP
One considers the smoothed current data and similar plant previous evolutions from Figure 2 (n points, cases 1-4). The MIP similarity analysis was applied to the pair of similar species [current S(t), X(t), and historic S'(t '), X'(t') concentrations; the historic process is denoted by apostrophe], by identifying the common reduced time domains [t o , t] and [t' o , t'] where the rate ratios are quasi-constant. Thus, the unknown p=4 kinetic parameters [p,K,b,Y] are estimated by using the historic process parameters [/.t',K',b ', Y'], and the integral form of the ratios decomposed to point out the dominant terms [1 ]. Under the hypothesis o f constant [ V, Q, q, Qw], [v', Q ', q ', Q 'w] in the analysed time sub-intervals, and neglecting the substrate formation, one experimental times t, t '): t K[AX-a IXdt] / K'[AX'-a' to
obtains the MIP over-determined linear algebraic set (for every t' X X' I X ' d t ' ] ,~ I S d X / I S ' d X ' t'o Xo X' o
q Q+q 9a = VQw+ q
Q+q --' V
t t' t t' Y K [ A S - fl] t t' [ A X - a I Xdt ] / [ A X ' - a ' I X ' dt' ] ~ b I Xdt / b' I X ' d t " ~ /~ I SXdt //~' I S' X ' d t " to t,~ to t,~ r ' K ' [ AS ' - p' ] to t,~
K[AX-a ,
t t' I Ydt] / K ' [ A X ' - a ' I x ' d r ] to t'o
t t' ~ ( / ~ - b ) ISXdt / ( p ' - b ' ) I S ' X ' d t ' ; to t'o
,
a ' -- Vq' Q'w+q Q'+q '
t
Q,
,
t'
Q'+q' ----7-" v f l = -Q v [Sin(t - t o ) - to~ Sdt]," fl'= -v 7 [ S i n ( t ' - t ' ~ ) - t,o~S'dt']"
(3)
Table 1. MIP and NLS kinetic estimates of the WWT process (smoothed, noise free data). Case Procedure ~(1 / day) ~ ( g / m 3) b(1 / day) ~.(g / g)
s.d.SSR
History MIP (q9 = 0.72) NLS (determinant criterion) History MIP (~o = 9.3) NLS (determinant criterion) History MIP ((p = 0.86) Exact solution History MIP (q)= 0.9) Exact solution
1 0.8946 0.8642 1 1.2761 1.2387 1 1.44 1.5 1 1.0653 2.0
300 130.3 128.7 300 354.7 354.4 300 198.9 200 300 208.2 200
O. 1 0.0988 0.1354 O. 1 0.0997 0.0956 O. 1 0.1017 O.15 O. 1 0.0928 0.50
0.4 0.2840 0.3627 0.4 0.4216 0.4413 0.4 0.4309 0.5 0.4 0.1023 0.20
1.59e-2 4.79e-3 1.18e-3 6.78e-4 1.97e-3 4.84e-5 3.22e-3 3.23e-5
(g/L) (g/L) (g/L) (g/L) (g/L) (g/L) (g/L) (g/L)
/11
Notation: s.d.SSR =
[ S , X ] e x p - [S,X]II2 ^ 2 / (2n- p) .
Generalisation of (3) for variable reactor volume and flow-rates can be easily obtained with no significant computational complications. To exemplify this MIP estimation rule, one considers cases 1-4 of known filtered data (noise free, Fig. 2), corresponding to the following common operating conditions: V = V' - 7750 (m3); S o = S' o - 0 (g/m3); X o = X ' o - 2000 (g/m3); Q - Q' - 150000 (m3/day); [Q'w-Qw, S'in-Sin, q'/Q'=q/Q] of [500 (m3/day), 350 (g/m3), 0.33] (cases 1 & 3), and of [700
845 (m3/day), 300 (g/m3), 0.33] (cases 2 & 4). The historic parameters [,u',K' b' Y~] = [1 (1/day), 300 (g/m3), 0.1 (1/day), 0.4 (g/g)], are adapted to the current data via MIP and compared in Table 1 with the exact NLS estimates. The NLS solution was obtained by using the MMA optimisation routine [20] and a determinant criterion because of the high intercorrelated observations. In all the cases the MIP estimate is very close to the exact NLS solution. Substrate evolution- Case 1
0.3
0 2l/------
0.2
. . . .
GO O.
OOOC,
0 5
5 .
10
O~,OC,
15
.
2'0
.
2~5
30
0
.
5
.
10
.
.
15 time, days
20
25
t
o.51 0
30
I/
~
OOOOOOOOOOOOOOO
o
~ .
1o
1'~
2'0
Biomass evolution- Case 3 .
.
.
2'~
I'5
20
0
.
15 time, days
20
5
~o
1~
time, days
2'o
2;
Oooo
25
~
,
5
10 15 20" Biomass evolution - Case 4 ,
,
2'5
,
Xt
o
30
3o
Substrate evolution - Case 4 . . . .
,
x
2'5
. . . . . . . . . ooOOOOOOOOOOOOOOOO. . . . . . . .
0
30
10
.
~o.2l r
2l/----
8
5
0.3
-
oo0.1
I'0
5
,., ,., o ,.., ~ u O
Subs~ate evolution - Case 3
0.3
n n n Q Q n n n n 0
1"
|
• 0
c~ n c~ o o 0 0 0
Biomass evolution- Case 2
oOOOOOOOOOOO~___1 ..__ __.- , - - - - - - - -
2~~-
~-n
.
4 -J
0~--r~.o'-d-~ ~ - ' ~ ' 0 ~
OO~OOOOu
Biomass evolution- Case 1 .
Substrate evolution- Case 2
0.3[
30
,
----_ OTO000000 .
30
0
.
.
5
.
10
15 time, days
20
2'5
30
Fig. 2. Current data (O), MIP predicted (-), and previous WWT plant smoothed evolutions (---). As for all the shortcut estimators, the MIP is sensitive to the data quality. However, due to the used prior information, the poor conditioning (from incomplete data or model form) is solved by adopting the same ratio for those parameters as from the prior information. To reveal the MIP solution robustness, noised data are 'generated' by uniformely alterating the case 3 data with a random relative error of max. + 15% (Fig. 3, case 5). The obtained MIP estimate from Table 2 are of very good quality, close to those obtained from using the noise free data (case 3). 6. CONCLUSIONS Quick estimation of the WWT biological kinetics by using modem shortcut techniques allows a rapid reduced model updating in on-line process identification and monitoring. Consistent information from databanks regarding the plant past similar evolution in removing various pollutants can be successful used to identify the current process characteristics. Various model structures can be approached in an effective, robust, simple-to-use model updating strategy via MIP [ 18]. The proposed rule does not use tuning factors, model linearisations, and have no convergence problem. The estimate is usually close to the NLS solution for moderately noised data.
846 Substrate evolution- Case 5 03 l
0.2 co 0.1
Oo,_,oonooonoooo
0(b
,
,
0
5
10
8 ~6
t
2
~
-
v
1~
2'0
2'5
Biomass evolution - Case 5 '
O
o
OO'-'--" Y "--"'--" "--"-'-"-'--
0
- ,
5
10
'
-~-'~
'o o 'GCC, O ,-, ,-, '
~
15 time, days
o
OO
o
Table 2. MIP and NLS kinetic estimates for the WWT plant noised data (case 5). Parameter MIP estimate History noise noised free data data 30 ft(1 / day) 1 1.4408 1.4448 ~2(g / m 3) 300 198.9 195.9
--''
20
2'5
30
[~(1/day)
0.1
0.1017
0.1022
f'(g / g)
0.4
0.4309
0.4302
~0 s.d.SSR(g/L)
-
0.86 6.35e-3
0.82 6.4e-3
-
Fig. 3. Current data (O), MIP predicted (-), and previous WWT plant evolutions (---, case 5). This paper was developed in the framework of the EU TEMPUS Project no. 5-JEP11219-96, from which partial financial support is acknowledged.
Acknowledgement.
REFERENCES
1 2 3 4 5 6
G. Maria and D.W.T. Rippin, Comp. Chem. Eng., 21 (1997), 1169. M.B. Beck, 1986, IEE Proc., 133 (1986), 254. G. Maria and C. Maria, Sci. & Technol. Environmental Protection (Bucharest), 4 (1998), 59. L.H. Hosten, Comp. Chem. Eng., 3 (1979), 117. G. Maria and T. Ognean, Water Res., 23 (1989), 175. A.S. Tomlin, T. Turanyi and M.J. Pilling, In: Oxidation Kinetics and Autoignition of Hydrocarbons, (Pilling, M.J., Ed.), Elsevier, Amsterdam, 1995. 7. G. Maria and D.W.T. Rippin, Chem. Eng. Sci., 48 (1993), 3855. 8. S. Vajda, H. Rabitz, E. Walter and Y. Lecourtier, Chem. Eng. Commun., 83 (1989), 191. 9. G. Maria, Canadian J1. Chem. Eng., 67 (1989), 825. 1O.K. Edwards, T.F. Edgar and V.I. Manousiouthakis, Comp. Chem. Eng. (1994)(submitted). 11.C. Filippi, J. Bordet, J. Villermaux, S. Marchal and C. Georgakis, Comp. Chem. Eng., 13 (1989), 35. 12.J. Fotopoulos, Ph.D. Diss., Lehigh University (1996). 13.G.T. Daiger and C.P.Jr. Leslie Grady, Water Res., 16 (1982), 365. 14.K.H. Bellgardt, W. Kuhlmann, H.D. Meyer, K. Schtigerl and M. Thoma, IEE Proc., 133 (1986), 226. 15.G.C. Goodwin and K.S. Sin, Adaptive Filtering Prediction and Control, Prentice-Hall, N.J., 1984. 16.P. de Valliere and D. Bonvin, Comp. Chem. Eng., 14 (1990), 799. 17.S.S. Jang, B. Joseph and H. Mukai, Ind. Eng. Chem. Process Des. Dev., 25 (1986), 809. 18.G. Maria and D.W.T. Rippin, Comp. Chem. Eng., $20 (1996), $587. 19.G. Maria, P. Terwiesch and D.W.T. Rippin, Chem. Eng. Comm., 143 (1996), 133. 20.G. Maria, FOCAPO98 Int. Conf., CACHE Corp., Snowbird (US), July 5, 1998. 21.J.B. Snape, I.J. Dunn, J. Ingham and J.E. Prenosil, Dynamic of Environmental Bioprocesses: Modelling and Simulation, VCH, Weinheim, 1995. 22.G. Tchobanoglous and F.L. Burton, Wastewater Engineering: Treatment, Disposal, and Reuse, McGraw-Hill, New York, 1991. 23.N.F. Gray, Wastewater Treatment: Theory and Practice, Oxford Univ. Press, Oxford, 1990. 24.D. Maquin and J. Ragot, DECHEMA Monographs 116 (1989). 25.G. Maria and O. Muntean, Chem. Eng. Sci., 42 (1987), 1451.
European Symposiumon Computer Aided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
847
Implementation of Flue Gas Cleaning Systems into an Object-Oriented Process Simulator for Practical Use G. Schuster, K. Weigl, A. Friedl Institute of Chemical Engineering, Fuel Technology and Environmental Technology, Vienna University of Technology, Getreidemarkt 9/159, A-1060 Vienna ABSTRACT Using flue gas desulfurization (FGD) in fossil fired power plant can be regarded as state of the art. Designing fossil fired power stations respectively calculating possible efficiencies of novel fossil fired power plant concepts this unit has to be considered. For this reason modules describing FGD are built up and implemented into a process simulation environment for thermal power plant engineering. With this modules complete thermal power plant schemes are calculated. 1.
INTRODUCTION
In developed countries the use of flue gas desulfurization (FGD) in fossil fired power plants, especially using coal, is state of the art since many years. The application was forced by enacting legal emission limits as well as by increasing environmental consciousness. The most used process is absorption with suspensions of calcium-compounds (caustic lime, slaked lime or limestone) and thereby production of gypsum. For example in Germany about 87 % of the electric power plants equipped with FGD use this process technique [ 1]. As FGD is part of the energy production process, it is usable to optimize the whole process with process simulation concerning energy saving potential and minimizing the demand of consumables. Simulation tools situated in chemical engineering are well established in estimating mass and heat/energy balances of the absorption process, but in the field of overall power plant modeling the application of such programs is not very common. For this reason IPSEpro T M is used, which has an flexible, object oriented structure, so it is possible to integrate modules for FGD into an existing power plant model library [2]. 2.
D E S C R I P T I O N OF THE PROCESS
Slaked lime or limestone is suspended in water (the solubilities of these compounds are in the range of grams per liter). In some cases caustic lime is used as educt, which reacts exothermically with water. In an absorber the suspension is in contact with flue gas in co- or countercurrent flow. SO2 from flue gas is dissolved in the absorbent (chemisorption occurs); due to the pH-value (in the range of 5 to 6) most of the dissolved SO2 exists as HSO3. The next step is oxidation to HSO4- by oxygen from excess air of combustion respectively by injected air into the absorber. HSO4 reacts to SO42 and precipitates as gypsum (CaSO4"2H20) as can be seen in the main chemical reaction scheme (Table 1). A part of the circulating absorbent is fed to the gypsum thickening process. In case of using limestone as educt CO32- degases as carbon dioxide in this range of pH-value. The solubility of SO2 in water ( 123.0 x 10 -4 , mole fraction at 50~ is by far higher than the one of CO2 (3.5 • 10 -4 ) so that the absorption of CO2 from flue gas can be neglected [3].
848
Table 1: Main chemical reaction scheme CaO + H20 ~ Ca(OH)2
HSO3" + 89 ---' HSO4"
Ca(OH)2 ~ Ca 2+ + 2 O H
HSO4" + H + ~
CaCO3 -~- C a 2+ + CO32-
SO42 + Ca 2+ + 2H20 ~ CaSO4"2H20~,
H+
5042- +
SO2 + H 2 0 ---" H2803
CO32" + H § ~- HCO3" + H + ~ H2CO3
H2SO3 -~- H S O 3 - + H + -~- 8 0 3 2 - + H +
H2CO3 ---* H 2 0 + CO2"~
3.
THE SIMULATION TOOL
IPSEpro TM is a commercial, equation-oriented simulation tool, that is specially designed for modeling thermal power plant processes. It has an object oriented structure, so it is possible to integrate a module for FGD into an existing power plant model library. Due to a special editor IPSEpro TM provides the facility to change the equations characterizing the units respectively to create completely new units. 4.
MODEL DESCRIPTION
Clean Water
Absorber
--
The following modules for wet FGD processes are created (in Figure 1 a possible arrangement for a FGD with caustic lime is depicted): Lime slaking If caustic lime is used, it has to be slaked before fed into the absorber considering the strongly exothermic heat of reaction of CaO + H20 ~ Ca(OH)2 (-65.166 kJ/mol). Saturator Before entering the absorber the flue gas stream gets saturated with water by injecting liquid water according to the dew point.
OxidizingAir
--
Absorber The steady-state model FGD (refer to Figure 1) has inlet connections for (raw) flue gas, oxidizing air and suspensions of CaCO3, Ca(OH)2 and CaSO42H20 at ~austic the top and at the bottom of the absorber. Outlet connections are implemented for clean gas and Gypsum suspension, which is withdrawn at the bottom. Due to intensive contact between flue gas and suspension Figure 1: FGD unit with caustic lime clean gas is assumed to be saturated with water. An overall mass balance as well as partial mass balances for the named elements are included. Heat balances are implemented as follows: heat capacities of inlet and outlet streams and heat of vaporization of water are taken into account as well as heat of reaction of Equation 1 in case of limestone respectively of Equation 2 for the case of lime (Data are taken from [4] and [5]).
~er
Slaking
Water
Beffilter[
AHR,298 SO2(g) + 2H20(1) + CaCO3(s) + 1/202 ~ CaSO4"2H20 + COffg) SO2(g) + H20(1) + Ca(OH)2(s) + 89 ~ CaSOn'2H20
-340.7 kJ/mol -453.9 kJ/mol
(1) (2)
The following parameters are used in the module in order to describe the absorption process in detail:
849 9
9 removal efficiency rlso2 =
~
n so2,fg
-
-
n so2,cg
hso2,fg
9 liquid to gas ratio i/g, pH- value of the absorbentpH superficial velocity [m/s] of flue gas in the absorber v stoichiometric factor Ca/Aso2(molar ratio of used calcium-compound to removed SO2) excess air ratio/3 of the reaction HSO3- + 89 ~ HSO4" ratio of oxygen introduced by flue gas and additional oxidizing air stream to stoichiometric necessary oxygen 9 pressure drops in the absorber [bar]: one for the absorption zone Apabs and one for the oxidizing air according to the static pressure of the suspension at the absorber bottom Apa~r 9 solid content of suspension at the absorber bottom 9 9 9 9
Giving a maximum gas velocity in the absorber the minimum absorber diameter is obtained by the volume flow of the flue gas stream. With a value for ]/g and a required residence time of the suspension in the bottom of the absorber, the height of the suspension in the bottom is estimated assuming a constant absorber diameter. Analogously a required residence time of the flue gas in the absorption zone gives the height of the absorption zone. Basic design of the absorption unit for an existing thermal power plant scheme can easily be derived with this module. Although the overall reactions are simple, the chemistry of the SO2-absorption process is quite complex. Furthermore in spray absorbers without internals the exchange surface is very difficult to determine exactly and depends on many different factors such as nozzle geometry respectively droplet diameter or gas velocity. For this reason the calculation of the separation efficiency is usually done by empiric correlations. Usually short-cut calculations are known for certain types of FGD systems, often formulated by manufacturers. One of the implemented correlations is shown in Equation 3 [6]"
/ rlso2
-
4/ 09 vo9el
pH+l 3 5 . 1 0 - 4 . C M
-0
58.10-4.Cs02+1 ....
45.10 - 5
CCl
//1
l - e
where Cso: means the concentration [mg/m3s, dry] of SO2 in the flue gas and CMg and Cct the concentrations [ppm] of Mg 2§ respectively Cl-ions in the absorbent. CMg is an input parameter and can be estimated from magnesium content in applied absorbent; Cct is calculated by the model from chlorine content in fuel and following in flue gas assuming complete absorption. In [7] sufficient accuracy for this correlation for the following operating conditions is stated: superficial velocities as high as 2.7 m/s, absorption height 11 to 15 m, pH values above 5.5 and SO2 inlet concentrations above 3000 mg/m3s, dry. Hydrocyclone The gypsum suspension is separated into an underflow rich in solids and an overflow poor in solids. The overflow stream usually is recirculated to the absorption process. Table 2" Main parameters of the process Beltfilter Analogous to the hydrocylone the Turbine inlet temperature 540~ suspension is splitted up to a stream with Turbine inlet pressure 200 bar high solid content and a recycle stream. Excess air ratio of combustion 1.4 Coal mass flow 174.3 to/h 5. RESULTS lower heating value 5440 kJ/kg water content 54 w% With the described models the Flue gas volume flow 515000 m3s/h calculation of complete power plant pH value of absorber suspension 5.5 schemes can be carried out. As an example a
850
scheme for a lignite power plant presented in [8] is taken and extended with flue gas desulfurization units using limestone (refer to Figure 2). Main parameters of the process are given in Table 2.
Fig. 2. Flowsheet of the thermal power plant
851 S02 concentration in clean gas [mg/m%] 600 I
800 I
>,, 34.5% o t-
400 I
200 I
In the following case an electric power output of 100 MW is assumed and lignite is used as fuel with a sulfur content of 2% (moisture-and-ash free maJ). The overall net electric efficiency rlet, ov can be calculated by considering boiler heat input and the power consumption of all pumps, compressors and of the coal mill. Figure 3 shows the effect of varying
140 120
34.0%
100
o E 9 33.5% SO raw flue g
~ 33.0%
~
8o
~
60
~
40 "~ 32.5%
20
> 0 32.0% 75%
80%
85%
90%
0 100%
95%
SO2 removal efficiency
Fig. 3. Overall net efficiency and l/g v e r s u s S O 2 removal efficiency (and correspondingly SO2 concentration in clean gas) SOz concentration in raw flue gas [mglm=,] 2000
0
4000
6000
8000
10000
12000
14000
r>~ 34.5% r= ._. ~ 34.0%
80 70 60
2L 9 33.5%
5o
0
40 ~'
O 33.0% ~
32.5%
0
32.0%
~
3(1 Ca 20 10 0 0%
1%
2%
3%
4%
5%
6%
7%
Sulfur content in fuel [w% moisture-and-ash free]
Fig. 4. Overall net electric efficency and 1/g versus sulfur content in fuel (and correspondingly SO2 concentration in raw flue gas) for SO2 concentration in clean gas fixed at 400 mg/m% S02 concentration in raw flue gas [mglm3s] 0
2000
4000
6000
8000
10000 12000 14000 12000 10000 ~ ' o'}
:_~ 34.0% 2]000 m
u =9 33.5%
g
/
m
%
~
8000
~
"-'
6000 33.0%
g/
32.5%
~
0
4000
g/
32.00/0 0%
. 1%
.
. 2%
.
o
3%
.
2000
. 4%
C~ 0
0 5%
6%
7%
Sulfur content in fuel [w% moisture-and-ash free]
Fig. 5. Overall net electric efficency and CaCO3 demand versus sulfur content in fuel (and correspondingly SO2 concentration in raw flue gas) for SO2 concentrations in clean gas fixed at 50, 400 and 2000 mg/m 3,
the value of fig leaving the variables residence time of flue gas and suspension respectively flue gas velocity in the absorber constant. This results in a higher SO2 removal on the one hand but on the other hand power consumption of the suspension circulating pumps and the oxidizing air compressor (due to a higher suspension level in the absorber bottom as suspension residence time is constant) is increased. Overall net electric efficiency is depicted versus SO2 removal and SO2 content in clean gas [mg/m3s]. It can be seen, that removing of SO2 from flue gas above a certain extent does not make sense with regard to economic as well as to ecological aspects, because with decreasing SO2 emissions CO2 emissions and others increase due to a higher consumption of fuel. In Germany e.g. the legal limitation for coal fired power plants is defined (according to 13. BImSchV) with 400 mg/m% [8]. Keeping this limit an increase in sulfur content in fuel leads to a decrease in rlet, ov as shown in Figure 4. In Figure 5 the values of r/el,or are depicted for three different clean gas concentrations and furthermore the demand of CaCO3 is drawn off.
852 6.
CONCLUSION
An existing power plant model library has been extended with modules for wet flue gas desulfurization according to the gypsum process. The modules handle mass and heat balances and allow a basic design of the absorber dimensions. The absorber module can be provided with empiric short-cut correlations in order to calculate SO2 removal efficiency. The implementation of one of these formulas is presented. The influence on overall net electric efficiency of a lignite power plant by varying SO2 concentrations in clean gas respectively by varying sulfur content in fuel and thereby SO2 content in raw flue gas is shown. With the presented tool a general planner of a thermal power plant can calculate its overall performance considering mass and heat streams. NOTATION C 1;'lso 2 AHR,298 v P
Ca/ASO2
concentration Mole flow SO2 [mol/s] Enthalpy of reaction at 25~ Superficial velocity in the absorber [m/s] Liquid- gas ratio [l/m3 (STP,wet)] Pressure [bar] Stoichiometric factor of used calcium-compound to absorbed sulfur dioxide
Greek rlso: rlet,ov
Excess air ratio in absorber (-) Removal efficiency (-) Overall net electric efficiency
Subscripts
fg cg
Flue gas Clean Gas
REFERENCES [1] M. Schtitz, VGB Kraftwerkstech. 77 (1997), 943-945. [2] E. Perz, A Computer Method for Thermal Power Cycle Calculation, ASME-Paper IGTI GT-351, 8p (1990). [3] E. Wilhelm, R. Battino, R.J. Wilcock, Chem. Rev. (Washington, D.C.) 77 (1977), 219-262. [4] I. Barin, Thermochemical Data of pure Substances, 2na ed., VCH, Weinheim, Germany, 1993. [5] R.K. Freier, Aqueous Solutions, de Gruyter, Berlin, Germany, 1976. [6] Bechtel Corporation (1977), EPA Alkali Scrubbing Test Facility: Advanced Program, U.S.Department of Commerce. [7] M. Eden, B. Heiting, M. Luckas (1997), VGB Kraftwerkstech. 77 (1997), 505-511. [8] K.Weigl, G. Schuster, G.N. Stamatelopoulos, A. Friedl, Comput. Chem. Eng. 23 Supplement (1999), 919-922. [9] H. Lehmann, Handbuch der Dampferzeugerpraxis, 3rd ed., Resch-Media Mail Verlag, Gr~ifelfin~unich, Germany, 1994.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000ElsevierScienceB.V. All rights reserved.
853
Dynamic optimisation of small size wastewater treatment plants including nitrification and denitrification processes B. Chachuat, N. Roche and M.A.Latifi* Laboratoire des Sciences du Gdnie Chimique, CNRS-ENSIC, B.P. 451, 1 rue Grandville, 54001 Nancy Cedex, France
Abstract- In this paper, dynamic optimisation of small size wastewater treatment plants is studied. The problem is stated as a hybrid dynamic optimisation problem which is solved using a gradient-based method. The aeration policy which minimises the energy consumption and satisfies discharge requirements under specified constraints (process and physical constraints) is then determined. The comparison between usual rule-based control policies and optimised aeration strategies showed that the optimised aeration profiles lead to reductions of energy consumption of at least 30%. 1. I N T R O D U C T I O N The pollution of water is mainly due to rain storm, domestic and industrial activities. The polluted water contains among others large quantities of organic and nitrogenous compounds. For many reasons (regulation constraints, salubrity, environment, water reuse . . . . ), the concentrations of these compounds must be reduced. This is achieved by means of wastewater treatment plants which are mainly based on the activated sludge process. For small communities, i.e. about 5 000 equivalent-inhabitants (in this case, small size wastewater treatment plants are more appropriate), the activated sludge process is low organic loaded. The plants typically consist of (i) a unique aeration tank (aerated and mixed using surface turbines), where a suspended microbial culture is used to treat the incoming wastewater, and (ii) a settler in which the microbial culture is separated from the liquid being treated. Most of the culture is recycled and mixed with incoming wastewater. Most of small size wastewater treatment plants (i) do not deal with any type of control, (ii) use very simple control strategies (time control, manual control, rules of thumb) or (iii) implement more "advanced" controllers (proportional controllers)[1]. Although biological removal of organic materials from wastewater is correctly handled in most cases by means of the aforementioned controls, nitrogen concentration in the treated wastewater may significantly exceed the allowed levels. In addition, the energy consumption may be very large. In the present paper, both economical aspects and discharge requirements are considered. The objective is to determine the optimal aeration policy which minimises the energy consumption and satisfies effluent and technical constraints. An illustration of potential benefits realised upon its application is detailed and a comparison between optimised and rule-based operation policies is presented. Corresponding author
854 2. T R E A T M E N T
PLANT MODEL
The biological processes involved in the aeration tank are modelled by the generally adopted IAWQt's activated sludge model No.1 [2] with two restrictions: (i) the state variable describing total alkalinity is not included, (ii) inert particulate material from influent and from biomass decay are combined into a single variable since they are of minor interest. The model for the aeration tank is derived from mass balance on each component S I, S s, X , X s, XB.N, Xs.,~, Suo, So,, SNO, XNO and S o. The mass balance equation related to dissolved oxygen concentration may contain an additional term which describes the oxygen transfer from the turbines. Hence, the functioning of the aeration tank is described by two models depending on whether the aeration process is on or off. The settler is modelled as a tank with 10 horizontal layers where each layer is assumed to be perfectly stirred. Clarification and thickening processes are described using Takfics traditional layer model [3]. The solid flux between two consecutive layers results from both sedimentation and liquid fluxes in the settler. Improvements were brought to the aforementioned model (i)to insure compatibility between both aeration tank and settler models and (ii) to describe soluble material fluxes in the settler [4]. Considering that soluble and particulate material concentrations in the recycling loop and in the bottom layer of the settler are equal, a global model can be defined for the plant as : dx dt dx dt
period)
-- f ( 1 ) ( X )
(aeration
- f(2~(x)
(non-aeration period)
(1)
where x is the 131-dimensional state vector representing state variables of both aeration tank and settler models. The process is thus described by two sets of differential equations. These systems, known as hybrid systems, are quite common in chemical engineering and their optimisation has attracted increasing attention since recent years [5].
3. OPTIMISATION P R O B L E M STATEMENT The activated sludge process consists of alternating oxic (aeration) and anoxic (non aeration) sequences to provide appropriate conditions for ammonia and nitrate degradation. Hence, the process can be seen as a succession of cycles defined as the duration between two consecutive starts of the turbines. Two parameters are then used to characterise a cycle: Ik the kth cycle duration and a k the aeration period within the corresponding cycle. Performance index. The optimisation objective is to determine the aeration and non-aeration periods which minimise the energy dissipated by the turbines. Considering that the power consumed is time-independent, the performance index J can be defined as :
; IAWQ : International Association for Water Quality.
855
J = ~la k =
Ik
(2)
k=l
Constraints. To ensure feasibility of the resulting aeration profiles, several constraints must be satisfied : (i) Maximum concentrations of total nitrogen (TN), chemical oxygen demand (COD), biochemical oxygen demand (BDO) and suspended solid (SS) are imposed :
TNmax = 10mg.L -1
CODn~x = 125mg.U l
BODma •
=
25mg.L -1
SSmax = 30mg.L -1
(3)
The resulting constraints are inequality path constraints of the following form :
N(x) < N . . . . Vt 6 [t0,t~ ]
(4)
(ii) Minimum and maximum lengths of aeration sequences t ~ and t~
and minimum length
off
of non-aeration sequences t m~n are constrained to prevent turbines from early wear. The following physical values were chosen: on = 5 mn
off t min
on = 60 mn
tmin
tma x
:
5
mn
(5)
(iii) A maximum duration for non-aerated sequences /'~off max is also defined to avoid too long non-stirred periods which may cause flocs sedimentation in the aeration tank and induce phenomena not described by the model. This value was set to : t~
"-
(6)
120 m n
4. OPTIMISATION METHOD Since the control variables a k and Ik are time-independent, the optimisation method used converts the dynamic optimisation problem into a non-linear programming problem (NLP). The inequality path constraints (4) are formulated as follows :
~l {f.tt~ max[O,g(x)- N =
~- 1
max
]z dt+ f,~/ ' m a x [ O , N ( x ) - N max ]2dt } = 0
(7)
The resulting non-linear programming problem can be solved by any gradient based method, e.g. a successive quadratic programming (SQP) m e t h o d - i n our case NLPQL [6]-. Therefore, the computation of gradients of the performance index as well as of the constraints with respect to parameters is needed. Gradients of the performance index (2) and of the constraints (5,6) are obtained by simple differentiation since they do not depend on the state x of the system. Three different methods can be used to estimate the gradients of the integral constraints (7): finite differences, sensitivity analysis and adjoint system [7]. For large process models, the most appropriate method is the last one and it has been chosen in this study. Its principal is detailed below. Two Hamiltonians ~1) and ~2~ are defined as :
H(1)(x,p,~,) = F + ~, r f(1) and H(2)(x,p,~,) = F + ~,r f(2) where F(x)
--
max[0,N(x)
m
N max
-] 2
(8)
and ~. is a 131-dimensional vector of costate variables
whose expression and boundary conditions are derived from Pontryaguin's maximum principal:
856
I~.~= OH<'>
[t~ -1 t'
0x r , t E OHm2>
~,r=
-~xr
,b]
(9)
, l
, t ~[t~,t~ ]
The associated boundary conditions are 9
)=0 (10)
+) The gradients of the constraints (7) with respect to parameters a t and l~ are given by 9
V g : ) ~ r ( f ~'' - -
f(2))t
(11)
~
M
Vlkg-'-- E
M-I
~T(f(l)--f(2))t ~
+E~.r(f),, + N . c
j=k
j=k+l
(12)
c
Note that both state and adjoint systems are integrated using DDASSL method [8]. Although those systems consist only of ODEs, it has been found that they may be stiff, mainly for the concentration of dissolved oxygen and its associated costate variables in the neighbourhood of switching times. 5. R E S U L T S A N D D I S C U S S I O N Typical optimisation results are shown in Table 1. Here, the optimisation was performed from 6 a.m. and over 42 cycles so that the time horizon is approximately 1 day. The flow rate and load variations were simulated using weighting functions (Fig. 1) [9]. The aeration rate first increases during 13 hours and is maximal by 7 p.m., then decreases and reaches its minimum during the night. These variations are in good agreement with the flow rate and load diurnal variations : low aeration rates are due to low influent loads and conversely, load peaks induce high aeration rates.
Table 1 Performance index Cycle 1~ 6 7 --+ 12 13 ~ 18 19 ~ 24 25--+ 30 31 --+ 36 37 --+ 42
vs. Time
Period
6h00 9h 16 12h02 14h29 19h46 23h55 3h27
--+ --+ --+ --+ + --+ ~
9h 16 12h02 14h29 19h46 23h55 3h27 6h27
Mean value"
Criteria 15,3% 18,0% 21,6% 23,9% 25,6% 14,2% 16,6% 21,3 %
5"
1.2
o.~ = .~ 9
0.6 O.4 0.2
........
0
.........
0
Flow rate
........................................ i ..................................... 5 10
15
(h) Fig. 1. Typical weighting functions weather conditions.
20
Time
for dry
857 Note that all effluent criteria are satisfied. Also note that effluent constraints with respect to COD, BOD and suspended solids remain inactive. Two operating modes are considered in this study: rule-based and optimised modes. A comparison between these two operating modes is given in Fig. 2. In the rule-based operating mode, fixed and equal cycle lengths are used and the turbines are switched-on until a specified condition is satisfied (maximal concentration of dissolved oxygen, minimal value of redox potential or minimal concentration of ammonia) within each cycle. Here, the specified condition is the minimal concentration of ammonia which was set to 0.5 mg.L-'. In the optimised operation however, both cycle lengths and aeration durations are determined. In both cases the constraints on total nitrogen are satisfied, but the use of dynamic optimisation methods reduces by at least 30% the daily aeration periods.
Optimised aeration sequences: Jopt = 21, 3%
Usual aeration sequences: J = 30, 0%
lllllllllllllllllllllllrll oill I oli BhO0
12h00
18h00
0hOO
6h00
Fig. 2. Rule-based
vs.
6h00
12h00
II
18hO0
0h00
6h00
optimised aeration profiles.
Since the optimised aeration profiles are being calculated through local gradient search methods, it has to be clear that global optimality of these profiles is not guaranteed. Indeed, computational results with different initial sets of parameters shows that the optimisation problem may exhibit several local minima. The search for global optimum is obviously of great importance. Two global optimisation methods are therefore under implementation: alpha branch and bound (~BB) [10] and iterative dynamic programming (IDP) methods [11]. However, even with the non-uniqueness of the optimum of the problem under consideration, the results obtained are very promising. Global optimum will probably lead to larger energy consumption reduction. 6. C O N C L U S I O N S In this contribution, the minimisation of energy consumption in the aerators for small wastewater treatment plant was analysed. The problem is a hybrid dynamic optimisation problem which was converted into a non linear programming (NLP) problem. The gradientbased method, SQP, was found to be an efficient method to solve this NLP problem. The comparison between the usual rule-based control policy and the optimised aeration strategy was shown to provide better aeration profiles with reductions of energy consumption of at least 30%. ACKNOWLEDGEMENTS The authors are grateful to 'Minist~re de l'Enseignement Sup6rieur et de la Recherche' (MESR) for its financial support.
858 NOMENCLATURE M number of cycles (-) S~ inert soluble material (g.m 3) s s soluble substrate (g.m-3) SNo soluble biodegradable organic nitrogen (g.m3) So, soluble ammoniacal nitrogen (g.m 3) SNo soluble nitric nitrogen (g.m -3) S O dissolved oxygen (g.m3)
XB,. heterotrophic biomass (g.m 3) autotrophic biomass (g.m -3) X, inertparticulate material and products (g.m -3) XNO particulate biodegradable organic nitrogen (g.m 3) X s slowly biodegradable substrate (g.m -3) tt,k turbine's switching-on time within cycle k (s) tck turbine's switching-off time within cycle k (s)
Superscripts (1) aeration sequences
(2) non-aeration sequences
XB, A
REFERENCES
1. Lindberg, C.-F. (1997). Control and estimation strategies applied to the activated sludge process. PhD thesis, Uppsala University. 2. Henze, M., Grady, C. P., Coujer, W., Marais, G. and Matsuo, T. (1987). Activated Sludge Model No. 1. Technical Report 1, IAWQ, London. 3. Takfics, I., Patry, G. G. and Nolasco, D. (1991). A dynamical model of the clarificationthickening process. Wat. Res. 25, 1263-1271. 4. Jepsson, U. (1996). Modelling aspects ofwastewater treatment processes. PhD thesis, Lund Institute of Technology. 5. Barton, P. I. and Park, T. (1997). Analysis and control of combined discrete/continuous systems: progress and challenges in the chemical industries. AIChE Symposium Series 93, 102-114. 6. Schittkowski, K. (1985). NLPQL: a fortran subroutine solving constrained non-linear programming problems. Annals of Operations Research 5,485-500. 7. Latifi, M. A., Corriou, J.P. and Fikar, M. (1998). Dynamic optimisation of chemical processes. Trends in Chemical Engineering. 4, 189-201. 8. Brenan, K. E., Campbell, S.E. and Petzold, L. R. (1989). Numerical solution of initial value problem in differential-algebraic equations. North-Holland, New-York. 9. Isaacs, S. and Thormberg, D.E. (1998). A comparison between model and rule based control of a periodic activated sludge process. Wat. Sci. Tech. 37(12), 343-352. 10. Androulakis, I.P., Maranas, C. D ; and Floudas, C.A. (1995). ~I3B : a global optimization method for general constrained nonconvex problems. J. Glob. Opt. 7, 337-363. 11. Luus, R. (1990). Application of dynamic programming to high-dimensional non-linear optimal control problems. Int. J. Control 52, 239-250.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
859
A New Procedure for Predicting NOx Emissions from Furnaces T. Faravelli ~, L. Buaa, A. Frassoldati a, A Antiforab, L. Tognotti ~, E. Ranzi a aCIIC Dipartimento di Chimica Industriale e Ingegneria Chimica Politecnico di Milano, Pza L. da Vinci, 32, 20133 Milano, Italy bAnsaldo Energia SpA, Piazza Monumento 12, 20025 Legnano, Italy CDipartimento di Ingegneria Chimica, Chimica e Scienza dei Materiali, UniversitA di Pisa, via Diotisalvi 2, 56126 Pisa, Italy This paper presents a postprocessor for the determination of NOx formation from industrial boilers. The flow and temperature fields within the furnace, obtained through CFD codes, are approximated by a network of ideal reactors. This approach allows using a very detailed and comprehensive kinetic model, without loosing the detailed thermal and fluidynamic information. The meaning of this work is to move a further step in the direction of a better description of both kinetics and fluidynamics aspects during combustion. The prediction of pollutant byproducts, whose amount is in terms of ppms or even ppbs, necessarily requires the use of a detailed description of the process chemistry. Moreover, in the case of NOx formation, the air and fuel staging, as applied to recent plants, implies the interactions between nitric oxides and hydrocarbon radicals. A detailed description of these reaction paths is extremely important to correctly reproduce the f'mal emissions. The comparison of global predictions against experimental industrial results shows a quite satisfactory, qualitative and quantitative, agreement in different operating conditions and plant configurations. INTRODUCTION The design of combustion systems for utility boilers requires compliance with stringent limitations concerning the pollutant emissions. This has increased the demand of analytical and computational tools, able to predict flow mixing and chemical kinetics in currently used furnace configurations. Even with the continuos increase of computer power and speed it is still impossible to couple detailed fluidynamics and detailed kinetics. Generally when considering industrial furnaces the main interest is devoted to the overall energy aspect and consequently the chemistry is sacrificed and reduced to a few reactions. Although the complexity of 3D simulations prevents the coupling of sophisticated flow and combustion models, in some cases, however, a detailed kinetics is absolutely necessary. This occurs, for instance, when: a) non-equilibrium radical concentrations make ineffective simplified approaches; b) large fuel-rich regions are produced as in the case of the reburning technique. This is also true in all the cases where the interest lies in the pollutant formation. Only detailed kinetic models are able to predict minor species whose concentrations range from ppbs to a few ppms. The predictions of NOx emissions may be decoupled from the simulation of the combustion environment. This is due to the different time scales, and to the fact that minor species affect only marginally the main combustion process and consequently do not influence the overall temperature and flow field. In this light, a "chemically oriented" approach, able to account for a detailed kinetic model of the NO formation mechanism, can be conveniently coupled to CFD results obtained by using a few main combustion reactions. The approach proposed in this work refers to the so called hybrid method (Ehrhardt, 1993). On the basis of the computed 3D results for flow,
860 temperature and stoichiometry fields, the volume of the combustor is "reduced" to a simplified network of ideal reactors, which can be modeled as "perfectly stirred" or "plug flow" reactors. Within each reactor the detailed kinetic models can be considered to predict the concentrations of the minor pollutant species and especially NOx (Benedetto et al., 1997). 1. THE SFIRN APPROACH The proposed 'chemically oriented approach', named SFIRN (Simplified Fluidynamic by Ideal Reactor Network), is based on a two step procedure. Starting from the complex flow, temperature and stoichiometry fields computed by means of the 3D CFD simulation, a simplification of the flow behavior is sought so as to represent the condition within the furnace through a network of ideal reactors ("perfectly stirred", PSR, or "plug flow", PFR) strictly interconnected. The concentrations of the minor pollutant species are then evaluated using a comprehensive detailed kinetic scheme, applied to the simplified flow field of the ideal reactors. The definition of number, type, and connections of the ideal reactors, together with their operating conditions, is the critical step of the procedure. In order to simplify the approach, the furnace is arbitrarily divided into different regions (f'trst and second burner levels, 25OO
2
2o0o
7
I=
J
1~'1~176176
i i..,'.i i ~,'~! , i'. ,~. :,.i
.... -'~[
[,.
Fuel o 0.0
I
~
..._Air
2 I..
Flue gas
9| 1.0 t 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 9 6.0 '~ 0.5
Fig. 1. Temperature vs. stoichiometry distribution relative to the second burner row zone of 75 MWe Cassano plant.
Fig. 2. Simplified ideal reactor network representing the second burner row zone.
reburning, over-fire air ports levels). The network of reactors is then built inside each region. It is important to consider first that the NOx formation is mainly influenced by the stoichiometry and temperature conditions. So, within each zone, the cells which can be considered equivalent from the "NO~ formation" viewpoint are those presenting similar values of stoichiometry and temperature. These are grouped together to form an ideal isothermal reactor without considering their relative position. Fig.1 shows, as an example, the temperature vs. stoichiometry plot of the CFD results relative to the second burner row region. The parameter ~. is computed as the ratio of the actual oxygen content and the stoichiometric value. The plot shows how the computed results are scattered around an adiabatic temperature distribution. The homogeneous cells are grouped in a single equivalent reactor, by dividing the stoichiometry axis in three parts (rich reducing zone, lean oxidizing zone and the mixing zone). Each zone corresponds to an ideal reactor characterized by average stoichiometry and temperature values. It is worth noting that the mixing zone is centered around the mean stoichiometric value of each region in which the furnace is divided. The amplitude and position of the different zones is the only degree of freedom of the proposed approach. The goodness and correctness of the assumed grouping is then verified on the basis of a comparison with the residence time distribution computed by using CFD, as it will be illustrated later on.
861 Fig.2 shows the reactors individuated by the procedure, applied to the second burner row zone of the furnace. From the geometrical point of view this separation technique enables to identify contiguous cells, which represent the fuel rich zone (reactor 1), the surrounding air flow from each burner (reactor 2), and the zone in which air and fuel are completely mixed (reactor 3). Once the reactors are identified, the further step requires the def'mition of the volume and temperature values for each reactor of the network, its type and f'mally the interlinking flows. The volume is directly evaluated as the sum of the single cell volumes. The temperature of each isothermal reactor is evaluated as a "kinetic average" of the cell temperatures, using a proper weight of the NOx formation contribution of each cell. fff(T) 9T. NO~(T, 2, r). r(T). dT = o
(1)
~f(T). NO,(T, 3,, ~:). ~'(T). dT 0
The NOx formation function has been previously tabulated as a function of temperature, stoichiometry and residence time. The reactor is assumed to be perfectly stirred (PSR) or plug flow (PFR) according to the flow characteristics. The reactor is considered as a plug flow when a prevalent streamline is present in the belonging cells. When this is not the case a perfectly stirred reactor is assumed. The mass flow rate entering each reactor is determined from the convective and diffusive mass flow rates exchanged between cells belonging to the different reactors. As an example, fig. 3 shows the resulting Fig. 3. Ideal reactor network network coming from the examined 75 MWe furnace, representing the furnace which will be discussed in the next paragraphs. Thirteen reactors are individuated. The PFR at the bottom represents the hopper, whereas 2 PFRs and 1 PSR are assumed for each burner row. The PFRs correspond to the potential core of the inlet jets of both fuel and air, while the PSR individuates the mixing zone. Two PSRs and one PFR were needed to represem the "rebuming" zone, while 3 PSRs were used in the postcombustion zone. The proposed approach does not correctly account for the effect of turbulence on the chemistry, due to the temperature fluctuations. However it is argued that this effect is important mainly for the reactions with high activation energies (usually higher than 80 kcal/mol). Indeed, in the actual conditions, these reactions are only a few and their contribution results marginal as evidenced by sensitivity analysis. 2. THE KINETICS AND THE IDEAL REACTOR NETWORK MODEL The simulations performed throughout this work have been carried out referring to a comprehensive detailed kinetic scheme (NOX9911) involving about three thousands reactions, with about two hundreds species (molecules and radicals). Rate constants of the reactions in the overall kinetic model are not reported here because of the huge dimensions of the scheme. NOX9911, likewise all the mechanistic kinetic schemes of some complexity, is basically founded on a strongly modular and hierarchical structure, in which the simplest reaction sub-mechanisms are necessary to investigate the more complex ones (Westbrook and l')rver_ IOR4~. The neellliarirv of" NC)3(O011
.~cheme i.~ it.~e.omnrehen.~ivene.~. The overall
862 mechanism includes all the subsystems to model the hydrocarbon oxidation in a wide range of operative conditions. The main oxidation mechanism, for the simulation of hydrocarbon mixtures containing up to twelve carbon atoms, is the core of the overall kinetic scheme (Ranzi et al., 1997). This model is based on a detailed sub-mechanism of Cl-C4 species. Assuming analogy rules for similar reactions, only a few fundamental kinetic parameters are needed for the progressive extension of the scheme towards heavier species. These parameters define the main classes of primary oxidation reactions appropriate to the temperature ranges. The nitrogen sub-mechanism was partially derived from the works of Miller and Bowman (1989), and the successive modifications proposed by Glarborg and coworkers (Glarborg et al., 1995). Thermochemical data were primarily obtained from the CHEMK/N thermodynamic database (Kee et al., 1989). Unavailable thermodynamics quantities were estimated by group additivity methods (Benson, 1976). The ideal reactor modeling was performed using a general program (DSMOKE) allowing the simulation of ideal reactor networks, anyhow interconnected. The presence of recycle streams is also handled. The large number of reactors required by SFIRN approach (up to 50), the dimensions of the kinetic scheme and, above all, the presence of several iteration loops among the reactors, makes the numerical solution of the problem a hard and heavy task. The large system of ordinary differential equations describing the governing equations in "plug flow reactors" is generally stiff and solved by implicit algorithms. 3. THE APPLICATIOI~I EXAMPLE The test case is represented by the gas/oil fired furnace of the 75 MWe unit of the Cassano power plant. This front fired furnace is equipped with two rows of Low NOx TEA burners, one row of rebuming ports (RB), and one row of Over-Fire Air (OFA) ports. Six different gas fired and two oil fired operating conditions were considered. In the gas fired case, the two
Fig. 4. 75MWe Plant: a) Temperature, b) Fuel, and c) Velocity (central plane) Fields. baseline configurations (Basel and Base2) differ for the excess air in the primary combustion zone. The two Single OFA conditions (OFA1 and OFA2) are characterized by the different content of flue gas mixed with the combustion air (respectively 3% for OFA1 and 7% for OFA2). In the Double OFA configurations (C4P2 and C8P2) some post combustion air is supplied also through the rebuming injectors, located immediately above the upper burner row. All the CFD calculations were carried out by the CINAR code (Lockwood et al., 1999; Antifora et al., 1999) based on a coarse mesh including 27000 cells with a burner discretization consisting of a (3x3) cells matrix. Only the C8P2 configuration was described using a freer mesh of 48000 cells, and a matrix (5x5) for the burners.
863 The two oil combustion cases are relative to a baseline and a gas-reburn configurations. The CFD calculations were performed by using the larger mesh. Methane was used as rebum fuel, 1.6 jr CFD.-. while the oil fuel was represented by a mixture of 1.2 SF IRN - n-heptane and benzene to match the C/H ~0.8 experimental ratio. The presence of bond nitrogen (0.32- 0.44 % weight) was taken into account by 1).4 adding HCN to the fuel, as suggested in the literature (Miller and Bowman, 1989). NOx 0 1 2 3 4 5 6 7 8 9 emissions were measured at the end of the t Isl convective section of the boilers. It was assumed Fig. 5. Residence time distribution for that the measured NOx concentration are equal to tracer injected in the 1st burner row the NO concentration at the furnace outlet. As an example, figure 4 shows the computed 3D profiles, which should be compared with the resulting ideal reactor network of fig. 3 for the C4P2 case. The front fired furnace exhibits the typical behavior with a preferential streamline in the rear wall region, which is as usual hotter than the front section and contains some unburned fuel. The temperature increase observed at the OFA level is due to the combustion completion of such fuel. The furnace is not operating with gas-reburn and the air flow entering from the reburning ports is not able to penetrate the main flue stream or better contributes to push the flue gas toward the rear wall. The individuated ideal reactors, even though in a quite small number, are able to take into account these thermal and mixing characteristics. Table 1. Mean Residence Time The four reactors describing the zone close Tracer Injection ..... CFD SFIRN to the rear wall (left side in fig. 3) exhibit I st Burner Row 1.537 1.518 higher temperatures and contain some I st and 2 *d Burner Row 1.028 0.969 unburned fuel. The presence of a by-pass 2 *a Burner Row 0.487 0.419 stream is evidenced by the PFR in the rebuming zone. The PSRs def'med by the air and flue gas fed both in the rebuming and OFA zones take into account the back mixing highlighted by the velocity profiles in fig. 4c. The appropriate description of the mixing characteristics of the SFIRN approach are confirmed by the comparison of the residence time distributions against CFD results. For example, figures 5 shows the residence time distribution curves for a tracer injected in the 1st burner row and monitored at the rebuming level. The agreement is quite good and the presence of a bimodal distribution, with the second "hump" led by the hopper delay, is also evidenced in both the calculations. Being the two curves close enough, the mean residence time results (table 1), relative to different tracer injection location, correctly agree. It is important to underline that even though strong simplifications have been introduced, the main fluidynamic aspects of the system are taken into account. Obviously further developments in the accuracy of the method can be pursued by increasing the number of ideal reactors, even though some evaluation on required CPU time should be made. The solution of the very large set of differential and algebraic equation system is quite expensive (about half hour for a typical ease on a Pentium II PC), but the continuous and fast development in the computing power will allow to easily handle larger networks. ,__
a,,
|ii
I
i
.
i
_
i
.
.
.
.
.
.
.
.
4. RESULTS The computed results are compared with experimental data coming from the already discussed industrial applications. More than the absolute predicted NOx values, which are also quite good, the main outcome is the capability of the proposed approach to predict the trends of the byproduct emissions and the environmental impact of the different process or operating
864
Table 2. NOx emissions [ m g ~ m 3 @ 3% 02, dry] ~as fired 75 MWe furnace Experimental .... Simulated Base I 212 223 Base2 273 28.8 % 289 29.6 % OFA 1 215 1.4 % 214 -4.0 % OFA2 149 - 2 9 . 7 % 157 - 2 9 . 6 % C4P2 107 -49.5 % 118 - 4 7 . 1 % C8P2 107 - 4 9 . 5 % 112 -49.8%
Table 3. NOx emissions [ m g ~ m 3 @ 3% 02, dry] oil fired 75 MWe furnace Experimental 'Simulated Baseline 420 412 Gas-reburn 225 - 46.4 % 213 - 48.3 % i
ill
|
i=l
i
condition alternatives. The comparison between the experimental data at the furnace outlet and the calculations performed with the new NOx predictive procedure is summarized in Tables 2 and 3. It is worth noting the very good matching of the general trends obtained. As an example, the increased pollutant emissions due to the increase of the excess air from Base 1 to Base 2, is correctly reproduced and the same applies to the effect of air staging and gas mixing for all the different cases. Also the NOx reduction of about 50% in the oil fired conditions when a gas rebuming is operated is correctly reproduced.
CONCLUSION This work presents a new approach (SFIRN) using complex kinetics for predicting NOx emission from furnaces of utility boilers. It takes advantage and moves from the flow and temperature fields obtained from CFD computations. A successive def'mition of a network of ideal isothermal reactors allows the use of detailed kinetic mechanisms, which characterize the pollutant formation. The resulting reactor network is able to take into account the main thermal and fluidynamic aspects as coming from CFD. The residence time distributions, as evaluated by CFD and SFIRN, are close and this further demonstrates the general validity of the approach. Despite the strong simplifications, the obtained results for the analyzed different operative conditions are extremely encouraging and indicate that this approach can become an interesting tool to optimize the furnace design and to attain the stringent law requirements for the pollutant emissions from thermal plant.
REFERENCES Antifora, A., Faravelli, T., Kandamby,N., Ranzi, E., Sala, M., and Vigevano, L. (1999). Proceedings of the .Fifth International Conference on Combustion Technologies for Clean Environment,Lisbona. Benedetto, D., Pasini, S., Tognotti, L., Lamarca, C., Faravelli, T., Ranzi, E. (1997) Proceedings of Fourth International Conference on Technologies and Combustion for a Clean Environment, Lisbona. Benson, S.W. (1976). Thermochemical Kinetics (2 nd ed.), John Wiley and Sons, New York. Ehrhardt, K.R. (1993). Development of a Hybrid Model for the Prediction of Nitric Oxides Emissions of Furnaces, Energy Lab. Report, M.I.T. Glarborg, P., Dam-Johansen, K., and Miller, J.A., (1995). Int. s Chem. Kin, 27:1207-1220. Kee, R.J., Rupley, F., and Miller, J.A. (1989). Sandia Report SAND89-8008, Sandia National Laboratories, Livermore. Lockwood, F.C., Abbas, T., Kandamby,N.H. and Sakthitharan, V. (1999). Special Issue oflJCAT on: Computational Reactive Fluid Dynamic Modeling, Software Tools and Applications
Miller, J.A., and Bowman, C.T. (1989). Prog. Ener. Combust. Sci., 15:287. Ranzi, E., Faravelli, T., Gaffuri, P., Sogaro, A., D'Anna, A., and Ciajolo, A. (1997). A Wide Range Modeling Study of iso-Octane Oxidation, Combust. Flame, 108:24-42. Westbrook, C.K., and Dryer, F.L. (1984). Prog. Ener. Combust. Sci., 10:1.
European Symposium on Computer Aided Process Engineering - l0 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
865
Liquid Effluent Properties Prediction from an Industrial Wastewater Treatment Plant Using Artificial Neural Networks C.A. Gontarski a, P.R. Rodrigues b, M. Mori ~and L.F. Prenem b. a Departamento de Engenharia Qufmica, Universidade Federal do Parang,- P.O. Box 19011, Zip Code 81531-990, Curitiba, PR, Brazil. b Rhodiaco Ind. Quim. Ltda, P.O. Box 071, Zip Code 13140-000, Paulfnia,SP, Brazil. c Departamento de Processos Qufmicos/FEQ, Universidade Estadual de Campinas, P.O. Box 6066, Bar&o Geraldo, Zip Code 13081-970, Campinas, SP, Brazil. This work presents a way to predict the environmental properties of the output stream of the wastewater treatment plant at Rhodiaco Ltda, one of the major chemical plants in Brazil. The industrial plant produces purified terephtalic acid and generates wastewater that should be treated in an activated sludge system. The influence of input variables is analyzed, and satisfactory predicted results are obtained for an optimized situation.
1. PROCESS DESCRIPTION Treatment of the process wastewater was done in three aeration basins and one sedimentation tank that operate in series (figure 1). All reactors also receive a stream of the untreated wastewater, and the sludge from the sedimentation tank is recycled. The solids removal stream flows from the effluent of the last reactor to a settler tank where the solids are properly eliminated. Influent wastewater~
~
"1reactor/
I
~ ' ~
reactor / Solids Removal Stream v
Tank
"1reactor/ "]reactor]
"]reactor| sludge recycle
Figure 1 - Process flowsheet for treatment plant.
v
Final effluent
866
2. APPLICATION OF NEURAL NETWORKS
The modeling and simulation of chemical processes have been developed using ever more complex deterministic models, due to the recent personal computers evolution. Furthermore, some circumstances turn the application of these models impossible to solve. For example, this happens when some data to be used in the model is too difficult to obtain, or when the model is very complex and requires a lot of simplifications [1]. Some works have been published [2-3] that uses neural networks to solve these kinds of problems. In this work, the back-propagation neural networks [4] are used to predict the elimination of the total organic carbon (TOC) in the treatment plant, using the deltabar-delta algorithm [5] for estimation of the weights, and the sigmoid function as the neuron transfer function. The original database, obtained from the plant control system and from the laboratory, has much unusable information and other unstable situations that must be eliminated. The average hydraulic residence time in each equipment is considered to establish relationships input/output in the data. Seven neural networks are used to simulate the system, one network for each reactor and another to predict the final TOC of the effluent based on the conditions for the last reactors. The training of the back-propagation neural networks considers the following variables in the wastewater treatment plant: a) the inlet wastewater TOC in each reactor; b) the inlet flow ratio of the liquid and of the recycled sludge, c) the concentration of suspended solids (sludge) in the reactors, d) the concentration of dissolved oxygen in the reactors, e) average sludge residence time, f) the parameters related to reaction kinetics, and others. The output of the neural network is the predicted TOC of the outlet streams. The overfitting, typical of back-propagation networks, is supervised by testing the results with a test data set [6]. When this type of neural network is trained with sufficient connections the RMS error of the test data set decreases in the beginning of the training, and then reaches a minimum point. The network is saved in the minimum point to be used in the model. Figures 2 and 3 show the results for the reactor 1A and 1B, respectively, for a condition of initial TOC in the reactor limited to 140 mg/I, and the results are shown in Table 1 [7]. They also show the relation between predicted and observed values for the TOC in the first reactors. In this case, the data are randomly distributed between test and train sets. Table 1. Results obtained for the TOC prediction to the wastewater treatment_plant. Network / RMS error Of RMS error of Correlation index CorrelatiOn index Equipment train data set test data set of train data set of test data set REACTOR i A 0.0155 0.0265 0.9928 0.9779 REACTOR 2 A 0.0445 0.0463 0.9709 0.9615 REACTOR 3 A 0.0244 0.0307 0.9825 0.9724 REACTOR 1 B 0.0187 0.0229 0.9879 0.9807 REACTOR 2 B 0.0285 0.0339 0.9825 0.9699 REACTOR 3 B 0.0309 0.0311 0.9769 0.9812 FINAL TANK 0.0396 0.0401 0.9730 0.9684
867
o x
180150
train data set test data set
~/,
150
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60
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Observed TOC Fig. 2. Comparison between the observed value of TOC for the reactor 1A and the predicted by the artificial neural network.
/
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/
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0
180
train data set test data set
o
ao
6o
9'o
I~o
Observed TOC
igo
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Fig. 3. Comparison between the observed value of TOC for the reactor 1B and the predicted by the artificial neural network.
Other works applying neural networks to actual data from a chemical process found a correlation index equal to 0.8 to a coke furnace [3]. For a wastewater treatment plant H~ck and K0hne [2] used a neural network which have a correlation index of 0.92. 3. ANALYSIS OF THE RESULTS
Identification of the main variables for the training processes is one of the objectives of this work. The sensitivity analysis can show some interesting information about the process. For this purpose, the variation of the correlation index compared with a base case is analyzed. The correlation index between the observed and calculated values of the j-th variable (RE) is calculated by,
where: N is the number of data points; x~ is the predicted value of the j-th variable; y~ is the observed value of the j-th variable; px is the average of the values predicted by the network; py is the average of the observed values; and Nv is the number of network inputs. The effect of the elimination of each input variable is analyzed by the effect on the correlation index expressed by,
( % influence)j
=
1-
* 1 O0
,(.j = 1 , N v ) .
,
(2)
868
where: Rc8 is the correlation index between predicted and observed values for the base case. Some variables are eliminated because the low value of their calculated influence, and the final configuration of the input variables for the first reactor is shown in Tables 2 and 3. All the variables in the Tables 2 and 3 are considered important for the neural network because the least value encountered for the influence is significant. Table 2. Influence of the variables on the trained neural network for by disabling the input of the variable. Variable R O.9782 BASE CASE -0.3341 PH in the inlet stream 0.7941 TOC in the inlet stream 0.1941 PH in the first reactor 0.1094 TOC in the first reactor 0.4289 Solids concentration (sludge)in the reactor 0.1216 Solids Concentration in the sludge stream -0.1533 Liquid flow rate for the first reactor 0.6637 Solids removal flow rate from the third reactor 0.7502 Sludge flow rate (recycled) Table 3. Influence of the variables on the trained neural network for by disabling the input of the variable. Variable R 0.9813 BASE CASE 0.6470 PH in the inlet stream 0.4077 TOC in the inlet stream 0.4890 PH in the first reactor 0.4364 TOC in the first reactor 0.7876 Solids concentration (sludge)in the reactor 0.2837 Solids Concentration in the sludge stream 0.0265 Liquid flow rate for the first reactor 0.2048 Solids removal flow rate from the third reactor 0.6113 Sludge flow rate (recycled)
the reactor 1A, Influence X 65.84% 18.81% 80.15% 88.81% 56.15% 87.56% 84.32% 32.15% 23.30% the reactor 1B, Influence X 34.06% 58.46% 50.16% 55.52% 19.73% 71.08% 97.30% 79.13% 37.70%
This analysis do not allow full conclusions about the influence of the input variables on the final TOC. Then, the sensitivity analysis is based on a small increment made on each input variable, as shown in the formula, V a
= Vj o:~w i
(j = 1 N v )
(3)
where: Vja is the utilized value in the model for the j-th variable to obtain the corresponding Rj ; Vf is the value that is used in the base case for the j-th variable; wj is a factor applied for each j-th variable. These values are shown in Table 4. The factor for each variable is defined, by eq. 4, considering the range of variation for the variable, allowing a better way to compare the effect between the variables.
869
( v~4AX wj=f*
MIN
" VME, -V,.i
)
,(j = 1,Nv).
,
(4)
where: F is a global factor for all variables (=1,05); VjMAx is the maximum value encountered for the j-th variable; V~MIN is the minimum value encountered for the j-th variable; VjM~D is the average value for the j-th variable. The values of the factors and the influence are small to avoid the problem of extrapolation on the neural networks. Table 4. Values of the factors used to change the original value of each variable. Variable Reactor 1A Reactor 1B 1.0088 1.0088 PH in the inlet stream 1.0114 1.0114 TOC in the inlet stream 1.0013 1.0010 PH in the first reactor 1.0261 1.0185 TOC in the first reactor 1.0077 1.0113 Solids concentration (sludge)in the reactor 1.0093 1.0106 Solids Concentration in the sludge stream 1.0063 1.0173 Liquid flow rate for the first reactor 1.0124 1.0199 Solids removal flow rate from the third reactor 1.0132 1.0234 Sludge flow rate (recycled) In Tables 5 and 6 are the results for the influence of each variable in reactor 1A and 1B respectively, and a classification of the variable as it has a high, medium or low influence. The values are different because the networks are independent and the range of the variables vary between them. Table 5. Sensitivity analysis for the reactor 1A Variable BASE CASE pH in the inlet stream TOC in the inlet stream pH in the first reactor TOC in the first reactor Solids concentration (sludge)in the reactor Solids Concentration in the sludge stream Liquid flow rate for the first reactor Solids removal flow rate from the third reactor Sludge flow rate (recycled)
0.9782 0.9792 0.9776 0.9795 0.9733 0.9783 0.9773 0.9791 0.9784 0.9777
Influence X -0.105% 0.053% -0.137% 0.496% -0.013% 0.089% -0.099% -0.023% 0.051%
Classification X High Medium High High Low Medium High Low Medium
4. CONCLUSIONS
The sensitivity analysis from both reactors (1A and 1B) shows common conclusions, that indicate that the liquid flow rate and the pH of the inlet stream are the most important variables to control the plant, if the values of all variables are in the range of the studied data.
870
The main conclusion of this work is that the use of neural networks can be used to establish a better operating condition, which has been defined by some variables such as the splitting ratio of the inlet stream for each reactor. Neural nets represent a possible aid to operations in order to predict upsets and proactively act to minimize output fluctuations. In the future, some work will be done to predict effluent conditions based on the actual operation data set. Table 6. Sensitivity analysis for the reactor 1B Variable BASE CASE pH in the inlet stream TOC in the inlet stream pH in the first reactor TOC in the first reactor Solids concentration (sludge)in the reactor Solids Concentration in the sludge stream Liquid flow rate for the first reactor Solids removal flow rate from the third reactor Sludge flow rate (recycled)
0.9813 0.9803 0.9803 0.9805 0.9809 0.9811 0.9814 0.9797 0.9814 0.9807
Influence X 0.106% 0.104% 0.082% 0.045% 0.015% -0.009% 0.167% -0.008% 0.057%
Classification X High High High Medium Low Low High Low Medium
REFERENCES 1. Bhat, N.V., Minderman, P.A., Mcavoy, T., Wang, N.S. "Modelling Chemical Process Systems via Neural Computation". IEEE Control Systems Magazine. v. 10, 1990. 2. H~ck, M., K~hne, M., "Estimation of wastewater process parameters using neural networks", Water Sci Technolvol. 33, no. 1,p. 101-115, 1996. 3. Blaesi, J., Jensen, B., "Can Neural Networks Compete With Process Calculations?" intech Appilying Technology, p. 34-37, 1992. 4. Rumelhart, D.E., McClelland, J.L., editors, Parallel Distributed Processing: Explorations in the Microstructure of Cognition. V. 1, Foundations. Mit Press, 1986. 5. Jacobs, R.A. "Increased Rates of Convergence Through Learning Rate Adaptation". Neural Networks. v. 1, p. 295-307, 1988. 6. Caudill, M., "Neural Network Training Tips and Techniques". AI Expert. Jan., 1991. 7. Gontarski C.A., Mori M., Bonifacio W., "Predig#,o de Comportamento de um Sistema de Tratamento de Efluentes Industriais Utilizando Redes Neuronais". Anais do II Congresso de Engenharia de Processos do Mercosul, Florian6polis, SC, Brasil (cdrom), 1999.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
Incorporating Production Scheduling Wastewater Treatment Plants
871
in the
Optimal
Operation
of
R. Gouveia and J.M. Pinto? Dept. of Chemical Engineering, University of Sgo Paulo, S~o Paulo SP, Brazil, 05508-900 The generation of wastewater is common to every industry, as a result of the production process. Whenever organic compounds must be removed, the most suitable treatment is through the activated sludge system. During the treatment, there is an increase in the mass of microorganisms, which needs to be removed. In multiproduct plants, production tasks affect the wastewater characteristics and therefore may have a significant impact on the sludge discharge. The purpose of this paper is to develop an optimization model to minimize the sludge discharge by incorporating scheduling decisions. Reductions of approximately 10 % are achieved with respect to the optimal dynamic operation of treatment plants. 1.
INTRODUCTION
As a result of the production process, every industry generate wastewater. Apart from the peculiar features of each effluent stream, the most suitable treatment to reduce the organic charges in wastewater is the activated sludge process (Nemerow and Dasgupta, 1991). This system consists of an aeration tank and a settling tank (Jenkins et al., 1993), as seen in fig. 1. TREATED EFFLUENT
[
Sector 1 ]
oz
I ..... !
r-
'
EFFLUENT _
[
1
EQUALIZATION TANK
>
EQUALIZED EFFLUENT
S>
AERATION TANK
o
..
RECYCLE
|
SLUD~ WASTE
:>
Fig. 1. Description of the system The aeration tank treats the effluent stream with an activated mass of microorganisms maintained in suspension and capable of stabilizing the substrate aerobically (Tchobanoglous and Burton, 1991). The soluble and insoluble organic compounds are removed from the effluent stream and converted into a flocculent microbial suspension, which is separated in a settling tank (Eckenfelder Jr., 1989). In many cases, wastewater is generated with variable flowrates and characteristics. In these situations, the wastewater system also requires an equalization system in order to make the operation more balanced. t Author to whom correspondence should be addressed. E-mail: [email protected]. The authors would like to
acknowledge financial support received from FAPESP under grant 99/05575-2.
872 During the operation of the activated sludge system there is an increase in the mass of activated microorganisms (sludge), which needs to be removed. The removal of sludge (dewatering, disposal, etc.) represents generally major investment and operating costs for wastewater treatment plants (Nemerow and Dasgupta, 1991). Gouveia and Pinto (1998) developed an optimization model for the optimal selection of the aeration tank configuration for steady state sludge discharge minimization. A dynamic model that accounts for daily variations in flowrate and concentrations of the effluent stream was developed with the addition of an equalization tank (Gouveia and Pinto, 1999). Scheduling usually considers the allocation of production tasks subject to resource constraints such as: equipment, labor, utilities, etc. In multiproduct plants, production tasks may have a significant impact on the wastewater characteristics since each product may generate effluent at different concentrations and amounts. Therefore, production schedule may have a significant impact on the sludge discharge. The purpose of this paper is to develop an optimization model to minimize the sludge discharge by incorporating production scheduling decisions. The system is composed of production lines as well as the wastewater treatment plant, as shown in Fig. 1. The minimization is obtained with the manipulation of the production schedule, outflow of the equalization system as well as the feed and recycle streams in the aeration tank. 2. OPTIMIZATION MODEL With the same assumptions considered in Gouveia and Pinto (1999) plus the fact that production sequence can be modified, we develop the equations for the scheduling decisions, mass balances around the equalization tank, cells of aeration tank and the settling tank. Moreover, the constraints represent the overall and component balances shown in discrete form. The main variables of the system are shown in Fig. 2, which also shows the interconnections among the several units in the system. Q,., I LINE 1 ' ] 11 q ie s i c
Yio J~,., ;
9
.
X,.,
[ Q,., '
~
SC'~: '
/
- - "
S,.:X"
Q ,a,, T R E A T E D
,/,"'//
' Q''
~,.,X':
"
'
- 7
~.
.
e
.
.
.
//~"/
,'
~ -.
S,:X'"
I X~.,
e n
Q:"
~
atr~ / Q , . . . . , , i ..... Air Supply % . 89 C O .... ....... %_2~ .................
.,r ~
Qk. . . .
, ,. . . .
[ ....... ~ CO,,,, ' ' ,. . . . . -~;. . . . . . . . . . . .
s
EFFLUET,~
ING
UaTli~l'~ ~ [ - - - ~ Q
~S,~,
Q*"
~
TANK
J
~ Q:,.,
Q.... I
Q....
I L
S,., "
Sd., '
S .... RECYCLE gO,,,, '
WASTE
Fig. 2. Model representation 2.1. Production lines
There are j parallel lines, for which a group of products is assigned. This allocation of products to lines is given by the set 1j. The scheduling constraints are as follows:
min{t.m-pi+l} Z
i~_Ij-
Z 0=rnax{t-pt+I,1}
Yi.e < 1
Vj, t
(1)
873 In (1), timing of operations is expressed in a similar form of the backward aggregation constraints developed by Shah et al. (1993). Note that y~,o is a binary variable that denotes if the start time of product i occurs at time 0. Each task lasts p i time intervals; the summation term avoids that the assignment occurs outside the scheduling horizon (m). Each process generates effluent at different volumes and substrate concentrations, as in (2) and (3). min{ pi}
Qj.t =
Z
Z
t~_Ij
0=max{t-pi+l,1}
qi,o "Y,,t-o
Vj,t
(2)
s~,o"y,.,-o
vj, t
(3)
rnin{ pi}
Sj.t
= Z tEI j
Z O=max{t-pi+l,1}
In (2), the amount of effluent generated depends on the product assignment. The parameter corresponds to the amount of effluent generated from process i after 0 intervals. Equation (3) is defined in a similar way.
qi.o
2.2. Equalization Tank As seen in Fig. 2, the j parallel production lines generate effluent that is collected in an equalization tank. Only global and substrate mass balances are considered, since there are no microorganisms neither consumption/transfer of oxygen in the equalization tank.
I
1
IJ~'E.t+l= I/7"E.t- Z Qj,t+l-Qe,t+l .At j=l
At
(4)
Vt
(Sj,t+l-Set+l)
Vt
VTE, t+ 1 j=l
2.3. Aeration Tank
We use generic mass balances in the kth cell (Springer, 1993). Since the first cell does not receive effluent from the previous one, for cell k=-1 terms with k-1 (- 0) must be deleted. Note that the aeration tank is open to the atmosphere and operates in overflow mode; therefore, there is no overall mass variation. Equations (6) to (9) describe global, microorganism, substrate and oxygen balances respectively. k
k-1
k
o=Qke.,+, +Q .... t+l +Qs.t+i-Qs.t+,
(6)
'v'k,t
' ) k - I y~"k -s,t+l 1 ~\ -ks,t+l -- Jfrks.t _}_~~ t . [ Q k t,+ 1 . X e d + l + Q k rec,t+l " X .... t+l + / XSs,t+l --
k k G..,x...~ +
g m "Xs,t+l "S;,t+l
+ Ks
~k
.s,+, 9 k
,k + At
=
(-Os't+l
,%
-v~ -"
k
k
-- k~ "Jfs, t+l
k
v?]
tOk-1 ~k-1
IQ;..~,L.., +Q.... t+l s .... ~+, + >Zs,t+l'Js,t+l
CO~,+~_~
=-
k S:,t+l
k
k
C
k
k
kmaxXs.t+l'~s,t+l
--Qs,t+lSs,t+l +
.,~k-l,,~k-1
j~k. [ Q e , t + l C O e , t+l + Q .... t+l "Orect+l -t- ~::(s,t +l t~"td s,t +l '
k
~-f--~c,k----
Ks + 5s,t+1 k
k
-Qs,t+lCOs,,+l + rof,t+l
"
v~l vk.t k
- rot,t+l
] Vk, t
(8)
(9)
In (9), oxygen supply ro;and consumption rot are described in Gouveia and Pinto (1999).
874 2.4. Settling Tank The settling tank also operates in overflow mode and therefore there is no accumulation term in the global mass balance, equation (10). We do not consider consumption and oxygen transfer. Global, microorganism and substrate balances are written as follows: O=Qsnt+l-(Qsdec, t+l +Qdes, t+l + Q .... t+l)
vt
(10)
At Xrec.t+ 1 : X .... t +-~d [QSn,t+l "Xsn.t+l-(Q,a~,t+, "X~a~,t+, +Qdes.t+lXdes.t+l +Q .... t+l x .... t+l) ]
'~-'.... t+l =S .... t +,-~f-(Q~,t+,ssnt+,--Qsdec,t+lSsdec, t+,-Qdes, t+l "Sdes,t+,-Q .... t+, "S .... t+l) vd
VI
vt
(11)
(12)
The efficiency of substrate removal is represented by: ( 1 - S~,,+,/S .... , ). 100 > E F F z
Vt
(13)
2.5. Additional Mass Balances
The model requires extra additional mass balances for total effluent and recycle streams Qe,,+l-- s k=l
Vt
(14)
Q .... t+l : ,~-, k t+l z...~Q....
~7~t
(15)
k=l
2.6. Objective Function
The total sludge discarded (to be minimized) is represented by the sludge waste plus the involuntary discharge of the sludge present in treated effluent. This is given b y DLT : DLo + s Al/lOOO'(Qsdec, t+l "Xsdec, t+l +Qdes,t+l "Ydes,t+l)
(16)
t:o
2.7. Variable Bounds
Values of MLVSS are bounded to x s . , <- 4000 (k = 1, .. . ,n- 1), since higher values may cause operating problems (Eckenfelder Jr, 1989). The cell that feeds the settling tank (cell n) has an upper bound of x S,t n _<2400 which is the concentration of microorganisms usually found in wastewater treatment systems. Oxygen concentration is maintained above 2.0 mg/1 in the outlet stream of each cell. We established a lower bound of 85% on the BOD removal efficiency, above the 80% required by the environmental legislation. 3. COMPUTATIONAL RESULTS The resulting model is a Mixed-Integer Nonlinear Programming Problem (MINLP), which is solved by the Outer Approximation / Augmented Penalty / Equality Relaxation method. The modeling system GAMS (Brooke et al., 1992) was used to implement the model and the solution is obtained with DICOPT++ (Viswanathan and Grossmann, 1990).
875 Consider a process with one production line (j= 1) with the wastewater generation profile as shown in Fig. 3 as a result of the processing of A-B-C-D in this sequence. This profile is equivalent to the one given in Gouveia and Pinto (1999). The aeration tank is sub-divided in four cells and the optimization horizon comprises 24 hs discretized in one-hour intervals. Parameters of the wastewater treatment system are shown in Table 1. Table 1 - Data for example (units given in the Notation) Vr m K~ Ka 1650
2.5
100
0.05
kmax
Xdes
EFF L
Va
5
50
85
150
A
8
mo _j
C
[_j
~
D
1200 A
B
' ,-
LL L
F, ~L1
[
'
Fig. 3. Wastewater generation
Suppose that each process (A, B, C and D) must be performed once along the horizon and the sequence of production can be accommodated~ The results for the optimal schedule, feed flow rate of equalized effluent (Qke,3, sludge k recycle (Q rec,3, efficiency, effluent substrate removal efficiency (BOD removal) and sludge discharge (Qd~,,3 are shown in Figs. 4 to 6. ~ ~ ! W l A T t
c
~
SUB~.......]-~,i~)[-[! ~
D
1:,
18
~121~t Lq~[ ~1 ~ ~[
24
o
12
....i J'~I~s~ .......... (%)
24
--400 EI}I [
21
o
Fig. 4. Wastewater to the equalization tank
.... (m~/d)
o
12
18
Fig. 5. Efficiency and sludge discarded
z_
0
18
24
0
6
12
18
24
Q
6
1:
!8
24
o
~)
!~,
24
Fig. 6. Main variables of the aeration tank The optimal results for the objective function (DL) and for the plant performance are compared to the optimal results with no scheduling, as shown in Table 2. Case A corresponds to the optimization of the system for a fixed sequence, while case B involves the scheduling of tasks. By incorporating scheduling decisions, the system can reduce the sludge discharge by 9.25% with respect to the case with fixed sequence. Table 2. Optimal Results
4.
Case
Sequence
max{S~dec,t } (mg/l)
mitn{EFFt} (%)
DL (kg/d)
A B
ABCD CABD
9.74 9.77
98.92 99.46
279.01 253.18
CONCLUSIONS
The proposed approach provides significant operational gains in the treatment plant when the scheduling production is considered simultaneously with the distribution of the feed into
876
the aeration tank. This approach is particularly important when the w a s t e w a t e r treatment system is a major resource constraint. Finally, although the approach is illustrated for a simple scheduling p r o b l e m for which no assignment o f tasks to unist is made, it is possible to apply the m e t h o d o l o g y to any discrete time scheduling problem. Nevertheless, combinatorics would b e c o m e a major issue in the solution problem. NOTATION Indexes and sets
i j k t, 0 /j
production task (i ~Ij.) processing line (1 ...... np) cell (1 ...... n) time (1 ..... m) set of tasks assigned to line j
Parameters EFFL lower bound efficiency of BOD removal (%) km~ maximtun rate of substrate utilization per unit
outflow of sludge waste (m3/d) total inflow effluent in aeration tank (m3/d) Qk inflow of effluent in cell k (m3/d) .e,t Qj,t effluent generated in line j at time t (m3/d) Qr~,t total flow of sludge recycle (m3/d) Qk inflow of sludge recycle in cell k (m3/d) . r e e ,t Qdes, t Qe, t
Q~,,
mass of microorganisms (d-~) k roy,, kd endogenous decay coefficient (d -1) /~\. half-velocity constant (mg/1) k rot,t pt processing time of production task i q,,o amount of effluent generated by task i at time 0 Sde~,t s,.o substrate concentration of effluent from task i at 0 Se,t I,,~ volume of settling tank (m 3) Sj;t Srec,t ~J,. volume of aeration tank (m3) k S~,, t,.k volume of cell k ( m3) /Zm maximmn specific growth rate (d -~) Variables
COe,, oxygen concentration of inflow effluent in aeration tank (rag/l) CO .....t oxygen concentration of inflow recycle (mg/1) k c O~,t oxygen concentration in cell k (mg/1) DLz sludge discharge (g/d)
outflow of cell k (m3/d)
Q~d~,t outflow of treated effluent (m3/d)
rate of oxygen supply at cell k (rag/l/d) rate of oxygen consumption at cell k (rag/l/d) BOD of sludge waste (mg/1) BOD of effluent after equalization tank (mg/1) BOD of effluent generated in linej at t (m3/d) biochemical oxygen demand of recycle (mg/l) biochemical oxygen demand in cell k (rag/l)
,%de~,t BOD of treated effluent (mg/l) Vre,t equalization tank volume (m 3) Xe, t MLVSS of treated effluent (rag/l) X~t
MLVSS of cell k (mg/l)
Xrec.t MLVSS of recycle (mg/1) yit binary variable that denotes if task i starts
processing at time t
REFERENCES
Brooke, A.; Kendrick D., Meeraus, A.; GAMS - A User's Guide (release 2. 25), The Scientific Press, 1992. Eckenfelder Jr, W.W.: Industrial Water Pollution Control, McGraw-Hill International Edition, 1989. Gouveia, R.: Pinto, J.M.: Sludge minimization discharge in activated sludge treatment plants, LV LatmIberoamerican Congress ofOp. Res., paper 101, Buenos Aires, 1998. Gouveia, R., Pinto, J.M.; Optimal Operating Policies in Activated Sludge Treatment Plants, Comp. & CThem. Engng. (Suppl.), V23, pp. $853- $856, 1999. Jenkins, D., Richard, M. G.; Daigger, G. T.; Manual on the Causes and Control of Activated Sludge Bulking and Foaming, Lewis Publishers, 1993. Nemerow, N.L.; Dasgupta A.: Industrial and Hazardous Waste Treatment, Van Nostrand Reinhold, 1991. Shah, N.; Pantelides, C.C.; Sargent, R.W.H.; A General Algorithm for Short-Term Scheduling of Batch Operations- II. Computational Issues, Comp. & Chem. Engng., V. 17, n.2, p.229, 1993. Springer, A.M.: Industrial Environmental Control Pulp and Paper, 2nd ed., Tappi, Atlanta, 1993. Tchobanoglous, G.: Burton, F.L.; Metcalf & Eddy - Wastewater Engineering - Treatment Disposal Reuse, McGraw-Hill International Edition, 1991. Viswanathan, J., Grossmann, I. E.; A Combined Penalty Function and Outer Approximation Method for MINLP Optimization, Comp. & (;hem. Engng, V. 14, n~ 7, pp. 769-782, 1990.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
877
A N A L Y S I S OF T H E O P E R A T I O N OF A N S D X P I L O T P L A N T
FOR Cr(VI) RECOVERY ELICECHE Ana. M. (1), CORVALAN Sergio M. (1), ALONSO Ana. I., Inmaculada ORTIZ Universidad de Cantabria. ETSIIyT. Dpto. Quimica. Avd. de los Castros s/n. 39005 Santander. Spain. (1) PLAPIQUI, Dpto. de Qca.e Ing.Qca., Universidad Nacional del Sur- CONICET. Camino La Carrindanga, km 7. 8000 Bahia Blanca. Argentina. Email: [email protected] ABSTRACT This work is focused on the quantitative analysis of the operation of a non-dispersive solvent extraction, NDSX, pilot plant and its application to the removal and concentration of chromium (VI) present in industrial wastewater. NDSX processes belong to the group of emerging clean technologies that accomplish two different objectives simultaneously, i) separation of the pollutant from a wastewaters stream and, ii) concentration-recovery of the valuable components. Thus, the environmental applications of the technology require the use of two membrane separation steps that are connected through an organic leading phase. There are two separation objectives related to the maximum Cr(VI) composition in the effluent treated and a minimum Cr(VI) composition in the concentrated solution, in order to be reused. The main objective of this work is to quantify the sensitivity of these two compositions and the effluent flow treated with respect to the operating variables. This information help to identify the relative incidence of the operating conditions in the performance of the plant. KEYWORDS: NDSX pilot plant, H F membranes, operation, Cr(VI) recovery, wastewater. INTRODUCTION Non-dispersive solvent extraction technology appears as a promising alternative for the treatment of polluted effluents, allowing at the same time the recovery of the valuable components in a simultaneous back-extraction step. Many decisions regarding the selection of the operating mode, module configuration, size of equipment and operating conditions need to be made in order to promote its industrial application. In this work the methodology for the selection of the operating conditions of a pilot plant in operation is developed. A large number of applications of this technology have been mentioned in the literature but there is very little information on the analysis and optimisation of these processes. As a motivating example, the removal and recovery of Cr(VI) from waste waters of a surface treatment plant has been studied. The NDSX process was experimentally analysed by Alonso et a1.(1994) working in a lab-scale. Alonso and Pantelides (1996) simulated the membrane modules in gPROMS. Alonso et al. (1999) presented the mathematical model and corresponding parameters for the pilot plant. Eliceche et al. (2000) formulated the optimisation problem of the pilot plant in the semicontinuous mode. In this work a sensitivity
878 analysis of the effluent flowrate and composition design specifications with respect to the operating variables is carried out. NDSX PILOT PLANT DESCRIPTION An extensive description of the pilot plant can be found in Alonso et al. (1999). The main components of the pilot plant are two membrane modules for the extraction and the stripping processes, as well as two tanks for the organic and the stripping streams. A schematic diagram of the pilot plant with both membrane modules operating in a concurrent mode is shown in figure 1. The organic phase extracts Cr(VI) from the effluent in the extraction module. In the stripping module Cr(VI) is removed from the organic phase and concentrated in the stripping aqueous phase. The organic stream is recycled to the extraction module via the organic tank. The stripping solution is recycled to the stripping tank. Thus, the system is formed by three liquid phases, two aqueous, feed and stripping and one organic phase. The aqueous phases run through the lumen of the hollow fibers and the organic phase flows on the shell side. These phases are contacted at the interface of the porous membranes, where the extraction and stripping reactions between the solute and the organic carrier take place.
Organic Tank
Stripping Module
Extraction Module Feed Solution
Raffinate
Figure 1: Schematic diagram of the semicontinuous NDSX pilot plant At the end of each batch, the stripping solution is replaced by a fresh solution. The organic solution, used as an intermediate carrier, is re-used in different batches and there are no organic effluents. In the semicontinuous mode, the effluent treated runs in a continuous mode. The pilot plant can also run in a batch mode, where a third tank for the extraction phase is added, as analysed and reported by Alonso et al (1999). Alonso et al (1999) proposed a mathematical model for the removal of Cr(VI) using a NDSX process. In this model, it is considered that the main resistance to the transport lies in the microporous membrane impregnated with the organic phase. It is assumed as well that the species are present in equilibrium concentration in the whole interface. Under these assumptions, the behaviour of the modules is described by partial differential equations where the Cr(VI) flux is considered to be proportional to the concentration gradient of Cr(VI)-carrier complex in the membrane. The value of the membrane mass transfer coefficient was calculated by Ortiz et al. (1996) from experimental data obtained on a lab scale and is equal to 2.2 10.8 m/s for the chemical system Cr(VI)-Aliquat 336. The stirred tanks are considered as ideal stirred vessels and are described by total differential equations. The extraction equilibrium expression is given by the algebraic equation reported by Alonso et al. (1997) where an ideal behaviour is assumed for the aqueous phase while a non-ideal behaviour of the
879 organic phase is taken into account in the equilibrium expression. The stripping equilibrium expression is described by a simpler algebraic equation suggested by Ortiz et al. (1996). In this case, the chemical equilibrium is described by a distribution coefficient defined as the ratio of the concentrations in the aqueous and organic phases based on the fact that the chlorine concentration in the stripping solution is always high in order, to favour the stripping process. The dynamic response of the pilot plant is represented by a system of differential and algebraic (DAE) equations describing the mass transport through the membrane modules and stirred tanks and can be found in Alonso et al. (1999). ANALYSIS OF THE OPERATING CONDITIONS The operating conditions of the NDSX pilot plant are selected solving the following optimisation problem, as formulated by Eliceche et al (2000): max
F~(u(t),v, tl )
u(t), v, t f
f (x (t), x (t), u (t), v) = 0
t ~ [O,t f ]
(P1)
](x(O),u(O),v) = o U(t o) : U(t I )
t min < t < t~ a~ f
--
_
V min ~_~ V ~_~ ]2 max
U min ~- uCt) < u m~x
t
E [O,tf ]
W mi" <_ W@)<__ W m~x
The objective function is to maximise the wastewater flowrate Fe, which is a function of the time horizon ts the time invariant parameters v and the control variables u(t). The pilot plant model is represented by the set of differential and algebraic equations f and the initial conditions /, where x(t) are the time derivatives of the variable x(t). The system of differential and algebraic equations describing the mass transport through the membrane modules and stirred tanks can be found in Alonso et al (1999). There are lower and upper bounds on the end point constraints w(t I), time invariant parameters and control variables. The aim of the process is to remove Cr(VI) from waste water and to concentrate it in the stripping solution for re-use. Therefore the maximum allowed wastewater Cr(VI) concentration (Ce,outlet) for disposal, 9.61 x 10.3 mol/m 3, and the minimum Cr(VI) concentration of the stripping stream required for re-use, 76 mol/m 3, are posed as inequality end point constraints w(t~. The same organic stream is used in different batches. Thus, to keep constant the initial Cr(VI) composition in the organic phase (Co,initial)an extra equality constraint on this control variable is assumed, imposing that the initial u(O) and final u(tl) Cr(VI) composition in the organic phase should be the same. The operating variables considered are the following time invariant parameters: volume of the organic and stripping tanks, organic and stripping flow rates. The initial Cr(VI) composition in the organic phase is also included as an optimisation variable in problem P 1.
880 The operating conditions were selected for a nominal Cr(VI) composition in wastewater equal to 1.234 mol/m 3. The objective function selected is the maximisation of the flow rate treated in the pilot plant. Problem P1 was formulated and solved with the code gOPT, the optimisation tool of gPROMS (1997). The main results from an arbitrary initial point, are shown in Table 1, where lower and upper bounds on flow rates and volumes correspond to available capacities of the pilot plant. The stripping solution reaches at the end of the batch time, the required concentration of 76 mol/m 3 of Cr(VI), being an active end point constraint. The Cr(VI) concentration in the raffinate, Ce, outlet, is equal to the upper bound of 9.61x10 -3 mol/m 3 required for disposal, which is also an active end point constraint at the solution point. The Cr(VI) concentration in the organic phase varies very little with time, recovering its initial value at the end of the batch. Table 1" Optimal operating conditions, solution of problem P 1.
Co,initial
Fe V, Vo Fo
F,
mol/m 3 m3/h m3 m3
m3/h m3/h
Batch time h
Initial point 65 0.0605 0.04 0.04 0.1 0.1 48
Solution point 81.8498 0.08767 0.0216 0.09 0.18 0.2 15.96
Lower bound 0.02 0.0216 0.012 2 X 10.2 2 x 10.2 1
Upper bound 300 0.4 0.06 0.09 0.18 0.2 107
A deeper insight in the plant behaviour requires a sensitivity analysis of the process objective with respect to the operating conditions. Changes in the objective function and end point constraints due to perturbations in the optimisation variables between the lower (LB) and upper (UB) bounds, except for the initial organic composition, are shown in Table 2. Percentage of changes of the objective function and endpoint constraints with respect to their optimum values is reported in Table 2. The percentage values clearly indicate the incidence that each operating variable has on the effluent flow, extraction and stripping compositions. The most important elements are indicated in bold numbers. The operating variables with the strongest influence in the effluent flow and stripping concentration are the initial organic composition and stripping volume. Next, the organic flow and organic volume exert a moderate influence and finally the striping flowrate has a very small influence in the three variables studied. For the raffinate composition the most important variables are the initial organic composition and organic flowrate, while the other operating variables have a very small influence.
881 Table 2 - Objective function and end point constraints versus optimisation variables
AGo,initial
AFo AVo AFs
Increments (UB- LB) 56.85 (81.85-25) 0.0384 0.16 0.0780 0.18
AF~
%
ACe,outlet
%
80.61
0.06106
69.65
0.007747
0.009000 -0.003073 0.002338 0.000688
10.27 -3.51 2.67 0.70
0.0000125 - 0.001037 0.0000050 0.00000082
mc s
%
52.7871
69.46
0.13 - 45.0202 10.79 - 2.1275 0.05 2.0595 0.01 0.6946
- 59.24 -2.80 2.71 0.91
Initial Cr(VI) composition in the organic phase This is the operating variable with the strongest influence on the effluent flowrate, extraction and stripping concentrations, with changes in the order of 70 to 80 %. They are monotonic increasing functions of Co,initial" The optimum value lies near the chemical equilibrium composition, which is 82.44 mol/m 3, an upper bound although it has not been explicitly formulated in problem P1. This is the maximum Cr concentration in the organic phase of the extraction module, that allows an outlet concentration lower than 9.61e-3 mol/m 3, when the inlet Cr(VI) concentration is 1.234 mol/m 3 . For this reason the range studied is from 25 to 81.8498, because at higher values of Co,initiaI the constraint on the effluent maximum composition is not satisfied. The stripping composition, in this interval, is lower than the minimum required value becoming feasible at 81.8498. Volume of the stripping tank The stripping concentration bs very sensitive and monotonic decreasing with respect to the stripping volume, 59% change. Thus the stripping volume lies at the lower bound, to facilitate the concentration of Cr(VI) in the stripping solution. Less stripping volume also implies less batch time. The effluent flow and the extraction concentration are monotonic increasing functions with respect to the stripping volume. The effluent flow is quite sensitive to the stripping volume with a change of 10 %. Flow rate of the organic phase The organic flow rate has an important influence on the raffinate concentration, 11% change, together with the initial organic composition. The effluent flow and the two end point constraints Ce, outlotand Cs are monotonic decreasing functions with respect to the organic flowrate Fo. Therefore to reduce the Cr(VI) raffinate concentration, the flow rate of the organic phase is equal to the upper bound. For lower values of Fo than the upper bound, the raffinate concentration is bigger than the maximum allowed concentration for disposal leading to and unfeasible operation. Volume of the organic tank The objective function F e and the two end point constraints are monotonic increasing functions with respect to the organic volume. The organic volume reaches the upper bound to maximise the effluent flow treated and the concentration of the stripping solution which are
882 sensitive to this variable, 3% change, while the raffinate that is not sensitive to this variable (0.05% change) satisfy the disposal requirements at the solution point. Flow rate of the stripping pha.se The strippingflowrate is the variable with the smallest influence in the effluent flow rate, stripping and extraction compositions, less than 1% change. The objective function F e and the two end point constraints are monotonic increasing function with respect to the stripping flowrate Fs. The sensitivity of the raffinate concentration, with a change of 0.01% in the interval studied, is two orders of magnitude smaller than effluent flow and stripping concentration sensitivity to F s. Thus Fs is equal to the upper bound to maximise the effluent flow treated and the concentration of the stripping solution. CONCLUSIONS
Insight on the dynamic behaviour of the pilot plant has been gained through the sensitivity analysis. A proper selection of the operating conditions leads to significant improvements in the performance of the plant. The numerical results quantify the potential that the methodology presented has in identifying the main operating conditions of nondispersive solvent extraction processes The removal and recovery of Cr(VI) has been studied as an application of the NDSX process, although other effluents have been treated successfully in the pilot plant. Working in semicontinuous mode with the simultaneous extraction and backextraction steps, the sensitivity analysis led to the following conclusions: i) the operating variables with the strongest influence in the effluent flow and stripping concentration are the initial organic composition and stripping volume; ii) the organic flow and organic volume exert a moderate influence, iii) the striping flowrate has a very small influence in the three variables studied, and iv) for the raffinate composition the most important variables are the initial organic composition and organic flowrate, while the other operating variables have a very small influence. REFERENCES Alonso, A.I., Urtiaga, A.M., Irabien, A., Ortiz 1.(1994). Extraction of Cr(VI) with Aliquat 336 in hollow fiber contactors: mass transfer analysis and modelling, Chem. Engng. Sci., 49, 901. Alonso, A.I. and Pantelides C.C.(1996). Modelling and simulation of integrated membrane processes for recovery of Cr(VI) with Aliquat 336, J. Membr. Sci., 110, 151. 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, 1997. Alonso, A.I., Galan, B., Gonz~lez, M., Ortiz, I.(1999) Experimental and Theoretical Analysis of a NDSX pilot plant for the removal of Cr(VI) from Galvanic Process waste waters, Ind. Eng. Chem. Res., p 1666, Vol 38, No.4. Eliceche A., Alonso A. and I. Ortiz, Optimal Operation of membrane separation processes for wastewater treatment, Comp. and Chem. Engng., in press, 2000. gPROMS Technical Document-The gOPT Dynamic Optimisation Tool, (1997). Process Systems Enterprise Ltd. Ortiz, I., Galan, B., Irabien, A.(1996). Membrane mass transport coefficient for the recovery of Cr(VI) in hollow fibber extraction and stripping modules, J. Membr. Sci., 31, 46.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
883
Optimum deNOx performance using inferential feedforward reductant flow control H.C. Krijnsen, J.C.M. van Leeuwen, R. Bakker, H.P.A. Calis and C.M. van den Bleek DelftChemTech, Faculty of Applied Sciences, Delft University of Technology, Julianalaan 136, 2628 BL Delft, the Netherlands To adequately control the reductant flow for the catalytic removal of NOx from diesel exhaust gases a tool is required that is capable of accurately and quickly predicting the engine's NOx emissions based on its operating variables, and that is also capable of predicting the optimum ammonia/NOx ratio for NOx removal. Two algorithms for non-linear modelling are evaluated: (1) neural networks and (2) the split & fit algorithm of Bakker et al. [ 1,2]. Measurements were carried out on a semi-stationary diesel engine. Results of the split & fit algorithm and the neural network were compared to (3) the traditionally used engine map and (4) a linear fit. Both the neural network and the split & fit algorithm gave excellent NOx predictions with a short computation time (0.3 ms), making them very promising tools in real-time automotive NOx emission control. With regard to the estimation of the optimum NH3/NOx ratio, the neural network predicts the effect of NH3/NOx ratio on the final NO2 emission very well. 1. INTRODUCTION One of the approaches for NOx emission abatement is selective catalytic reduction (SCR), which involves adding a reductant to the exhaust gas to catalytically remove NOx. The reductant flow rate that has to be injected into the exhaust gas depends on the NOx concentration and the exhaust flow rate, the required NOx reduction and the catalyst conditions. On-line NOx analysis equipment is not only very expensive but is also susceptible to soot plugging and has to be frequently calibrated and serviced. Therefore, an alternative for NOx concentration measurement is needed. Furthermore, the optimum reductant/NOx ratio has to be determined in order to keep the slip of both reductant and NOx through the system as low as possible. This could be done by modelling the catalyst system. However, almost all models describing the reaction between ammonia and NO over vanadia/titania monolithic catalysts 3-9 were only evaluated for a small temperature range (typicallly a range of 100~ and for simulated exhaust gases, containing only nitrogen, water, NO and ammonia. The effect of the SO2 concentration was only taken into account by Tronconi 6-9, while the effects of soot and hydrocarbons were not taken into account so far, neither in the experiments nor in the models. Furthermore, the effect of ammonia oxidation was neglected in the models even though ammonia oxidation is significant at temperatures above 350~ Only one author 4 considered a range of NO concentrations within model and experiments. The result is that for all other models, the reaction order in NO is not verified within the model and therefore the model may show discrepancies when other NO concentrations are considered. Earlier experiments have shown that the NOx emission from our engine is a function of the intake air temperature, the intake air pressure, intake air humidity and the engine load, which is in agreement with literature 1~
884 Air ..I The aim of this paper is to .I9 DeNOx I i Fuel~ Di, sel show that it is possible to Catalyst Cleaned ! "7 En(.line predict both the NOx emission I Exhaust Gas" i s i and the optimum reductant 1 ! i Optimum /NOx ratio based on the diesel Reductant !............................ "1~' ReductantJ engine's and catalyst' s Engine and Tank NOx ratio Intake Air operating variables. For this soEstimation ! Conditions ! ! i called 'soft-sensoring' or T I i Inferential inferen ti al measure men t Dosing ~ .........j NOx application, a non-linear blackControl Estimation box modelling algorithm is used that takes a large set of Figure 1" Schematic representation of the inferential feedforward NOv control svstem measurements to learn how to predict the NOx emissions based on the operating variables. Two candidates are evaluated in this paper: 1) An artificial neural network (ANN). 2) The split & fit algorithm of Bakker et al. [1]. This split & fit algorithm (s&f) splits the input data into a number of disjoint regions, to each of which its own local linear model is assigned. The control system is given in Figure 1. These two black-box modelling algorithms are compared to an engine map based on engine load only and alternatively to a linear fit through the measurement data.
2. E X P E R I M E N T A L SETUP The engine used for the experiments was a Lister-Petter LPW3, a three cylinder water cooled diesel engine fitted in a Wilson LD 12.5/W4 generator with a rated power of 9.9 kW and a constant engine speed of 1500 rpm. The fuel used during all the experiments was a summer quality diesel fuel containing 0.04wt% sulfur. Downstream of the diesel engine a Coming EX80 wall flow monolith was located in order to filter off the soot before deNOx experiments were performed in the deNOx reactor. The reactor contained a 1.05 litre monolithic catalyst. The catalyst considered is a fully extruded V2Os-WO3-TiO2 catalyst (Frauenthal). In order to keep the inlet conditions of the combustion air constant during the experiments, dried air was used for deNOx experiments; to assess the influence of intake air temperature and intake air humidity, these variables were varied by injecting steam into the intake air and heating the intake air. Part of the exhaust gas of the engine was used for deNOx experiments, the rest of the exhaust gas was led to a vent so the exhaust flow rate through the deNOx system could be varied independently from the engine load. Sample gas streams to be analysed by a chemiluminescence NOx analyser (Signal), were led through washing bottles containing 85wt% phosphoric acid and 35wt% sulfuric acid to remove water and ammonia. In addition, a NOx sensor was located directly downstream of the engine. The ammonia concentration in the exhaust gas was measured by a microwave NH3 analyser (Siemens). The four input variables (intake air temperature, pressure and humidity and engine load) as well as the response variable (exhaust NOx concentration, measured close to the engine) were logged by a data acquisition system running on a PC. For reductant dosage
885 Table 1 Prediction errors of the black-box algorithms considered Off-line error Black box model Training % ANN 2.8 s&f 2.7 Engine map 8.1 Linear fit 6.0 a average of three tests
Test % 3.4 3.2 6.3 6.0
Real-time error %a 4.5 4.9 9.4 4.8
control experiments the four input variables were made input to a 486 PC on which the dosing control program ran. 3. RESULTS 3.1. NOx emission prediction - off-line
13 hours of data with a frequency of 1 Hz were used for training and testing the fit algorithms. Two third of the data was used for training the algorithms and one third for validating the fit algorithms. The variables that affect the NOx emission of the LPW3 engine are engine load, intake air temperature, intake air pressure and intake air humidity. These variables are taken into account in the fit algorithms. Based on cross-correlation data, the initial time delays of the variables were estimated. The time delays were further optimised by trial and error within the split and fit algorithm. The optimum set of input variables for the black-box models was found to be (a) intake the air temperature, (b) intake air pressure and (c) intake air humidity, all three at the current time only and (d) the engine load at current time but also 2, 5, 8, 11 and 14 seconds in the past. The prediction errors of the training and test set are given in table 1. The neural network consists of one hidden layer containing 9 nodes. The s&f model comprises 19 linear clusters. 3.2. NOx emission prediction - real-time
After black-box modelling, all tools were applied in a real feedforward control algorithm. To determine the real-time performance of the models, measurements were performed on the LPW3 engine and the commercial vanadia catalyst at low gas velocities at the optimum conversion temperature of the catalyst, using ammonia as reducing agent. In order to estimate deviations between NOx emission prediction and actual
Figure 2: Real-time NOx emission prediction results using the s&f model at 300~ 5 mn3/h, NH3/NOx ratio of 0.7, relative humidity of 50%, intake air temperature of 20~
886 180 150
t.O D o3
E O z
120
120
I
-~--NO2 from NH3 --o-NO2 from NOx NO2 total
--.- 250oC 300~ ~ 350oc ~ 400oC -u- 450oc -B- 500oc
100 _l J:
I I
90
Minimum acidification
60
effeit
~. 9 80-
~.~.
tO
6o-
E 40
z 30
20
0 0.30
0 0.50
0.70
0.90
NH3/NOx ratio [-]
1.10
0
0.5
1
1.5
NHa/NOx ratio [-]
Fig 3: Acidification effect of the NOx and NH3 Fig 4: Final NO2 emission as a function of reaction as a function of NH3/NOx ratio, the NH3/NOx ratio for at 250-500~ at 260~ 5 mn3/h, 1600 ppm NOx 5 mn3/h, 1000 ppm NOx NOx emission, the temperature of the catalyst was kept at 573K and the NHflNOx ratio was kept at 0.7 and was not yet optimised. The gas hourly space velocity was kept at 4,700 h l. The prediction error was estimated by measuring the NOx concentration directly at the engine and comparing this to the NOx emission prediction. Three tests were performed at several humidities and intake air temperatures, while varying the engine load. Results for the real-time performance of the neural network are given in figure 2. The averaged absolute error estimated by comparison of real-time NOx prediction and NOn measurement are given in table 1. The reason for the linear fit model and ANN tool to have about the same error is that the relationship between engine load and NOx emission is linear up to about 7 kW (see also figure 2) while the highest engine load used during the tests was 8 kW, in order to prevent problems due to soot plugging the equipment. As soon as the humidity and temperature vary over a test the linear fit performance is expected to decline significantly.
3.3. Optimum NH~/NOx ratio estimation To minimise the emission of pollutants (NOx as well as reductant), the reductant/NOx ratio resulting in the lowest impact on the environment should be estimated and used for real-time reductant control. For that reason, the inferential NOx estimation method determines the NOn emission from the engine while another estimation method, based on the catalyst and exhaust conditions, estimates the optimum reductant~Ox ratio (See figure 3). The acidification of the environment is due to NOn that is rapidly converted to NO2 at the vent, and to NH3 that eventually is converted to NO2. The sum of these NOn and NH3 emissions is thus a measure of the acidification effect. The NH3/NOx ratio at the minimum acidification effect is the optimum NH3/NOx ratio and is a function of the NOx concentration, the gas velocity and the catalyst temperature. The effect of the catalyst temperature is displayed in figure 4. Likewise, the optimum ratio is also a function of the NOn concentration. Measurement data as shown in figure 4 are used for training (99 data points) and testing (41 data points) a neural network and
887 160
9target 500 ppm A test 9target 1000 ppm o test 9target 1500 ppm D test
~ ' 120 133 tO f.O ~
"1
80
E 0 z
40
o 9
0.4
9
O
[]
O 0
i
i
0.7
1
[]
1.3
NH3/NOx ratio [-]
Figure 5" NO2 emission prediction by the ANN for estimation of the optimum NH3/NOx ratio at 250~ 5 mn3/h
a s&f algorithm. The average absolute errors of the training set are 2.4% and 1.6% for the ANN (1 hidden layer, 10 nodes) and s&f algorithm (24 clusters) respectively, while the average absolute errors of the test set are 12 and 8.6% for the ANN and s&f algorithm respectively. The large error of the test set is due to the fact that data around the minimum NO2 emission was used for testing the algorithms. Even though the trend was well predicted by both algorithms for both the training and test data, the location of the minimum NO2 emission predicted by the s&f proved to be incorrect for an evaluation data set. Prediction results of the ANN are shown in figure 5.
4. DISCUSSION Off-line NOx emission prediction by using the s&f and ANN tool showed very good agreement between prediction and measurement. Preliminary real-time test results performed somewhat less. The difference between off-line and real-time results may be ascribed to the time dependency of the engine and deNOx equipment. The engine map performs much worse than the ANN, s&f and linear fit. With respect to training time, the linear fit and the s&f algorithm perform best. Both blackbox models need in the order of one hour to be optimised as soon it is known which input variables should be considered for NOx emission and with what time delay. The neural network may take up to several days on a Pentium PC to be optimised. The time for real-time calculation of one NOx emission prediction, once a black-box model has been optimised, is in the order of 0.3 ms. The optimal NH3/NOx ratio region flattens at increasing catalyst temperature due to the oxidation of ammonia: part of the ammonia fed to the catalyst is no longer used for converting the NOx but is itself converted to either NOx or N2. As a result, the optimum shifts to higher NH3/NOx ratios while the ammonia slip lowers at overstoichiometric ammonia dosing due to ammonia oxidation. The result is a broadening of and an upward shift in the optimum NHJNOx ratio range. Training the NO2 emission as a function of gas velocity, catalyst temperature, NOx emission and NH3/NOx ratio showed that the ANN, unlike the s&f algorithm, is very capable in finding the optimum NO2 emission. In case the ammonia emission is considered more harmful than the emission of NOx, one may decide to weigh the emissions in another way and the catalyst measurement results already gathered can be used. To do so, no additional measurements have to be performed.
888 5. CONCLUSIONS ANNs and the s&f algorithm can be used to create models that are able to predict the NOx emission from a diesel engine with sufficient accuracy and to estimate the optimum reductant/NOx ratio based on easily measured input data. The calculation time (up to 0.3 ms) that is needed for the prediction of the NOx concentration and optimum reductant/NOx ratio is short enough to allow for real-time control. Together with the high accuracy of the predictions, this makes the concept of using one of the algorithms for real-time automotive NOx abatement control very promising. Applying an engine map gives significantly larger errors in NOx emission prediction. With regard to the estimation of the optimum NH3/NOx ratio, the ANN predicts the effect of NH3/NOx ratio on the final NO2 emission very well. The NOx emission prediction and optimum NH3/NOx ratio prediction should be combined in order to get the most beneficial impact on the environment. FUTURE WORK More measurements for testing the NOx emission have to be performed. After these tests and possibly further improvements in the control system, the prediction of the optimal NH3/NOx ratio will be integrated with the NOx emission prediction and tested real-time. ACKNOWLEDGEMENTS The authors would like to thank Sander Baltussen for the implementation of the described fit algorithms and Frauenthal (Frauental, Austria) for supplying the vanadia deNOx catalyst. REFERENCES [1] Bakker, R., Takens, F., Schouten, J.C., Giles, C.L., Coppens, M.-O., Takens, F., van den Bleek, C.M., Advances in Neural Information Processing Systems 12; Solla, S.A., Leen, T.K., Mtiller, K.-R. Eds.; MIT Press, Five Cambridge Center, Cambridge, MA, (2000). [2] Bakker, R., Schouten, J.C., Giles, C.L., Takens, F., van den Bleek, C.M., submitted (2000) [3] Buzanowski, M.A. and Yang, R.T., Ind. Eng. Chem. Res. 29 (1990) 2074. [4] Pinoy, L. J. and Hosten, L. H., Catal. Today 17, pp. 151-158 (1993) [5] Svachula, J., Ferlazzo, N., Forzatti, P., Tronconi, E., and Bregani, F., Ind. Eng. Chem. Res. 32 (1993) 1053. [6] Tronconi, E., Cavanna, A., and Forzatti, P., Ind. Eng. Chem. Res. 37 (1998) 2341. [7] Tronconi, E., Forzatti, P., Gomex Martin, J.P. and Mallogi, S., Chem. Eng. Sci. 47 (1992) 2401. [8] Tronconi, E., Catal. Today 34, (1997) 421. [9] Tronconi, E., Lietti, L., Forzatti, P., and Mallogi, S., Chem. Eng. Sci. 51 (1996) 2965. [10] Lin, C.-Y. and Y.-L. Jeng, J. Ship Res., 40 (2), (1996) 172. [11] Lin, C.-Y., Y.-L. Jeng, C.-S. Wu and K.-J. Wu, J. Environ. Sci. Health 31 (4) (1996) 765. [12] Boot, P, Int. Congr. Combust. Engines, Proc. 20th, London, May 1993, CIMAC, G6teborg (1993) D67 1. [13] Kondoh, H., T. Kawano and K. Masuda, Int. Congr. Combust. Engines, Proc. Conf. 1998, CIMAC, Copenhagen (1998) 803. [14] Juva, A., A. Rautiola, P. Saikkonen and D. Le Breton, Fuels-gas oils, 16 (2) (1996) 1. [15] Dodge, L.G., D.M. Leone, D.W. Naegeli, D.W. Dickey, and K.R. Swenson, SAE P962060 (1996). [16] Rakopoulos, C. D. and D.T. Hountalas, SAE P-981021 (1998).
European Symposiumon Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
889
Software tool for waste treatment selection using economic and ecological assessments Laurent Cavin, Olivier Jankowitsch, Ulrich Fischer*, and Konrad Hungerbtihler Safety and Environmental Technology Group, Laboratory of Technical Chemistry, ETH Zurich, CH 8092 Zurich, Switzerland {cavin, janko, ufischer, hungerb } @tech.chem.ethz.ch
Abstract For two reasons waste emanating from chemical production is of particular importance for chemical companies. First, waste treatment cost significantly contribute to the total production cost, and second, increasing environmental awareness urges the companies to reduce waste streams and emissions from treating them, thus reducing environmental impact. We developed a software tool that, taking an existing chemical plant as an example, for a given waste stream calculates the cheapest feasible combination of treatment operations that satisfies legal emission limits. A special feature of the software is that uncertainty in waste stream composition and separation efficiencies (e.g. in early process development phases) can be easily propagated through the model leading to probability distributions of treatment cost and selected treatment paths. This computer tool is now being extended for assessing the environmental impact of chemical waste treatment. As indicators for this assessment the method of environmental scarcity as well as eleven indicators selected by chemical industry as part of its responsible care program are used. To demonstrate its capabilities the model was applied to two made-up waste streams. The results show that rather small variations in one or several input parameters might result in pronounced differences in treatment cost because certain treatment options are unfeasible and emission limits are exceeded rendering a scenario legally non-compliant. Furthermore, possible conflicts between cost and environmental impact are demonstrated, and the major contributions with regard to environmental impact of the different steps of a treatment path are highlighted. In summary, the model supports the recognition of technical, legal, financial, and environmental problems in chemical waste treatment already at early phases of process design. Introduction In fine chemical industry, in particular by spent solvents in pharmaceutical production, large amounts of waste are produced per mass of product. Due to waste disposal legislation and emission laws many expensive treatment steps might be required before allowing the release of such streams into the environment (atmosphere, rivers, deposits). The investigation of case studies from fine chemical industry revealed that waste treatment cost can amount for 10-30% or even more of total production cost [Schlegel, 1993]. But much higher cost might arise when treatment of a given waste stream is considered feasible during early phases of process development while later on it is recognized that no capacity or technology is available for treatment. As a consequence it is important to check whether emissions from waste treatment would be legally compliant and to obtain reliable estimates of the corresponding treatment cost already at early stages of chemical process development in order to evaluate the economic potential of new products or processes. As the relative contribution of waste treatment cost varies significantly from project to project even within one company [Dimmer, 1999] they should be estimated specifically for each project. Furthermore, because at early phases many parameters are still uncertain this uncertainty should be considered in the estimation of waste treatment cost.
Petrides et al. [1995] presented a research prototype, EnviroCAD, that supports the design of new waste treatment processes. Linninger and Chakroborty [I999] presented a methodology for
890
Figure 1: Concept of the software tool calculating cost and environmental impact of chemical waste treatment under uncertainty.
pharmaceutical waste management through computer-aided synthesis of treatment policies. The method identifies treatment tasks through examination of waste properties against a set of environmental targets. We present a software tool that automatically calculates the cost of all possible treatment scenarios that are available at a given plant and that are legally compliant for a given waste stream. In addition to cost, also environmental indicators are calculated. The indicators selected are the method of environmental scarcity [BUWAL, 1998] as well as eleven indicators selected by chemical industry as part of its responsible care program [CEFIC, 1998]. Because changes in legislation might have direct (e.g. introduction of CO2-taxes) or indirect (e.g. additional treatment cost) economic implications, this perspective of integrated process design might improve the longterm profitability of a process. In this approach uncertainty can be considered by assigning any type of probability distribution to any of the input parameters. A Monte Carlo approach is used for uncertainty propagation. As a consequence the results are not single values but also probability distributions representing the information needed for strategic business decisions. In addition, the most sensitive parameters are highlighted. Research effort can then be concentrated on determining these parameters with high precision. To demonstrate its capabilities with respect to economic and ecological assessment, the model has been applied to two made-up waste streams considering uncertainty in stream volume and composition.
Model Description The concept of the computer tool presented here is shown in Figure 1. The aim of the model is to determine the best possible treatment path from either an economic or ecological point of view for a chemical waste stream with uncertain volume or composition. The uncertainty specified might be the result of considering uncertainty in modeling the chemical process itself [e.g. Dimmer, 1999], or for a given waste stream to be treated it might be stipulated on the basis of heuristic knowledge. In two steps technically infeasible and legally non-compliant treatment paths are eliminated. For all remaining possibilities the treatment cost as well environmental impact is
891
CWastestream) C" Liquid ~
~
I
(
)
.......
% Figure 2" Treatment operations and decision structure considered in the model. The specific operations are stripping (mainly halogenated compounds), NH3-recovery, heavy metal precipitation, and decontamination.
calculated. The aspects considered in the environmental assessment are energy and utility demand, emissions, and the mass to be deposited. The results from the economic and ecological assessment can be used in several ways: - to identify the cheapest or the environmentally most benign treatment option for nondistributed input parameters - to calculate the probabilities that certain treatment paths would be the cheapest or the environmentally most benign when distributed input parameters are stipulated - to calculate probability distributions for either minimal cost or eco-units - to display a pareto plot comparing the values of both objectives for one treatment path - to identify the most sensitive parameters for each path The treatment operations considered in the model represent the waste treatment facilities of the Valais Works of Lonza Group at Visp (CH) [Righetti, 1990]. Figure 2 summarizes the most important treatment operations and the decision structure as defined in the model. The complete decision structure and a detailed description of the cost calculation is given by Cavin [1998], while the ecological assessment is outlined by Jankowitsch [2000]. The model was implemented in MATLAB so that the Monte Carlo simulation could be realized by vector and matrix operations. Further details on the implementation of the model are given by Cavin [1998] and Jankowitsch [2000].
Results
and
Discussion
To demonstrate the capabilities of the model with regard to economic and ecological assessment the results obtained for two made-up waste streams are discussed in this paper. The first stream WS1 consists of water and varying amounts of 1,1-dichloroethane and 1,1,2,2-tetrachloroethane. For these two compounds uniform distributions of their amount were assumed. Furthermore, it was assumed that only 1,1,2,2-tetrachloroethane can be recycled with a benefit. For the amounts stipulated, dichloroethane and tetrachloroethane were assumed to be completely soluble and insoluble in water, respectively. The stipulated uncertainties in stream composition and separation efficiencies reflect empirical knowledge.
892
These assumptions resulted in the following three most economic treatment paths: Path #1: Tetrachloroethane is recycled by rectification and the remainder is burned. Path #2: Tetrachloroethane is recycled by phase separation and the remainder is burned. Path #3: Tetrachloroethane is recycled by phase separation. The aqueous phase is stripped to remove dichloroethane, and the remainder is sent to the sewage treatment plant. In Figure 3 the results obtained for each 1000 Monte Carlo runs are shown when only either path #1, #2, or #3 were considered feasible, or when the cheapest of the three treatment paths was selected for a given set of input parameters. In the latter case 872 times path #3 was the cheapest, while path #1 and #2 were selected only 38 and 90 times, respectively, when path #3 could not fulfil the chlorine limit specified for waste water input to the sewage treatment plant. The resulting mean cost are 342, 340, and 188 CHF for path #1, #2, and #3, respectively, and 209 CHF when each time the cheapest path is selected. The sensitivity analysis for this problem revealed that in addition to the amounts of chlorinated compounds the stripping efficiency had a major influence on the cost because a low stripping efficiency in combination with high amounts of chlorinated solvents renders the treatment in the sewage plant (path #3) infeasible. In summary, the results obtained for stream WS 1 show that for a certain constellation rather small variations in one or several input parameters might result in pronounced differences in treatment cost because a distinct treatment option is rendered technically or legally infeasible. However, the resulting probability distribution indicates that it is rather unlikely that path #3 cannot be applied for treating this waste stream. Waste stream WS2 consists of water, toluene, a small amount of chloromethane, and a minor amount of ammoniumsulfate. Figure 4 depicts the results obtained for the ecological as well as the economic assessment of treating this stream. The different treatment paths are ranked according to their environmental impact measured using the environmental scarcity method [BUWAL, 1998]. The on average environmentally most benign path #4 is not the cheapest one (the latter being path #21). Thus, a multi-objective situation arises as presented in Figure 5. In this plot the uncertainties on the economic and ecological assessments are indicated as ranges of cost and eco-points as obtained from the Monte Carlo simulation. The three paths #1, #17, and #19 can be eliminated from further evaluation because these are clearly worse with respect to both objectives when compared to the other paths, even when the underlying uncertainty is considered. In principle, path #2 cannot be eliminated because it might be possible that this path is
4S
,
35
,
,
J
Cheapest Treatment Paths
I,
8o
I ~o
,
7o
~o
Cost ,
~o
[CHF/stream]
~......
,
,
Path #1
:~1~
15o
i
2~o
2~o
Cost
3~. .' . .
'
[CHF/strearn]
3
Path #3
Path #2 iI
1 ~
~Lo
Cost
4o
[CHF/stream]
"
~o
,oo
~5o
I
! 2oo
2~o Cost
~o
3Lo
[CHF/stream]
Figure 3: Treatment cost calculated for waste stream WS1 for each 1000 Monte Carlo runs considering either the cheapest feasible treatment path or only path #1, #2, or #3, respectively.
893 + . . . . .
+
+
I
Ranking:
+ ..... +
[Ecoi Paths flags] ] [ - ] I [index + ..... + I i t [ i
i 2 3 4 5
+ . . . . .
I 1 I i i
~
4 2 23 21 1
- 00110110 - 00110010 - 01011100 - 01011000 - 00110000
~
5 Cleanest +
Paths
I Eco Impact I [UBP/stream]
+
1 i i ] [ ~
2.360e+004 4.017e+004 4.534e+004 5.323e+004 1.508e+005
] mean price i [CHF/stream]
+
1 1 i 1 [
+
1007.0 1015.0 638.7 631.7 1033.0
llegal i #
~. . . . .
II000 Ii000 11000 i 447 ]i000
+
Figure 4: Ecological assessment (mean values) using the environmental scarcity method (UBP = Umwelt-belastungspunkte - environmental impact points) as well as cost obtained for 1000 Monte-Carlo runs and different paths for treating waste stream WS2.
environmentally more benign than path #4 for some combination of input parameters. Nevertheless, when neglecting path #2, three paths (#4, #21, and #23) remain. Their mean values are not dominated by other points with regard to both objectives. The difference in environmental impact between path #4 and paths #21 and #23 is rather small, while the difference in cost between the former path and the latter two is large. Therefore, probably path #4 would not be considered in future planning. Between the other two paths a trade-off situation exists. Path #21 might be considered because it is the cheapest and its environmental impact is only slighty higher than that of path #23. On the other hand, Figure 4 shows that for this path from 1000 Monte Carlo runs only 447 were legally compliant, while the rest did not fulfil the N-emission limit to rivers. Therefore, there is a 50% probability that path #21 would not be feasible at all, and one might therefore consider path #23 in future evaluations. In Figure 6 the contributions in environmental impact of the different treatment steps of the cheapest treatment path (#21) are visualized. The highest environmental burden is due to the emissions from the sewage treatment plant. Another significant contribution results from the stripping operation. A significant environmental benefit is obtained for the recovery of toluene by phase separation.
Figure g" Pareto plot of results presented in Figure 4.
894
Figure 6: Cumulative contribution in environmental impact resulting from the different treatment steps of the cheapest treatment path (#21) for waste stream WS2.
Conclusions and Outlook A model for assessing economic and environmental implications of chemical waste treatment operations under uncertainty has been presented. The model was applied to two made-up waste streams to demonstrate its capabilities. The results show that rather small variations in one or several input parameters might result in pronounced differences in treatment cost because certain treatment options are considered unfeasible or emission limits are exceeded rendering a scenario legally non-compliant. Furthermore, possible conflicts between cost and environmental impact have been demonstrated. The program also highlights the major contributions with regard to environmental impact of the different steps of a treatment path. Thus, the model is a valuable tool for recognizing technical, legal, financial, and environmental problems of treating waste emanating from chemical processes already at early phases of process design. In future work the model will be applied to case studies from chemical industry. In particular the ecological assessment will be evaluated with regard to the question, which set of indicators results in a comprehensive assessment of waste treatment operations in chemical industry. References B UWAL, 1998. Methode der 6kologischen Knappheit - Oekofaktoren 1997. Schriftenreihe Umwelt, Berne, Switzerland. Cavin, L., 1998. Tools implementation for computational cost estimation of waste treatment in chemical processes. Diploma thesis. ETH Zurich, Switzerland. CEFIC, 1998. http://www.cefic.be/activities/hse/rc/guide. Dimmer, P., 1999. Unsicherheits- und Sensitivitfitsberechnung in der frt~hen Kostensch~itzung von Batchverfahren unter besonderer Berficksichtigung der Abfallentsorgung. PhD thesis #13337, ETH Zurich, Switzerland. Jankowitsch, O., 2000. Environmental assessment of waste treatment process alternatives. Diploma thesis, ETH Zurich, Switzerland. Linninger A.A. and A. Chakraborty, 1999. Syntesis and optimization of waste treatment flowsheets. Comput. Chem. Eng. 23: 1415-1425. Petrides, D.P., K.G. Abeliotis, V. Aelion, E. Venkat, and S.K. Mallick, 1995. EnviroCAD: A computer tool for analysis and evaluation of waste recovery, treatment and disposal processes. J. Hazard. Mat. 42:225-246 Righetti, B., 1990. Umweltschutz- und Entsorgungskonzept im Werk Visp der Lonza AG. Chimia 44: 246-247. Schlegel, B., and R. Vouillamoz, 1993. Analyse der Entsorgungskosten. Internal Report, Lonza AG, Visp, Switzerland.
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rightsreserved.
895
Distributed Information System for Environmentally Conscious Process Design Yasuhiro Fukushima and Masahiko Hirao Department of Chemical System Engineering, The University of Tokyo, Tokyo 113-8656, Japan When we design a chemical process system, consideration of environmental impact is needed in addition to other conventional criteria.
We need to introduce product lifecycle
evaluation into the decision-making process, since changes in a single process indirectly influence the overall environmental impact throughout the entire lifecycle systems. product lifecycle evaluation brings several new difficulties.
However,
Data collection and estimation
in the lifecycle system is a time- and cost-consuming process.
Diversity of evaluation
methodology is increasing, and they should be dynamically selected and integrated to reflect the decision-makers' strategy.
In this paper, the current development of our information
system, which supports product lifecycle evaluation in the course of process design activities, is presented.
Our system aims to enable decision makers build their own iifecycle evalua-
tion systems by integrating information services supplied over the network.
Keywords: Process design, environmental impact, product lifecycle, information system INTRODUCTION Cost minimization has been the primary objective in process design.
This objective also
seemed reasonable from an enviromaaental perspective, since most of the operational cost was composed of energy consumption, resource consumption, and waste and effluent treatment. However, environmental concerns have only been incorporated as constraints for economic optimization.
Recently, design methodologies, which introduce environmental considera-
tions as a part of objectives, are studied and many environmental performance measures have been introduced for this purpose (Cano-Ruiz & McRae 1998).
By applying life cycle as-
sessment (LCA), these measures could also include consideration of environmental impact from other processes in the product lifecycle.
Using such measures, several process alter-
natives would be designed, optimized under a certain objective function, and the best one would be chosen.
896 However, the process optimization discussed above may still result in less than optimum design.
This is because the designed process should be operated as a part of the local manu-
facturing system.
If the measures did not reflect specific situations, such as the existence of
special equipment or the flow rate change in competing processes promoted by the introduction of designed process, designers might overlook these local factors.
Evaluation of post-
consumer plastics recycling process would be a good example involving this issue.
Even if
the recycling processes reduce the consumption of virgin materials, they usually entail extra energy and resources for conversion and additional transportation. affect other existing recycling processes.
These processes also
We have proposed a method that deals with this
issue and evaluate the environmental impact derived from entire process system (Hirao & Fukushima 1999). Consideration of environmental impact from a product lifecycle system requires a large amount and variety of datato be collected. In addition, the data should be kept up-to,date. However, data collection and maintenance are time- and cost-consuming processes, since activities in a product lifecycle are calTied out not by a single company but by many companies, governmental agencies, and citizens.
While many software tools such as Gabi
(Gabi
Software WWW page), SimaPro (SimaPro WWW page), and TEAM (Ecobilan Group WWW page) have been developed to help decision makers apply various valuable evaluation methods, fewer challenges are found in establishing effective data collection mechanisms. In this paper, we propose a new information system, which supports product lifecycle evaluation in the course of process design activities.
This system includes a distributed da-
tabase mechanism to reduce data collection cost and to share data management cost among the organizations involved in the product lifecycle system. also be placed ubiquitously in the network.
Data-processing modules can
These characteristics reduce software mainte-
nance costs as well, and moreover, enhance the flexibility of the decision-making process. SYSTEM ARCHITECTURE We designed our system in an object-oriented distributed-computing environment JiniTM (Jini TM WWW page) to cope with time-consuming procedures derived from the product lifecycle consideration, and to achieve the desired flexibility.
The essence of this paradigm is
to develop a useful module to function as a shared server "object" to increase its likelihood of being used.
Objects can be distributed and combined dynamically to realize a desired func-
tion throughout the network.
These server objects provide standardized services.
As
shown in Fig. 1, a service is defined as an interface between the server and client object. Interfaces for a service should be standardized so that anyone can develop either server or client objects, which can provide or make use of services over the network.
Therefore, in-
terface definition of a service is an essential part of the entire system development process.
897 Standardized Interface for "XXXService"
Client object can be developed expecting XXXService servers implement all the functions described in the interfi~ce.
Figure 1
Server object prepares allJi~nctions described in the interface and announces itself as a XXXService server,
Role of interface in describing a service
Figure 2 shows the entire evaluation process which is to be implemented in our system. We have started developing two sets of cliem applications: Lifecycle Modeler and Lifecycle Analyzer.
These applications are constructed as an imegration of four services, indicated as
double-lined blocks in Fig. 2.
The lifecycle model is the output from the Lifecycle Modeler
and is used in the Lifecycle Analyzer. >
17
Figure 2
Evaluation process using lifecycle model and relevant software modules
Lifeevcle Model We construct a lifecycle model (Hirao & Fukushima, 1999) which includes all activities that are to be evaluated.
The lifecycle model consists of two sets of data.
flow data, which includes a list of links, and the other is unit data.
One is system
Unit data describes the
relation between input and output of the unit, and is not necessarily linear, although our model currently assumes a linear unit class. Entire flow rates in the model are calculated by determining parameters for the model. The flow rate at any one point in the model can be a parameter which determines the entire
898 flow rate in the model, as shown in Fig. 3. are the parameter in our model.
Flow rates and flow dividing/combining ratios
Appropriate flow rates and flow dividing/combining ratios
are chosen as parameters for this lifecycle model to express possible scenario changes, and number of parameter must satisfy the degree of freedom.
Let us consider a scenario where
the demand, f~, changes from n~ to n2, as shown in Fig. 3.
All the flow rates in the lifecycle
model are calculated and total input-output index of the model are obtained for both cases. The difference between two total input-output indexes, which is called total-effect index, indicates the influence of the change of parameter values. By determining one flow, all flow rates can be calculated. Degree of freedom --- 1
ri I I t I i I
I 'a
I I I
I
Choose ft as pm'ameter of the model
fl--n| f f
r ~
ff
Total Input-Output "]
Total.
fl-- nz ~
[
~
Total Input-Output
Index
Figure 3
Parameters and degree of freedom
, _ _ bo},ndary 2
in
Figure 4
Possible system boundaries for
lifecycle model
lifecycle model
When constructing a lifecycle model, we must detemaine the system boundaries.
Note
that we do not have to model every relevant product lifecycle from "cradle to grave".
For
the processes included in the lifecycle model, we can use our own data, such as an actual operating data and calculated data.
The processes excluded from the model will be assumed to
have an averaged performance.
This is because each index in the total-effect index table
changes according to the boundaries and these indexes are translated into various impacts using some representative data in the following procedures.
The boundaries of the system
should be decided from the viewpoint of the data source.
If representative data is used for a
unit, the unit can be placed outside of the boundaries.
If your own data is used, the unit
must be inside the boundaries. Suppose we design a new process P~, as shown in Fig. 4.
In the conventional design, en-
vironmental performance measures are incorporated into the objective function to make P1 an environmentally conscious chemical process.
However, results from the conventional pro-
cedures are optimum only when we can assume that every actual activity, such as P2 and P3, which are to be connected with the new process, have an averaged performance.
Using the
lifecycle model of boundary 1, we can evade this strict limitation and use our own data for P2 and P3, thereby reflecting the situation of the local region in which the new plant is going to be installed.
If you find P4 unique as compared with averaged process and also need to use
actual data for P4, the lifecycle model should be extended from boundary I to boundary 2.
899
Lifecvcle Modeler Lifecycle Modeler is software that helps the investigator construct a lifecycle model using a graphical user interface.
We have already developed a primitive version of this software.
You can either input unit process data directly by yourself or collect data from distributed databases using the process data service.
Lifecvcle Analyzer Lifecycle Analyzer is software which support the analysis of environmental impact according to a specified scenario.
By changing parameters in the lifecycle model to simulate
the scenario, the total effect derived from the change can be analyzed by examining the totaleffect index.
We then need to interpret this result according to our interest.
First, the total-effect index is converted into values representing the emissions throughout the product lifecycle using emission factor. damage values using the potency factor.
Next, the emission values are converted into
Then, various damage values are integrated into an
objective parameter, usually called an indicator.
Seta~ices Within the system design, we found that the following modules are valuable as services in the distributed-object framework. 1) Process data service The process data service provides a distributed database mechanism to our system. are stored in any type of database distributed in the worldwide network.
Data
We expect that
these databases are maintained by each organization or group of organizations carrying out each process.
Client programs can use these databases directly through the standardized
interface, so that no database maintenance at the client side is needed.
We use this service
in the Lifecycle Modeler, to gather data for lifecycle model construction. 2) Emission factor service The emission factor service will be used in the Lifecycle Analyzer to interpret the totaleffect index into an emission value of the entire lifecycle.
There are a great variety of emis-
sion factors for a product depending on calculation methods, time, and representing levels. There are several different levels of emission factor data, such as worldwide, domestic, regional, and local.
Uncertainty of the data decreases as the applicable range narrows.
The
local emission factor for each specific material can be useful as a driving force for a manufacturer to produce a product with better performance, because the consumer will select materials for their product considering the emission factor.
Since many organizations are ex-
pected to provide this service, it will be valuable as a service in our system.
900 3) Potency factor service The potency factor service will also be used in the Lifecycle Analyzer to interpret emission values into damage values such as global warming, acidification, and resource depletion. An ordinary method for aggregating various type of emissions into a damage value is to convert each emission into one representative emission. One of the most well-known potency factors is GWP, for global warming damage, used to convert various gas emission values into an effective CO2 emission value (IPCC 1990). 4) Weighting method service The weighting method service will be used to determine weighting factors for each damage values. Using weighting factors, an objective parameter for decision making can be produced. Note that weighting is not a completely science-based procedure, since the importance of each damage is different for each decision maker who has independent sense of values. Weighting methods help users determine weighting factors by translating the problem into another form. The distance-to-target method determines the weighting according to a user's target. Eco-Indicator 95 (Goedkoop 1995) and Swiss Ecopoints (Ahbe et al. 1990) are indicators for various products calculated using the distance-to-target weighting method. CONCLUSION We proposed an information system, which aims to support product lifecycle evaluation in chemical process system design. An object-oriented distributed-computing environment was applied for this purpose. The main concept of this system design is based on services used over a network, which will help service users save time and cost, and the construction of user-dependent evaluation systems. Our system will be a trigger to start developing the information infrastructure for environmental management activities. ACKNOWLEDGEMENT This work is partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture of Japan (No. 10650738). REFERENCES
Ahbe S., Braunschweig A., Mtiller-Wenk R. (1990), Methodik fiir Oekobilanzen auf der Basis Oekologischer Optimierung, Buwal, publication 133, Bern, Switzerland. (in German) Cano-Ruiz, J. A., McRae, (J. J. (1998). Enviromnentally conscious chemical process design. Annual Review of Energy and the Environment, 23,499-536 Ecobilan Group, http://www.ecobilan.com/ Gabi Software, http://www.gabi-software.com/ Goedkoop, M. (1995), The Eco-lndicator 95 Final Report Hirao, M., Fukushima, Y. (1999), Evaluation of Environmental Impacts of Product Lifecycle tbr Process Design, Comp. Chem. Engng. 23 $823-826 ESCAPE-9 IPCC, CLIMATE CHANGE: The IPCC Scientific Assessment (1990) Jini TM,http :// ww.sun.com/jini/ SimaPro, http ://wv~v.pre.nl/simapro.html
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
901
Decision Making for Batch Manufacturing Sites under Uncertainty Andreas A. Linninger and Aninda Chakraborty Laboratory for Product and Process Design Department of Chemical Engineering, University of Illinois at Chicago e-mail: {linninge, achakr 1 }@uic.edu
Abstract Due to seasonal differences in production demands, effluent streams from batch manufacturing sites may exhibit considerable variations in number, amount as well as composition. Pharmaceutical and free chemical production distinguishes itself also in recycling practices since stringent purity requirement often prohibit direct recycle of raw materials into the next batch. Further complications stem from ever changing production campaigns, where each may last only for a few months. In such a dynamic environment, selection of recovery and treatment options as well as assessments of benefits and costs constitutes a formidable task. A deterministic superstructure generation and optimization approach for automatic waste management was presented earlier by Linninger and Chakraborty [1999]. Designing plant-wide waste management policies assuming perfect information may not be satisfactory given the variability of the production campaigns. Therefore this presentation addresses the problem of finding optimal waste management policies for entire manufacturing sites in the presence of uncertainty. It will offer a mathematical programming framework for implementing three contrasting strategies. Small case studies exemplify the impact of different perspectives on the structural design decisions in the overall plant design.
Keywords Robust Policy, Cost Effective Policy, Flexibility Index
Introduction Automatic waste management is concerned with the search for efficient recovery' and treatment policies for entire manufacturing sites. A deterministic two-phase hybrid methodology for the synthesis of recovery and treatment options in batch pharmaceutical production has been developed by Linninger and Chakraborty [1999]. In step one of this methodology, a lmowledge-based monotonic planning algorithrn evaluates the waste properties against regulatory limits and a relaxed set of technology selection criteria in order to generate a tree of feasible treatment options. Repeated application of the reasoning mechanism for all wastes and their residues constructs a superstructure of feasible treatment options. Phase two of the methodology optimizes a desired performance function subject to plantwide capacity, environmental, emission and logical constraints. This combinatorial
902 optimization yields the optimal waste management strategy as the best selection of one treatment path per waste on a plant-wide level. In this paper we will discuss impact of uncertainty as a result of varying waste loads but at unchanging composition. Section 1 of the presentation explains the methodology adopted for discrete representation of the uncertain parameters. In section 1, individual waste treatment policies are examined for their adaptability to waste load variations by means of a flexibility test. Section 2 discusses three different solution strategies for finding waste treatment policies for changing waste loads. In section 3, the different solution strategies are illustrated by two simple case studies from a single stage of a batch pharmaceutical plant.
l.Discrete Representation of Waste Loads First, we assume that the variations of each effluent, wk, can be specified by a finite number of possible states, Wka, WkI~, Wk~, Wk~ ...Wkv . Each outcome occurs with the likelihood, P(wkv). Here, Wk~ is the load of state v (v ~ N) for the k th waste stream (k s I). A waste scenario, W s, is an event formed by individual outcomes for each waste stream, Wkv. The probability of a scenario, P(W~), is given by the compound probability of the individual states. When the waste load variations are independent events (Bard, 1974), this compound probability is the product of the likelihood, i.e. P(W~)-I-[P(w~,). k
Obviously the sum of the compound probabilities of all distinct waste scenarios should add up to unity. In our discrete probability model, N waste streams in I discrete states need to be considered. Then, the size of the uncertain space is given by N I. Thus, for a set of 6 waste discrete streams each incurring 5 states, a total of 56 = 15625 different waste scenarios exist. Explicit enumeration of all possible scenarios for numerous waste streams would lead to a massive amount of uncertain parameters. Hence, a random Monte-Carlo simulation was used to reduce the uncertain space. A large randomly selected sample can statistically represent the entire uncertain space. More sophisticated techniques [e.g. Diwekar and Kalagnanam, 1997; Chakraborty and Linninger, 1999] could further reduce the quality of the sample with less computation effort.
1.1. Measure of Flexibility of a Treatment Policy Variations in waste loads may not only impact the cost function but may render the entire design infeasible due to design constraint violations. Hence, economic optimality at nominal conditions is insufficient as a performance criterion. Practical policies must also exhibit appreciable flexibility to waste load variations. To select among alternative designs, a quantitative measure of the process flexibility is necessary. The Flexibility Index, F, measures the elasticity of a waste treatment policy to changing waste loads. The definition o f F is depicted in equation 1. F = (Wk- Wk*)/Awk. (1) The concept of flexibility index has been previously discussed by [Swaney and Grossmann, 1985; Pistikopoulos and Grossmann, 1988]. This flexibility measure requires no discrete probability information on the waste variations. The nominal amount, Wk*, and expected deviation, AWk, suffice.
2. Solution Strategies for finding different Optimal Policies: The flexibility index can now be used to quantify the level of robustness of a given design. Hence, the flexibility will be included as an objective or a constraint in the
903 combinatorial search for plant-wide superior recovery and treatment strategies. In this section, we will present three competing strategies each pursuing a slightly different tradeoff among process performance and elasticity of the design.
2.1 Robust Policy- A Conservative Approach: A waste treatment policy that maxhnizes the flexibility index subject to the plantcapacity, environmental, emission and path constraints is termed a robust policy. This policy tolerates the largest waste load deviations among all the policies. The robust policy can be found through the MINLP formulation given by equations 2.1 - 2.5. In this formulation, the binary decision variables, xi's, determine the treatment paths of candidate policies, re. Each binary variables stands for a distinct treatment step in the superstructure, D. Note that the load to each treatment technology, c(wi), depends on the original waste load, Wk. Capacity constraints as expressed in Equ. 2.3 safeguard that the demand for a treatment does not exceed its available capacity, Cmax. Similarly, e(wi) accounts for the emissions caused by the treatment, xi. Clearly, permit levels, Emax, may not be exceeded, cf. Linninger and Chakraborty [1999]. Each policy has to obey to the path constraints, Equ. 2.5, which also ensure one path per waste stream. Objective" Max
(F,x~,wk)
F s.t.
(2.1)
w k > w~ +FAwk
VkeI
Z X i C(Wi) --
(2.2) (2.3) (2.4)
ZX j =Xi V children of i
VX i e D
(2.5)
2.2 Cost Effective Policy The robust policy does not consider the cost for obtaining this degree of flexibility. It will not surprise that the policy with the highest flexibility makes generous use of resources perhaps leading to poor cost performance. Hence cost consideration must also enter uncertainty analysis. The Cost Effective Policy ensures a reasonable cost performance alongside robustness with respect to waste load variations. This strategy can be implemented by minimizing the expected "average" cost subject to the plant-wide constraints, Equ. 3.1. Furthermore, this policy should be feasible in all discrete waste scenarios, Equ. 3.2 and 3.3. Each discrete waste scenario and their probabilities are obtained by Monte Carlo Sampling. The problem 3.1 - 3.4 describes the MILP search tbr a robust policy: Objective" Min ~ P(W~) ~xicost(W~) (w',x~) s~S[_
s.t.
xi c D
Z xi c(W?)---Cm,x
VseS
(3.2)
VseS
(3.3)
Vxi eD
(3.4)
x i cT
Y~xi e(W?)_<Em,x V leaves
Z x j =x i V children of i
904
2.3 An aggressive trade-off between flexibility and cost efficiency A cost-effective policy offer good economic performance and is feasible in all discrete waste scenarios. However, strict constraint enforcement may often not be necessary. As an example consider oversupply to a furnace. If the likelihood of an overload is very low, one could simply collect the surplus, store it and process it later without jeopardizing the feasibility of the candidate waste treatment policy. Therefore, very unlikely scenarios may be tolerated in favor of a more competitive overall performance. In earlier work, the concept of penalty function has been deployed by Bazaara and Shetty [1979]. For each constraint violation, a penalty term is added to the objective function, Equ. 4.1. The probability of constraint violation is termed failure probability, K s. It is equal to the likelihood of the scenario as computed by the Monte Carlo sample. The entire penalty contribution is the product of failure probability (KS), the specific weight for the penalty ( ~ ) and the maximum level of violation (dSmax). Minimization of the sum of expected cost and penalties renders the best trade-off design, cf. Equ. 4.1 - 4.5. This approach is known as Chance Constrained Optimization [Weisman and Holzman, 1972].
(W~,xi.d~x ) s~SL
~xic: D
EXi c(Wi~)+ds :Cm~x
(4.1)
min(O,x)l st VseS
(4.2)
d~x ->ds
VseS
(4.3)
d~x __.0
V se S
(4.4)
VxiED
(4.5)
xi c T
EX j =X i V children of i
where, K s = P(Ws) 3. Case Studies:
The proposed methodology and the different solution strategies are illustrated by two simple case studies. The first example uses a set of waste streams composed of three effluents from a single stage of a batch pharmaceutical plant producing a maleate salt. The recipe was simulated using the BDK software, [Linninger et. al., 1994]. It studies the flexibility index and the Robust Policy. The second consists of seven waste streams and illustrates the Cost Effective Policy and the Trade-offPolicy. The production site is assumed to have a bottleneck in the solvent recovery plant in both illustrations. This may be a likely situation in plants with advanced pollution prevention and waste reduction efforts.
905
Wl
W3
T ~ 2 L ~ T13TELl.Tl ! L4..~' ~
Fig. 1 Superstructure of the three exemplar waste streams in the case study (OnR- Onsite Recycle, LEA - Leaching, INC - Incineration, REU - Reuse, EVA - Evaporation, WW - Waste Water, SCR - Scrubbing, DRY - Drying, LF - Landfill, SCR- Scrubbing, ATM - Atmosphere)
3.1. Case Study I- Flexibility Analysis: Fig. 1 illustrates the superstructure generated for the three waste streams of the second case study. It has a total of 44 different treatment steps. For waste streams, W1 (Load = 200 kg; Expected deviation = 50 kg), and W2 (Load = 100 kg; Expected deviation - 10 kg), six distinct treatment paths are proposed, while waste stream W3 (Load = 100 kg; Expected deviation = 30 kg), offers only two treatment paths. The total number of feasible treatment policies is the cross product of all treatment paths. The superstructure for this simple case study embeds a total of 6 x 6 • 2 = 72 treatment policies. The Base Case Policy for the superstructure of Fig. 1 is W l {T2-T6-T9}, W2 {T20}, W3 {T40, T42}. This design yields a flexibility index of 0.24 for the given set of system constraints. On the other hand, the most robust policy is W1 {T1-T4-Tll-T12- T16- T17-T18-T19}, W2 {T20}, W3 {T41, T43, T44}. This robust policy has a flexibility index of 4.0 for the same set of system constraints.
3.2 Case Study I I - Cost Analysis Superstructure synthesis for the seven waste streams from the single stage of a batch pharmaceutical plant gives rise to a superstructure of treatment options having a total of 6912 distinct treatment policies. A state chart and the associated waste loads for the seven waste streams is shown in Table 1. The uncertainty space contains 3 x 5 x 3 x 5 x 3 x 4 x 5 = 13500 different waste scenarios. 5000 discrete waste scenarios using Monte Carlo Sampling were used. The Cost Effective Policy obtained by Equ. 3.1 - 3.4 has an expected average cost of 53858.36 S/yr. Since all system constraints of a cost effective policy are holding in all the discrete waste scenarios, solvent recovery is not opted for. This is due to the fact that the capacity of the solvent recovery plant is exceeded in some high waste load scenarios. The capacity constraints do not have to be satisfied in all discrete Monte Carlo Samples of the Aggressive Trade-off Policy, in 4.1 - 4.5. The penalty function approach of 4.1-4.5 yields an optimum with an expected average treatment cost o f - 6 7 6 1 6 . 3 0 S/yr. Here negative cost implies benefits obtained due to solvent recovery. The aggressive trade off policy recovers more valuable solvents when compared to the cost effective policy.
906
Wastes
w1 w2 w3 w4 w5 w6 w7
State l
State 2
P1
45000 0.25 59000 0.1 72500 0.3 60000 0.15 61600 0.2 34500 0.1 21000 0.05
,,
....
60000 71000 77000 70000 7000O 50000{ 40000
P2
State 3
P3 t State4
0.5 0.2 0.4 0.2 0.6 0.2 0.2
65000 78000 84000 75000 85000 60000 50000
0.25 [ _ 0"41870000.3 0.3 85000 0.2 _ 0.4180000 0.5 6000(
State 5
P5
_ -0.2 94000
0.1
P4
0.2 94000 0.15 0.3 0.2 7OOO0 0.05
Table 1 The amounts (Kgs) of the different states for each waste stream and their likelihoodof occurrence
Conclusions and Significance:
The assumption of perfect information as this is the premise of a deterministic design methodology may lead to flowsheets that are vulnerable to influences of uncertain parameters. In severe cases this could even lead to infeasible waste management strategies. Three different strategies to identify best plant-wide policies were presented. The discussion was based on simplifying assumptions such as linearized cost models, residual computation and varying waste loads at unchanging compositions. The waste loads were also assumed to vary independent of each other. These simplifications may be adequate for modeling uncertainty at multipurpose plants. The methodology addresses the need for evaluating uncertainty rigorously and its impact on decision-making for solvent-recovery and treatment selection. Stochastic changes to the amount and composition of wastes will be the subject of future work. References:
Bard, Y.; "Non-Linear Parameter Estimation", Academic Press, New York, 1974. Bazaraa, M. S. and Shetty, C .M; "Non-Linear Programming", John Wiley & Sons Inc., New York, 1979. Chakraborty, A. and Linninger, A. A.; "Discrete Sampling using Monte Carlo Simulation", UIC-LPPD Report (08-99), Chicago, IL, Sept. 1999. Diwekar U. M. and Kalagnanam, J. R; "An Efficient Sampling Technique for Optimization Under Uncertainty", AIChE Journal, 43, 440, 1997. GAMS - The Solver Manual; GAMS Development Corporation, 1996. Linninger, A. A., S. A. Ali, E. Stephanopoulos, C. Han and G. Stephanopoulos; "Synthesis and Assessment of Batch Processes for Pollution Prevention", AIChE Symposium Series, 90 (303), 46-53, 1994. Linninger A. A and Chakraborty, A; "Plant-Wide Optimal Waste Treatment Policy", Computers and Chemical Engineering, 23, $67 -70, 1999. Pistikopoulos, E. N. and Grossmann, I. E.; "Optimal Retrofit Design for Improving Process Flexibility in Linear System", Computers and Chemical Engineering, Vol. 12, No 7, pp. 719-731, 1988. Swaney, R. E. and Grossmann, I. E.; "An Index for Operational Flexibility in Chemical Process Design" 31 (4), pp 621- 630, AIChE Journal, 1985. Weisman, J. and Holzman, A. G. "Optimal Process System Design under Conditions of Risks", Ind. Eng. Chem. (Proc. Des. Dev.), 11,386-397, 1972.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
907
MINIMIZATION OF WATER CONSUMPTION AND WASTEWATER DISCHARGE IN THE SUGAR CANE INDUSTRY R. Pastor a, L. Abreu b, A. Espufia a and L. Puigjaner a
aChemical Engineering Department, Universitat Politbcnica de Catalunya, E.T.S.E.I.B., Diagonal 647, E-08028 Barcelona, Spain email :[email protected] bSchool of Electrical and Computer Engineering- FEEC,- UNICAMP. Campinas - SP, Brasil A comprehensive model has been developed for water minimization and wastewater discharge in the sugarcane industry. As starting point, it is considered that each production unit has a specified consumption of water that must be supplied from the fresh water sources or from one of the sources of regenerated water. In agreement with the concentration of contaminants either the water streams can be reused without treatment, or the best treatment alternative can be selected (taking into consideration its cost and contaminant removal efficiency). The streams are assigned to a treatment system and then the details of the quantities of water that will be reused are calculated. The application of the proposed methodology a case study based on a industrial situation is presented. 1. INTRODUCTION Most of food industrial processing is using large volumes of water in the industrial process. Due to the high cost of water and environmental regulations, there is a great interest in the minimization of freshwater consumption and wastewater discharge into the environment. The objective of this study is to minimize the water consumption by considering the opportunities of reuse/recycling wastewater with and without regeneration. The reuse/recycling without regeneration allows used water to be transferred from one production unit to another depending on water quality. Therefore, different wastewater treatment units are also considered and both the overall cost of water consumption and of wastewater production are subject to minimization. A different approach to face this subject is to involve economical criteria, and in the present work the following costs have been considered: cost of freshwater, cost of wastewater treatment, cost of discharge. 2. MOTIVATY EXAMPLE
2.1. Process description The typical raw sugar processing layout, consists of several (i.e. 3-6) mills tandem after cane preparation with two sets of knives and using a compound imbibition of about 20-40% water on cane. Husks of sugar cane are sometimes used as fuel in the boilers. Clarification layout is usually, with lime, sending the liquid cachaza from clarifiers to vacuum filters from were the filtrate is sent back to mixed juice before the addition of lime and the mud is used in agriculture as a fertilizer. Clarified juice with a solid concentration (Brix) 14% to 15% is sent from the clarifiers to the evaporation station, where it is evaporated to 63-65 Brix syrup. Syrup is sent to the crystallization where it is processed.
908 2.2. Water and wastewater streams identification The following main water streams have been identified: Imbibition, Filter, Cake filter water, Barometric condenser water (evaporation station, raw sugar, refinery), Raw sugar centrifuges, Refinery centrifuges, Cooling crystalliser water, Vacuum filter water Cooling system water. In the milling process the objective of this is to extract saccharosa as much as possible. Although bagasse is pressed at very high pressures, some juice remains, it is necessary to add water to the milling operation. This water is called Imbibition water. In Figure 1 a simplified diagram of water layout is shown. It indicates the main distribution of water in sugar process. Freshwater J i
9575,8
t~
i
I i
............................................................................................................................................................................
55
coolins r
i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425 Cooling System
140
E,,~.st~0n 445
I i
Barometric
4756
................................................................................... ii 9,6 Refinery Boiling
] R.r~.~/ J " 2560
{..~5~0 ci~:,~:::.w:i::................l....Filters
1707
4654
9868
i
i
1,,:~i. . . . . . . . . . . . . . . . . . . .!.5 . . . . .. ....... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .i. . . . . . . . . . . . . . . . . . . . .
i
ii
2617
Discharge ]
ar Boiling
I
......................................................................................................................
Condensers
....;/.I,~ ..........,.
!i i
R?~.S
1684
RawSugex-~tation
ii
31,2
L 1 2 o ...............
I,~bibiti~v,at=
i
Filters
:
iso
} c,x, ra........
t/]k
:~, FertirrigatienChannel
Fig.1. Simplified diagram of water distribution in sugarcane industry 3. MODEL DESCRIPTION The proposed model represents mass balance in each production unit. The user must specify all units that generate wastewater and diverse wastewater treatment options. The model automatically identifies the following recycle and reuse options for each water stream: i) reuse with regeneration, ii) reuse without regeneration and iii) recycle with regeneration. The mathematical model is based on linear programming. Several models associated to different objective functions are defined: Minimum cost associated to freshwater supply, minimum cost wastewater treatment and minimum discharge taxes. The model can be formulated as follows:
909
Objective function, reflecting treatment, freshwater and discharge costs min
= ~. C w t j
* A T j , i + ~i C f w * F W i + ~. C d * D i t
j
Constraints, reflecting mass balances, concentration of contaminants a) The concentration of contaminants in the water that is supplied to the processes must be within specified limits. This restriction is expressed mathematically as:
FWi+ Z A T j i q)cj+ZREri Ycr <- ini~bci V c,i. j
(1)
i'
b) The flow of water in and out of each process i is stated by the following two equations:
FW i + ~_~ATji + ~ REri = in i V c, i j
(2)
i'
Di + ~-" TV~j + ~-" REir = out i V c, i j
(3)
i'
c) It is assumed that all the water sent to treatment is transformed into treated water as:
Z TVji = Z ATji i
vj
(4)
i
d) The water treatment produces a reduction in the concentration of contaminants as stated: (1-a~a.)~--'TV~j 7'c, = ~ A T j , q~jV c,j i
(5)
i
3.1. First linear approximation This problem has been presented in the literature, as a nonlinear problem because contaminants content it should be expressed as a product of two variables: A r j i (Pcj. A first attempt to cope with this problem was to consider 9cj as a constant. That is the concentration of contaminant c after the water treatment j is forced to be bellow or equal to a pre-defined value. This approach is only acceptable if the real concentrations of all contaminants are very close to their pre-defined values. It can be argued that the LP problem obtained is so easy and fast to solve - as shown in the presentation of r e s u l t s - that solving this problem several times by means of a heuristic to define the new value of q~j may be worthwhile. This approach is also useful in what-if situations when a facility is operating with pre-defined values of q~cj and wants to check the consequences of the definition of new values for q~r 3.2. A MILP approach to the problem A new approach consisting of a MILP model was developed to avoid the definition of rough values for q~cj. Considering that every water treatment has a process tolerance that must be taken into account whenever using water treatment, it is proposed the definition of bands of operation for the concentration of contaminants. For each band k of the concentration of contaminants c, minimum (q~minckfl and maximum (q~max~kj) values are defined preferably according to tolerance of the water treatment process j. As a consequence a binary variable (X~kji) and an auxiliary variable (XA~kji) are introduced in the model, where: XAckji "-- Xckji , A rji V c, k, j, i (6) Therefore equations 1 and 5 are changed in the proposed formulation forcing the concentration of contaminants to be within minimum and maximum values.
FWi + ~ ~ YAckji q)maxckj + ~ REi'i Yci' <- ini t~ci V C, i k
j
i'
(7)
910
(1-a~j)~--'TV o yc~ _< ~ _ , ~ X A c k j i q~maxckj i
k
V c,j
(8)
i
(1-ac~)~-'~TVo Yci >- ~-'~,XAckji q)minckj V c , j (9) i k i The resulting linearization of equations 1 and 5 by expression 6 forces the introduction of new equations to enforce the linearity of the model. When the binary variable Xckji is equal to 1 it is stated that there is a flow of treated water with concentration k of contaminant c from water treatment j to process i. Equation 10 was introduced in the model to guaranty that when Xckji is equal to 1 we will have A Tji - XAckji. - ini (1 - Xckji) < ATj.~ - XAckj~ < ini (1 - Xckj~) V c, k, j, i (1 O) When the binary variable Xckj~ is equal to zero there is no flow of treated water with concentration k of contaminant c from water treatment j to process i. Therefore the auxiliary variable XA~kj.~ is also equal to zero as stated in equation 11. This equation also forces the amount of flow expressed in the auxiliary variable to be within desired minimum (Lj.) and maximum values (ini). Lj Xckji ~ YAckji ~ ini Xckji V C, k, j, i (11) If there is a flow of water from water treatment j to process i then the concentration of each contaminant c will be within one of the defined bands k. Equation 12 expresses this statement mathematically. Xck;i < 1
V c,j, i.
(12)
k
It is defined in equation 13 that if there is a flow of contaminant c from water treatmentj to process i, then there will also be a flow of the other contaminants c' from water treatment j to process i. XAc kji = ~ XA~ ,k+i V c, c'-r c , j, i k
(13)
k
4. R E S U L T S and D I S C U S S I O N
The optimized results obtained are: i ) R e c y c l e f r o m one equipment to another: In this case study it is possible to reuse the water, because the main contaminant is just sucrose. Imbitition water can be reused up to 51% of the sweet water (condensates with positive determination of sucrose) in an amount of 123.1 m3/h. In the refinery better results are achieved because 100% of the water can be reused without treatment : 6.7 m3/h (Barometric evaporation) and 2.9 m3/h (Barometric raws). ii) Wastewater Treatment: In fig 2, it is detailed which streams are regenerated and in which wastewater treatment units. Of the five treatment units, two were selected: WETL1, whose treatment is the most economical (raw sugar station condensate, cooling water) and MENBR1, because it is the most efficient for the removal of contaminants (Barometric condenser water: evaporation station, raw sugar, refinery, cooling crystallizes water, Vacuum filter water, Cooling system water). iii)Treated water: Treated water present in the treatment units and the units of production are assigned. Almost 50% of the water that is used in the milling, filter and bar refinery is treated in wastewater treatment plant I (WETL 1). iv)Fresh water: the quantities of fresh water necessary for each unit of production are determined, and shown in table 1. v)Discharge" Finally, the effluent that could not be either treated or reused and that therefore must be discharged. In this particular case only two stream is discharged." bar raw sugar (2000 m3/h), and cake filter (50 m3/h).
911
Freshwater ]
1757,3 t/h Freshwater
,_.... 42,8 Coo~g0~,,t,~=,, ,, I ! 12,2 I !..... ~, 108,1 M m ~ n e = &f~ctoe/
__•
425
Vacuum
3~,1
Filters
l
~.......... ~ i
Wastewater Treatment plant
Imbibitionwater
]
"'" .... Boiling <. ...... 25_,53 Raw"~ugar ~ .... 5-'62---i ......
: 839
= - - ........ t---i
t
i I
i
< ....... 29 % ' '
ii
I
,,
"~- ........................... i 19931~[----: F: .................. 67> ' : '
......... '- ...............
| i t i
.297
i ' ................. .
Refinery. Boidng
]~-]Filt~is ) ~'ii 'i " '
iq'-
................
I 2050
Water reuse with Treatment
'I' §
1386,8 * ........
I I ..........
I
[DISCHARGE
..... *"
Cooling System
Evap. Station
i-.9-
Water reuse without Treatment
1
140 445
.... ->
fi
_
I
i
i::.....................
..........'....
12-' i
g ..........................
t/h
Fig. 2. Diagram of water and wastewater distribution in sugarcane industry (optimized) Table 1. The optimized results obtained using GAMS/OSL. Optimized water Freshwater (t/h)
,,
Optimized
flow (t/h) Fresh Water
Reuse
Wastewater
Reuse
without trat with treat
eric
Wastewater
(t/h)
Discharge
efic 100
(%)
(%)
IMBIBI
275
0
123
159
103
1
0
FILTER
15
0
7
8
100
1
0
0
CAKEFILTER
50
11
0
39
78
50
50
0
BARO EVAPO
1684
297
0
1387
82
1707
0
100
RAW SUGAR
31
0
26
6
100
1
0
100
BARO RAW
4756
839
0
3917
82
4854
2000
59
REFINERY
10
0
10
0
100
1
0
100
BARO REF
2560
567
0
1993
78
2617
0
100
55
12
0
43
78
55
0
100
COOLING CRYST VACUUM
140
31
0
109
78
140
0
100
COOLI NG_WAT
445
445
0
0
0
445
0
100
10021
2202
165
7661
80
9872
2050
78
1473055
323763
227056
47150
Total Cost (u.m)
967256
912
Conclusions This study revealed that the optimization strategy envisaged for the freshwater consumption and wastewater minimization in the sugar cane industry leads to substantial water/cost savings in this type of industry. The model proposed identified significant results in the flesh water consumption. Moreover, the combined effect of lowering freshwater cost reducing simultaneously the effluent contaminants leads to obtain the maximum reuse or recycled water at minimum cost. Furthermore, the automated wastewater treatment unit selection that has the most efficient methods of contaminant removal enhances the reuse opportunities and contributes to additional savings. Further work is underway to extend this optimization model to contemplate other type of industries in the food sector.
Notation: Water flow from wastewater treatment unitj to production unit i Discharge taxes Cd Cfw Fresh water cost Wastewater treatment cost Cwt Discharge (wastewater flow from production unit i without treatment/reuse) Di Freshwater flow to production unit i FWi Inlet water to production unit i ini Outlet water from production unit i out~ REi, i' Water flow reuse from prod unit i to i' TV~j Water flow from prod unit i to wastewater treatment unitj Maximum inlet concentration of contaminant c to unit production i ~c,i Outlet concentration of contaminant c from wastewater treatment unitj (Pcj Removal efficiencies (%) of treatment processes (Zcj Oulet concentration of contaminant c to unit production i 7c, i Type of contaminant C Production unit i i Wastewater treatment unit j J Concentration of level contaminant c k Acknowledgements Financial support from the European Community is gratefully acknowledged (project IC18-CT98-0271). Rosario Pastor acknowledges a grant ( Training and Personal Research Sponsorship in Spain) from Ministry of Education and Culture - MEC. Luiz Abreu acknowledges FAPESP - Fundacao de Amparo a Pesquisa do Estado de Sao Paulo. REFERENCES 1. Almato M., Sanmarti E., Espufia A., Puigjaner L., (1998). AIChE Annual Meeting, Miami. 2. Galfin B., Grossman I.E., (1999), Comp. Chem. Eng. Sup., pp. S 161-S 164. 3. Pastor, R., Espufia, A., Puigjaner., (1999). AIChE Annual Meeting. Dallas. 4. Rossiter A. P., (1995). Edit. McGraw-Hill, New York. 5. Wang, I., R. Smith., (1994). Chemical Engineering Science, 49:3127-3145.
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rightsreserved.
913
HYDRODYNAMICS AND CHEMICAL MODEL TO EVALUATE ENVIRONMENTAL RISKS IN PROXIMITY OF RIVER MOUTH M. Di Natale *1, G. Merola*, D. Musmarra** (*) Dipartimento di Ingegneria Civile, Seconda Universith di Napoli, via Roma, 29, 81031 Aversa (CE), Italy (**) Istituto di Ricerche sulla Combustione, CNR, p.le Tecchio 80, 80125 Napoli, Italy
ABSRA CT
Pollution of natural river networks is a growing environmental problem, which produces more and more alterations to complex and fragile aquatic ecosystems. This problem is not limited to the river networks but extend all along the coastal area involved with the mouth apparatus. Anthropic emissions of pollutants both as uncontrolled discharge stream and due to accidental sources (i.e. treatment plant failure or chemical disaster in proximity) are the causes of pollution. Integrated numerical models are necessary to evaluate the environmental risk magnitude. These models have to take into account both the natural characteristics of the environment considered and bio-degradation kinetic of the pollutant. In the present paper a numerical unsteady 2D model, able to describe the complex interaction between the hydrodynamic field and biological and chemical processes, has been developed. The model has been applied to the analysis of the mouth of Sarno river (south of Italy) which well represents a case of extensive pollution. INTR OD U CTION
Marine pollution, which is growing in intensity, mainly involves coastal zones and in particular river mouths. In fact, in a broader view of superficial watercourse pollution, rivers have actually become the means of delivery for all domestic and industrial sewage produced by the anthropic activities diffused on the territory; they therefore represent the vector for the direct transport of all the polluted and toxic substances collected during the water course, to the big final recipient, the sea. In order to evaluate the scale of coastal pollution processes and set the framework for reliable prevision methods, besides monitoring activities (expensive and anyhow time-limited), the use of more or less sophisticated mathematical models is increasing more and more. In particular, non-stationary models have received great interest to describe the region involved into an accidental disaster (i.e. the rupture of a tank). In the present work a numerical model to evaluate the diffusive-convective transport is presented; it analyses the processes, that occur near the river mouth, due to an emission of known concentration produced by the river current flowing to the sea. The flow field
I Corresponding author: Te1.:+[39](81)5010236; fax: +[39](81)5045804; e-mail: dinatale@sunap'unina'it"
914 determination refers to a 2D schematization, in shallow water conditions, with depth averaged horizontal and vertical velocity components (Falconer, 1994; Di Natale, 1996). The transport of the pollutant specie has been described by using a stochastic approach, recently utilized also in other hydrodynamics fields (Uffink, 1988); it works through a Lagrangian scheme, in which the diffusive component of transport is studied with a random walk model, based on the formal analogy between the differential transport equation and the well known stochastic equation of Fokker-Planck. In addition, the model is improved to consider the reactivity of the pollutant by using the very simple Steeter's and Pelps's approach (Steeter and Pelps, 1925). This model considers the BOD concentration as representative of the pollutants and postulates a linear dependence between the BOD oxidation kinetics and its concentration. The model has been applied to the Sarno river mouth (southern Italian coast, Tyrrhenian Sea). A square wave concentration has been used to describe the immission of pollutant (BOD) and a long shore stream has been considered. Model results, in terms of iso-concentration lines are reported in the case of reactive and non reactive pollutant. FLOW FIELD DESCRIPTION
Within the hypothesis of 2D flow field (in shallow water condition) with vertical velocity components which are negligible in comparison with horizontal ones and hydrostatic pressures distribution, the classical equations of motion and continuity reduce to only three differential equations (Csanady, 1973) holding the following unknowns: u,v (horizontal components on averaged local velocities) and 1"1 (hydraulic raise). In the presence of a wave flow field the aforesaid equations can be lately simplified thereby eliminating the oscillatory component of flow field (not very significant for transport phenomena) by means of time averaging on T, the wave period (Falconer, 1994). Moreover, through an integration of the equations on the z axis, one can obtain a set of motion and continuity equations, in which the horizontal velocity components U and V are invariable along the z axis and equal to the averaged value (depth integrated equations). A great number of studies (Abbott, 1978" Brebbia, 1983" Kowalick, 1993) have provided the numerical integration of these equations system. In the present work we use a numerical explicit scheme where the velocity values u, v and hydraulic raise values 1"1 are evaluated through a forward formula (for time derivatives) and a centered formula (for derivatives of the shearing stresses) (Koutitas, 1988) Boundary conditions on the calculation grid, with steps Ax and At, are the classical time dependent conditions, in which the velocity components orthogonal to the solid boundaries and the spatial derivatives near the free open sea edge (open sea conditions) are null. In order to describe the physical aspects of the problem one has to consider the interaction phenomenon between current and wave motion that requires the addition, in motion equations, of radiation stress terms R;j (Longuett and Higgins, 1974), due to the wave motion itself and by the phenomenon of wave refraction in the presence of a current, which opposes the waves' propagation direction. In particular, in order to evaluate wave numbers , k x and k,.,. and height H in a generic depth, we refer to equations of non-rotationality wave number k, Doppler effect equation and energy conservation. In the flow field used in this case, there is a long-shore current, with oc=15~ (angle with the coastal line), h = 2 m , T= 6s, that invests the river current in proximity of river mouth.
915 MODEL EQ UA TIONS Mathematical transport model The convective-diffusive equation in the 2D turbulent flow field, which has been considered, is the well known expression (Fischer et al., 1979) bC OUC OVC ~ ~ - t - ~ + ~ =~D ~t ~x ~y ~x
bC 3 x ~+~Dy ~x Oy
bC ~y
(1)
The numerical solution for eq. (1), using the classical schemes at finite difference, is notoriously fairly complex because of some alterations due to the numerical diffusion, produced by numerical truncation errors. The proposed resolutive approach is based on a Lagrangian description of the transport process. As well known (Csanady,1973) this model analyzes the path of a generic solute particle that is characterized, in the case of turbulent flow field, by a random walk. In particular, referring to a generic particle, one can say that the displacement S i, after a time nAt, is a casual variable and its probability density G(S i) is yielded by stochastic equation of Fokker-Planck (Risken 1996). The determination of function G(x,y,t)=G(Si, t) is however complex and it is still a subject of specialized mathematical studies. (Risken, 1996) A remarkable simplification of the problem can be made if one postulates that the flow field is uniform (U=const.,V=const.), the dispersion coefficients constant (Dx=D=D) and the solute emission punctual. In that case one can prove (Uffink, 1988) that G(S i) is the well known expression of Gaussian probability density; the elementary displacement components As x and As y, during the generic time interval At, are: As x = UAt + R 1~/6DAt
(2)
AS y = VAt + R 2 ~/6DAt
(3)
in which R is a random number characterized by a uniform distribution into the interval [-1,1]. Using the illustrated procedure we can describe the Markovian stochastic process that characterized the solute particle trajectory in space and time dominion. To evaluate the instantaneous concentration C(x,y,t) in a generic point of spatial dominion one has to follow these steps: 9 the 2D dominion is divided in rectangular cells with Ax and Ay sides; 9 in the cell corresponding to the initial pollutant font N particles are disposed; 9 through the aforesaid random walk method one determines the displacement S i of each of N particles at fixed time t; 9 the concentration C in a generic cell is expressed by means of m//N ratio, between the m particles present in it and the N initial ones. Kinetics effects on pollutant substances transport In order to describe the oxidation kinetic influence on convective-diffusive transport phenomenon the Steeter and Pelps simplified model (Steeter and Pelps, 1925) has been used. This model is based on the following hypothesis: 9 BOD concentration, CBoD, and dissolved oxygen concentration Co2, are sufficient to define the whole system from a chemical point of view;
916 9 BOD oxidation velocity rBoDis proportional to 9 oxygen consumption rate coincides with rBo,,;
CBoD;
9 oxygen flux across the free surface is proportional to oxygen deficit
(C O -CO, ), in which 2
:g
C02 denotes the dissolved oxygen concentration in saturation conditions. Basing on these hypotheses we need to write, in each domain cell, two mass conservation equations: the first one concerning the "pseudo-compound" BOD and the other one concerning the dissolved oxygen. The first equation substantially states the non-reactive mass balance, to which a consumption term P,, concerning with the oxidation reactions for BOD reduction have to be added. This last term, in the aforesaid simplified model, is:
Pi = -kCsoD
(4),
in which k is the chemical kinetic constant. The second equation takes into account both the dissolved oxygen consumption rates, by oxidation reaction, and the oxygen flux crossing the free surface, given by an oxygen deficit. The Steeter and Pelps model assumes that the dissolved oxygen consumption rate can be equal to P;, while the flux is expressed by:
No2 = ~(C02 -C02) being
(5),
Co,- the dissolved oxygen concentration in saturation conditions (for a fixed value of
temperature) and /3 the mass transfer coefficient, expressed by (Vesilind et al., 1982):
fl - 3. 9~i 5 ~l O97)(T~
]5/dil5
(6)
in which T o is the temperature, d; is the water depth, and V, is the cell volume. The solution for these two balance equations is rather easy because they can be solved separately. In particular, by knowing, through the random walk method, the theoretical number m of particles present in the i cell at generic time nAt, one can determine the real particles number m ' , using eq. (4). This number is obtained subtracting from m the number of particles that have reacted in At. Then one can determine dissolved oxygen concentration through oxygen mass balance equation.
RESULTS Figure 1 report model results obtained by using a square wave immission law as sketched in top of figure 1. Such a figure refer to model calculation performed by neglecting the reactive term, therefore they can be regarded as diffusion of pollutant only. On the other hand, model results shown in figure 2 include the reactive term and in particular, k- l x l 0 6 s -~ has been used. The comparison between results in figure l and 2 shows that the reactive term can, significantly modify the concentration of pollutant. In this case, the concentration is decreased as consequence of BOD degradation. In order to reach a deeper insight of the local effect on the Sarno mouth,
917 the model results are reported, in Figure 3 over the map of the Sarno mouth. This figure well shows the propagation of the pollutant spot reaching the Castellammare di Stabia harbor.
C [g/mc]
Immissionlaw 500
40 3
Fig. n.1
Fig. n.2
4
8
16
T [hi
918
Fig. n.3
REFERENCES Fischer., List E.J, Koh R.C.Y, Imberger J., Brooks N.H. Mixing in Inland and Coastal Waters Academic press, Inc., 1979 Holly F.M. Jr, Dispersion in Rivers and Coastal water-Physical principles and dispersion equations, Developments in Hydraulic Engineering 3, Elsevier Pub., New York. 1985 Falconer R. A. An Introduction to nearlyhorizontal flows, Coastal, Estuary and Harbors Engineers Reference Book, FN Spon, 1994. Di Natale M. Sulla dispersione di inquinanti superficiali in aree costiere per effetto di marea XXV Convegno di Idraulica e Costruzioni Idrauliche Torino (Italy) 1996 Uffink, G.J.M. Analysis of dispersion by the random Walk method Ph. A. Thesis Deeft University, 1988.
Csnady, G.T. Turbulent Diffusion in the Environment Reider Publishing Company, Holland, 1973. Soulsby R. L. Tidal current boundary The Sea-Ocean Engineering layers Science Volume 9 Part A 15,551, 1990. Koutitas C.G. Mathematical Models in Coastal Engineering, Pentech Press, 1988. Abbott M. Computational Hydraulics, Pitman ed., 1979. Brebbia C.A. Computational Hydrodynamics, Butterworth ed., 1983. Kowalik, Z.- Murty, T.S. Numerical Modeling of Ocean Dynamics Advanced Series on Ocean Engineering -World Scientific, 1993. Longuett-Higgins M.S., Stewart R.C. Radiation Stresses in water waves: a physical discussion with applications Deep Sea Res vol 11 1964. Di Natale M. Non linear Hydrodynamic Effect of opposing Jet-Current on waves ISOPE '98, Montreal 1998. Risken H. The Fokker-Plank Equation Springer ed., 1996. Vesilind P.A, Peirce J.J. Environmental Engineering , Butterworth Publisher, Boston, 1982 Steeter, H.W., Phelps E. B.: A study of the pollution and natural purification of the Ohio River Public Health Bulletin n~ 1925 (Reprinted by US Dept of Health Education and Welfare, 1958) Nardini A. Soncini Sessa E., Bacci M Inquinamento fluviale: realizzazione ed uso di modelli matematici ed Marsilio (Italy) 1990.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
919
Simulation and optimization of the reactive absorption of HF/HNO3 during pickling acid regeneration W. Wukovits a , W. Kamer b, A. Lebl b, M. Harasek a and A. Friedl a a Institute of Chemical Engineering, Fuel and Environmental Technology, Vienna University of Technology, Getreidemarkt 9/159, A-1060 Vienna, Austria
b Andritz AG-Ruthner Surface Technologies, Eibesbrunnergasse 20, A-1120 Vienna, Austria The optimal operation of pickling acid regeneration is very important for its applicability and economical feasibility. This paper describes the simulation and optimization of the process conditions with the commercial process simulation programm ASPENplus 9.3. Beside the calculation and regression of physical properties, an important step was the implementation of reactions occuring during NOx-absorption in ASPENplus. The simulation model was improved by fitting with data from a pilot scale plant. Subsequently a sensitivity analysis was done to find process parameters to be optimized. 1. I N T R O D U C T I O N Surface cleaning is an important step in the production of steel to remove oxide layers built during heat-treatment processes or storage. This cleaning process, called pickling, is usually done by treatment with inorganic acids, mainly hydrochloric acid or sulfuric acid. For cleaning stainless steel a mixture of nitric acid, hydrofluoric acid and water is used. Because of economical reasons and environmental protection, it has become increasingly important to regenerate the used pickling acid. In order to ensure the economical operation of a pickling acid regeneration plant, it is necessary to optimize process conditions to: 9 maximize HF regeneration 9 maximize HNO3 regeneration, with regard to a high degree of NOx-oxidation 9 control the dilution of the regenerated acid through condensation of water, built during the burning of the spent acid (see description of the regeneration process) This paper presents the results of a sensitivity analysis of a pickling acid regeneration process done with the commercial process simulation programm ASPENplus 9.3. 2. R E G E N E R A T I O N PROCESS
Depending on the used pickling acid, regeneration can be performed by rectification, crystallization, extraction, absorption or membrane processes [1]. In case of a mixture of nitric acid and hydrofluoric acid, applicable processes are liquid-liquid extraction, fluoride cristallization, electrodialyses with bipolar ion exchange membranes, retardation (a process using ion exchange resins) [2] and absorption. In this study, the regeneration process involves burning and thermal decomposition of the spent acid / salt mixture, by which the salts of the metals are converted into their
920
Fig. 1. Flowsheet of the pickling acid regeneration plant corresponding oxides (PYROMARS | process). Unfortunately, also decomposition of a part of the nitric acid occurs during this process step. The flue gas passes a venturi scrubber, where it is contacted with the used acid (Fig.l). In this scrubber the used pickling solution is preheated and preconcentrated before entering the burning unit. Simultaneously the flue gas is quenched and metall oxides, which have passed the cyclone are removed. After the venturi scrubber, the gas enters the first of two packed absorption columns, where gaseous nitric and hydrofluoric acid are condensed or absorbed. In these columns, also oxidation and absorption of nitrous oxides occurs. The absorption step is modeled as adiabatic. To remove remaining HF and NOx from flue gas after the absorption step, the gas passes a scrubber and a catalytic DeNOx. 3. SIMULATION MODEL Compared with other absorption processes, the absorption of NOx is very complex. There exist various gaseous species which are in equilibrium and are absorbed simultaneously. In the liquid phase two oxy-acids are built, whereby one of these acids decomposes to NO. NO is desorbed from solution and enters the absorption cycle again. The important reactions in NOx-absorption are shown in Fig. 2. R3
N204
N204 + H20
~> HNO 2 + HNO3
2 NO 2
2 NO 2 + H20
> HNO2 + HNO3
NO + NO 2 + H20
R1
~
2 HNO 2
3 HNO 2 ~_- HNO3 + H20 + 2 NO R4
2 N O + O2/
x~ O + NO2 ~
N203
N203 + H20
> 2 HNO 2 R6
NO
Gas
Interface
Liquid
Fig. 2. Mechanism and reactions of NOx-absorption [3, 4, 5]
921 Models for the calculation of NOx-absorption during nitric acid production are presented for example by Wiegand et al. [6] and Suchak et al. [7]. A general view of the features of different models is given by Pradhan [8]. Because of the large number of species involved in the process, it was decided to use a commercial process simulation program for calculating the absorption step. Within this work, calculation and simulation was done using ASPENplus Version 9.3. The implemented calculation algorithms as well as models and databases for physical property calculation should allow a fast determination of process parameters to be optimized. In doing so, it was accepted to calculate NOx-absorption based on equilibrium calculations, not considering heat- and mass-transfer effects on the absorption step. From the four paths of NOn-absorption shown in Fig. 2, the path via N204 and via N203 was implemented in ASPENplus. Absorption of NO2 and formation of HNO2 in the gas phase were neglected. Calculations showed, that the absorption of N203 is of minor importance. 4. PHYSICAL PROPERTIES
The standard thermodynamic model to handle electrolyte systems in ASPENplus is ElecNRTL. It calculates the activity coefficients for molecular and ionic species using binary and pair parameters [9]. Adjustable parameters are the Born radius of ionic species and the NRTL interaction parameters for molecule-molecule, molecule-electrolyte and electrolyteelectrolyte pairs. The comparison of the calculated vapor-liquid equilibrium with data from literature gives good accordance in the interesting concentration range from 0 to 15 mole% for the system HF/H20 (Fig. 3). The difference using the ElecNRTL-model is even less than using the model ENRTL-HF, which takes into consideration the hexamerization of HF in the vapor phase. Although the deviations in the system HNO3/H20 are bigger, no data regression is necessary. The situation is different concerning mass density and heat capacity. Fig. 4 shows the mass density of the system HNO3/H20. The deviation for a solution of 10 wt% HNO3, calculated via Clark Aqueous Electrolyte Volume, the ASPENplus electrolyte mixture standard model, is considerable and increases with increasing HNO3 content. Density data obtained by the 115
1130 1110
110
.............................................................................................................................................................................. ~.:. I~onc.in wt% HNO3
....
%-C. ~-.
1090
~'105 P.
~ 1070 ,x
~
'~ 1030 c
"A~i,
'-'1050
100
~ . .
"o 1010 m 990
~ 95
E
90
930 0
0,1
0,2 0,3 0,4 m o l e f r a c HF [-]
Fig. 3. Vapor-liquid equilibrium of the system HF/H20 [ 10]
0,5
0,6
~
~'A ~
9 literature (10 wt%) [] -,i,-9 Clark (10 wt%) - 4.--o9 .Clark (30 wt%) ~ ~ f i t t e d data (10 wt%) - - -
970 950
85
~^
. 0
20
.
. 40
. 60
"-, ~'~,
literature (20 wt%) Clark (20 wt%) Costald (10 wt% fitted data (20 wt%)
. 80
temperature [~
100
120
Fig. 4. Density of the system HNO3/H20 [11]
922 Tab. 1. Gas composition [wt%] after column 2 obtained from experiment and simulation Pilot Plant 3,6 0,48 0,34 0,28 74,1 13,0 8,2
H20
HNO3 NO NO2 N2
02 CO2
Simulation 2,7 0,44 0,46 0,00 76,6 12,0 7,8
Costald model show better correspondence. But even these data require data regression. A comparison of fitted data with literature is also given in Fig. 4. Simulation results with the described model showed good accordance to data from a pilot scale pickling acid regeneration plant (Tab. 1). Nevertheless, it was necessary to fit adjustable simulation parameters to process data, to optimize the solubility calculation of nitrous oxides during nitric acid formation, because simulation gives a higher NO content than experiment, while the content of NO2 is zero in the results of simulation. The parameter to be fitted is the solubility of N204. ASPENplus uses Henry's law to calculate gas solubility in liquids [9]" P~ = H , * x i
Henry's constant Hi is obtained by the following relation: B i
lnH, = At +--~ + C~ * lnT + D r *T 1
After fitting the solubility of N 2 0 4 with data obtained from the pilot plant, the ratio of NO:NO2 in the gas stream after the absorption step corresponds well with the ratio observed during the experiment. Fig. 5 and Tab. 1 summarizes kinetic- and solubility-parameters used in further simulations. kl
2
krl - - ~ - ~ =
4,321'
* 5421,8 10_,5 T_~,exP(R,TI
RI"
r~ = - - ~ P o, P No
R2:
K 2 = ~2 PNO2
lnK 2 = -32,6 + - - - ~
R3"
r3 =-k3CN~o4
k~3 = k3 = 12994022,5 * exp(26298'4 ]
R4:
r4 =
R5:
Ks =
_ k~
kmol ] ,) [Pa~*m----3*s I
6866 E11
PN204
4
k 4 CHN02 2 2 He(No) CNo PN:o3
kra k4 = 2027,9' exp(515,49;3 / Ikmmol,sl H,No 4 7 4 0 1 1~] a lnK 5 =-28,1+--T---
PNo * PNo2
r6 6C o
'6: 40 Ill
9'rateconstantsaregivenintheformimplementedinASPENplus."
kr,=kp~.*T"*exP(R--@T I [12]
Fig. 5. Kinetic-parameters used in the simulation model [4, 5]
923 Tab. 2. Henry's constants Hi [atm/mole fraction] used in the simulation model (25~ in water) [ 13, 14]
N2 02
CO2
ASPEN 86530 43980 1610 29200 17E06
Literature 86400 43610 1630 28700 0,71 **)
NO N204 *) *) fittedsolubility **) valuefrom literature describes "bulk solubility"; all species of the gas and its reaction products with water are included 5. RESULT OF THE SENSITIVITY ANALYSIS Finally, the developed simulation model was used to find and estimate optimization possibilities for the described pickling acid regeneration process. A sensitivity analysis was realized to obtain the degree of HNO3 and NOx precipitation as well as the concentration of the regenerated acid as a function of different process parameters. Tab. 3 shows the results of the sensitivity analysis. The varied parameter, the area of variation (usually +/- 30% of the value used in the process) and the effect on precipitaton and concentration are given. An increase in HNO3 and NO• precipitation is given by an increase of column pressure and a decrease of column temperature. The reduction of the inert gas flow also leads to a better precipitation. But the sensitivity analysis also shows that all arrangements cause just a small increase in nitric acid concentration. The reason is the increase of water condensation at lower column temperature. The strongest influence on NO• results from the reduction of the inert gas flow and the increase of column pressure. The inert gas flow in the process is given by the energy demand for the evaporation and decomposition of the used pickling solution. Thus a reduction of the inert gas content is coupled with the optimization or change of the energy supply. An increase of the column pressure is only possible in column 2, because the high acid content of the gas stream entering column 1 would cause corrosion problems in the fan. Tab. 3. Results of the sensitivity analysis Precipitation [%] Parameter Temp. Col 1 Temp. Col 2 Gas-Holdup Col. 2 Input Inertgas Input Oxygen Pressure Col. 2
Unit Variation ~ 30-60 ~ 20-45 m3 10-28 m3/h 1930-4500 m3/h 230-530 mbar 963-1963
HNO3
NOx
78-70 80-86 No effect! 84-65 73-71 71-82
12-7 14-7 8-10 23-5 8-9 9-29
Conc. Reg. Acid [wt%] HNO3 HF 11,6-10,6 7,1-7,2 11,5-10,7 7,1-7,2 10,8 7,2 12,2-9,3 7,0-7,5 11,0-10,6 7,2 10,8-12,3 6,8-7,2
924 6. CONCLUSION Sensitivity analysis with the obtained absorption model shows the complexity of the optimization problem. It was found, that in the discussed process most variations of process parameters which give an large increase in precipitation of nitric acid or in NOx-oxidation result in no or only a small increase in the concentration of the regenerated acid because of water condensation. In further simulations special attention will be given to the rearrangement of apparatus to obtain a process, where it is possible to adjust the acid concentration nearly independently from the degree of acid- and NOx-absorption. The work with the developed simulation model in ASPENplus shows that it is possible to find and estimate optimization possibilities in NO• even in using the equilibrium approach. However, for a detailed simulation and process design mass- and heat-transfer calculations have to be taken into consideration. NOTATION ci
mole concentration [kmole/m3] rate constant rate constant (as implemented in ASPENplus) Ki equilibrium constant pi partial pressure [Pa] Pi partial pressure [atm] R gas constant ri reaction rate [kmole/m3*s] T temperature [K] xi mole fraction [-] Hc Henry's constant (mole concentration basis) [atm*m3/kmole] Hi Henry's constant (mole fraction basis) [atm/mole fraction] Ai, Bi, Ci, Di parameters for calculation of Henry's constant
ki kri
REFERENCES [ 1] Ullmann's Encylopedia of Industrial Chemistry, Vol. A 14, 5th Ed., VCH/Weinheim, 1989 [2] C.J. Brown; Iron Steel Eng., 67(1) (1990) 55-60 [3] D. Thomas, S. Brohez, J. Vanderschuren; Trans. Inst. Chem. Eng. Part B, 74 (1996) 52-58 [4] Ullmann's Encyclopedia of Industrial Chemistry, Vol. A 17, 5th Ed.,VCH/Weinheim 1991 [5] F.T. Shadid, D. Handley; The Chemical Engineering Journal, 43 (1990) 75-88 [6] K.W. Wiegand, E. Scheibler, M. Thiemann; Chem. Eng. Technol. 15(5) (1990) 289-297 [7] N.J. Suchak, K.R. Jethani, J.B. Joshi; AIChE J., 37(3) (1991) 323-339 [8] M.P.Pradhan, N.J. Suchak, P.R.Walse, J.B.Joshi; Chem. Eng. Sci., 52(24) (1997) 4569-4591 [9] ASPENplus Reference Manual Release 9.3, Vol. 2, 1996 [ 10] Dechema Data Series I/1 b Suppl.2, Dechema/Frankfurt a. Main, 1988 [ 11] Landolt-B6mstein - Neue Serie, Bd. 1; Teil B, Springer Verlag, 1977 [ 12] ASPENplus User Guide Release 9.3, Vol. 2, 1996 [13] J.M. Kasper, C.A.Clausen, C.D.Cooper; J. Air&Water Manage. Assoc., 46 (1996) 127-133 [ 14] CRC Handbook of Chemistry and Physics, 75 th Ed., CRC Press/Boca Raton, 1994-1995
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
925
Trend recognition of process data of a refinery using wavelets B. Bitzer and J. Richters University of Paderbom, Automation Engineering, Steingraben 21, D-59494 Soest/Germany, e-mail: fat@ibml 5.uni-paderbom.de The goal of the project "Forecasting of state parameters in refineries", which is supported by the Federal Ministry of Education and Research of Germany, is to reduce the emission of a refinery by use of intelligent methods and logic instruments for analysis of the refinery-wide gas-network using the real-time expert system G2. Exceptional production events and changes in continual use of the systems often lead to an increased emission of fuel gas. A premature trend recognition with event localisation and representations to exploit alternatives of intervention is especially necessary if the attention of the production staff is turned towards malfunction in primary production processes. A trend recognition of process data using wavelets is described in this paper. I.
INTRODUCTION
The fuel gas system represents a combination of consuming and producing units, each of them playing a part in the overall system balance. Fuel gas is used for process heaters, electricity generation and steam production. Because of the complexity of the fuel gas systems, it is difficult to manage with varying individual conditions. Changing weather and production conditions may result in excess fuel gas production, such that gas flaring can be necessary to maintain the balance of the system. Knowledge-based gas dispatching systems with a trend recognition could prevent excessive flaring of potential fuel gas and unnecessary simultaneous import of additional natural gas. 2.
WAVELETS
A real-time trend recognition of process data is especially problematic with noisy signals. Conventional methods often eliminate too many characteristics or the noise is insufficiently reduced, so that recognising the essential characteristics of the signal becomes very difficult. It is also very usefull for an easy real-time trend recognition to reduce the bulk of data. The wavelet transformation has all these features. The wavelet is compared with the signal and then shifted on the time axis to be compared with the next part of the signal (Fig. 1).
H Fig. 1: Shifting the wavelet
926 With the wavelet transformation a signal is split into a high- and a low-frequency part using scaled and shifted wavelets. This method could be compared to the Fourier transformation, but instead of cosine and sine function the wavelet transformation uses wavelets for analysing the signal (Fig.2). Depending on the type of wavelet a better scaling of the time and frequency is achieved. !
.
.
.
.
.
I
:1 A I 0,
0.5 0 -05 -1 0
0
0.6
1
2
db2
Haar (dbl)
0
1
2
$
4
0
db3
5
I0
1$
db9
Fig. 2" Debauchies-wavelets For the next dissection-step the wavelet W(t) is scaled, that means that the wavelet will be deor increased and then compared again with the signal x(t). The results are the wavelet coefficients X(a,b) which is a scale of the approximation of the wavelet to the part of the signal (1). (1)
Every calculation step is then defined by three values: the position on the time axis, the scale of the used wavelet and the wavelet coefficients. This method is called continuous wavelet transformation. 3.
DYADIC WAVELET TRANSFORMATION
The Dyadic Wavelet Transformation (DWT) is a special case of the wavelet transformation, which is also called discrete or fast wavelet transformation. With the DWT no complete frequency resolution will be achieved but dyadic graduated frequency bands are generated. The DWT could be described as a convolution of a discrete-time signal S with a pulse response of a high-pass filter H and a low-pass filter L (Fig. 3). I-I
~
.
.
.
.
H'
H' 7~
L
Fig. 3" Scheme of the DWT
(~ 1000
[-'
927 The signal is split with the two filters into two parts, a high-frequented part called details and a low-frequented part called approximation. The number of values of every part is bisected. The transformation could be repeated with the approximation until the number of values falls below the number of filter coefficients. A further transformation of the details is normally not useful. The signal could be reconstructed with the corresponding approximation plus the details. If every detail and approximation of every transformation step is used for reconstruction then the reconstructed signal is equivalent to the original signal. The DWT could be realized atter a formulation from Stefan Mallat with a pyramid algorithm [ 1,2]. This algorithm depends on the repeated use of a Conjugate Quadrature Filter (CQF) which is a special pair of Finite Impulse Response filter (FIR-filter) and could be realized as a convolution with a matrix. X - W-x
(2)
In this example the four coefficients of the db2-wavelet are used. With the signal vektor Y"
X --[Xl,X2,'",Xn] T
(3)
and the quadratic matrix W"
gl
g2
g3
go
gl
g2
g3 ~
......
W=
_g2__ _g_3_.. . . . . . . g3
-g2
gl
-go
g3
-g2
gl
go
gl
g3
-g2
(4)
-go "..
~gl
-go
the approximations a and the details d could be calculated with formula (2) as the result vektor 2 [3 ]. X : [a 0, al,
9o
", a(~_l)
,
oo
do, d l, ., d(2_l)
]T
(5)
The boundary value problem is solved in this matrix by using the first values which could be seen in the last line of the matrix W (4). By doing this the first values of the signal influence the last values of the details and approximations which is only correct for periodic signals. For analysing non periodic process data it is better to take the mean value of the last values of the signal instead of the first values and put them at the end of the signal vektor. The important requirement of the algorithm is a high speed of operation because the algorithm must be coupled with a real-time expert system where the trend recognition is executed. On the other hand the algorithm must be flexible for a different number of signal values and a different
928 number of transformation steps. In this case the algorithm is written in C and coupled as a remote procedure with the real-time expert system G2. 4.
PROCESS DATA AND APPROXIMATIONS
A wavelet transformation is not usefull for every signal. Signals with only few changes in the value have only few high-frequency parts in the signal, so that other methods could supply better results more easily and faster. On the other hand the result of the DWT of very noisy signals could not be so good, so that a change of the transformation depth could improve the results depending on the signal characteristics that will be detected.
I v'tItv;t,,
Vt}vlt'vt,vlvt {
%
{
{
{
{
I
[
Fig. 4: Signal with 8 approximations
,~
%
I
, I
I
Fig. 5: Signal with 4 approximations
A typical process signal (temperature) with 128 values and the result of the wavelet transformation with the depth 4 (= 8 values) is shown in figure 4. The approximations are placed to the corresponding values in the charts. In figure 5 the approximations which are shown in figure 4 are transformed once more. It can be seen that the approximations are more detailed in figure 4, but if the noise or other disturbing signals increases a transformation depth more could be necessary. 5.
TREND RECOGNITION
Based on the approximations of the DWT a trend recognition is executed in the realtime expert system. Depending on the preceding sign of the first and second differentiation of the approximation a trend sequence is calculated. The differentiations are defined as follows
[4]: X n' = X n - X n _
"
'
'
X n =X n -Xn_ 1
1
(6)
(7)
Mostly the differentiations of the approximations will not be exactly zero so that it is more usefull to take an interval for it. It is also helpful to integrate a hysteresis for every trend to reduce the changes in the trend type. For some sensors it is not necessary to determine all
929 trend typs but only an increase or decrease. Also a change in the trend direction from increase to decrease or vice versa could be wrongly interpreted, so that the computered trends must be checked. Based on this trend recognition a trend sequence of every signal could be computered. Nine trends could be distinguished generally depending on the results of the differentiations which could be positiv, negativ or zero (Table 1). Table 1" Trends trend number
first differentiation
second differentiation
1
sign[x:]
= 1
sign[x~] = 0
2
sign[x:]
= !
sign[x:] = 1
3
sign[x:]
=
1
sign[x;]
= -1
sign[x'] = -1
sign[x~] = 0
sign[x" ] = - 1
sign[x;] = 1
sign[x'~] = - 1
sign[x~] = -1
sign[x:] = 0
sign[x;] = 0
sign[x:] = 0
sign[x~] = 1
sign[x" ] = 0
sign[x~,] = - 1
trend form
J f
/
/
Mostly the differentiations of the approximations will not be exactly zero so that it is more usefull to take an interval for it. It is also helpful to integrate a hysteresis for every trend to reduce the changes in the trend type. For some sensors it is not necessary to determine all trend types but only an increase or decrease. Also a change in the trend direction from increase to decrease or vice versa could be wrongly interpreted, so that the computered trends must be checked. Based on this trend recognition a trend sequence of every signal could be computered. 6.
RESULTS
For optimizing the fuel gas system with this application only the long-term trends are interesting, because the duration of potential flaring must be long enough before actions are taken in the fuel gas system of the refinery. As an example 512 values of a signal at different times are analysed with a transformation depth of 7 steps so that the results are 4 approximations. The sensor whose curve is shown in the following figures has a sampling
930 period of 30 seconds so that the 512 values represent a period of time of about four hours. With the four approximations a sequence of two trends could be computered (Fig. 6).
J
o.o o.o
550.
0.0 0.0
i
Fig. 6: Signal at different times with approximations 7.
SUMMARY
The wavelet transformation is a method which could reduce the bulk of values in a suitable way. The number of steps to reduce the signal could be selected depending on the type of signal and the characteristics which must be extracted. If the wavelet transformation is realised as a remote procedure, it could run on different computers for a signal preprocessing to relieve the real-time expert system. The result of the method called approximation could easily be used to recognise trends in the signal in a very simple way. Based on the results of this trend recognition IF-THEN-rules could be developed with methods of knowledge engineering and then integrated in the real-time expert system to describe the processes of the refinery. Based on these types of rules a forecast of the fuel gas production and consumption could be made and advices to the staff for a better use of the fuel gas could be generated. REFERENCES
.
.
M.Vishwanath, The recursive Pyramid Algorithm for the Discrete Wavelet Transform. IEEE Transactions on signal processing, Vol. 42, No. 3 (1994) 673 -676. S. Mallat, A Theory for Multiresolution Signal Decomposition: The Wavelet Representation, IEEE Transactions on pattern analysis and machine intelligence, Vol.2, No. 7 (1989) 674 - 693. T. Strutz, E. Mtiller, The Dyadic Wavelet Transformation using orthogonal Filters Implementation in C, Frequenz, 50 (1996) 51 - 59. G. Locher, B. Bakshi, G. Stephanopoulos, K. Schiigerl, A method for an automated rule extraction from raw process data, Automatisierungstechnik, 44 (1996) 61 - 70.
European Symposiumon ComputerAided Process Engineering- l0 S. Pierucci (Editor) 9 2000 ElsevierScience B.V. All rights reserved.
Comparison
931
of methods for assessing human health and the environmental impact
in e a r l y p h a s e s o f c h e m i c a l p r o c e s s d e v e l o p m e n t G. Koller, U. Fischer*, K. Hungerbi.ihler Safety & Environmental Technology Group, Laboratory of Technical Chemistry, Swiss Federal Institute of Technology (ETH), Z~irich, Switzerland 1. Abstract In chemical process design it is important to consider potential safety and environmental problems as early as possible. In this study a number of different methods for assessing human health or environmental impact are compared on a theoretical and a practical basis. Although developed for the same goal of assessing human health or environmental impact, these methods vary significantly in the number of effects considered and the way of processing data. However, the application to a case study resulted in similar assessments for most methods with regard to total impact while rather small deviations in the contribution of individual substances were found. Comparably large differences were found in particular for inorganic substances depending on the way methods consider non-degradability. The highest differences between assessment methods arise from the default assumptions in case of missing substance data. This fact shows the importance of using all available information, ranked according to its quality, in automated assessment. Furthermore, automatization speeds up process assessment and facilitates the comparison of different methods. 2.
Introduction
It is commonly agreed that the best approach towards developing both cost effective and environmentally friendly processes is to consider environmental aspects as early as possible during process development. Therefore, much effort has been contributed from regulatory, industrial, as well as scientific side, to develop methods and tools assisting this design process. A number of attempts have been made to classify and analyze the large variety of methods for environmental assessment of substances and processes. Methods were reviewed from the viewpoint of process design and optimization (Cano-Ruiz and McRae, 1998), of Life Cycle Assessment (Hertwich et al., 1997), of substance ranking (Swanson et al., 1994), or they were classified according to their degree of sophistication (Jia et al., 1996; Pennington and Yue, 1999). Guidelines for developing or selecting methods for chemical ranking and scoring have been proposed by Swanson and Socha (1997). These reviews either describe a large number of methods or present a systematic approach for classification, but none of them presents a classification of existing methods. Practical comparisons of different environmental assessment methods for chemical processes are still missing, probably because this is a dataintensive and time-consuming task. In this paper a general scheme is proposed for comparing methods assessing human health and environmental impact. This scheme combines and extends the concepts of Pennington and Yue (1999) and Swanson and Socha (1997). It is applied to a number of methods currently used for process assessment including the method recently presented by Koller et al. (2000a). Furthermore, selected methods are applied to a process from pharmaceutical industry. Differences between the methods and problems arising from applying existing methods in automated assessment of chemical processes during early design phases are discussed. 3.
Classification of methods In the following paragraphs the most important characteristics of methods assessing human health and environmental impact are discussed. Table 1 summarizes the characteristics of a number of methods de-
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scribed in literature. Most abbreviations of Table 1 are explained in the text below (keywords of Table 1 in bold). Goal: The first important part of any assessment method is a clear definition of its goal and scope. Methods can be developed for substance or process assessment. They can be designed as a first filter for preliminary screening (p = preselection in Table 1 column goal), for instance for selecting substances to focus on during a detailed assessment. Another goal can be a relative ranking of substances or processes (r = ranking) based on a single numerical index. Finally, methods can be developed to identify and quantify all problems associated with substances or processes (a - assessment). Type of method: Pennington and Yue (1999) classified assessment methods into 5 different types. The simplest way of considering environmental impact is to sum the mass of waste or of specific compounds (direct data summation). The atom efficiency of Trost (1991), the mass loss indices presented by Heinzle et al. (1998), or the original WAR algorithm (Mallick et a1.1996) are examples of such an approach. The next level is effect normalization, where some kind of equivalence factor based on legal thresholds or benchmarks is used for weighing different substances according to their relative contribution to a certain aspect. Scoring and ranking approaches first characterize the objects in different effect classes, usually on some kind of index scale. Then they combine the different indices based on expert judgement. A number of methods of the effect normalization or the scoring and ranking type are listed in Table 1 (type of method = 2 or 3). These models were finally selected for application in this study (see below) because they are particularly suited for early phases of chemical process design. In model based approaches the basis for combining different aspects is some kind of environmental model rather than pure expert judgement. The last and most sophisticated type is a detailed impact assessment including site and case specific considerations. However, this approach is data and time intensive and its application might be difficult in early process design phases. Considered aspects: The core of any assessment method is the variety of aspects considered and the exact way (degree of detail) of evaluating them. A full impact assessment of a process should cover the whole chain from the process itself (handling of substances, output streams, process details such as waste (pre)treatment) over exposure (release scenario, environmental fate, bioaccumulation, routes of intake) to the final adverse effects. The way of considering the different aspects largely defines the accuracy of final T a b l e 1: Comparison of assessment methods for human health and environmental impact general
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considered aspects
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method author year ], ~ EI?/~ atom efficiency fi.e. mass) ITrost 9 I 1991 lap 1 emission limit |sw,ss LRV , 19'85~aP PNEC (Predicted No Effect Conc)|European Commission ~ 1996,1.aS PNEC&halflife ,~European Commission I 1996,~aS EURAM .~Hansenetal. 11999JrrS 2 KEMI |Swed. Nat. Chem. Insp.1 1995~pS CHEMS .~S. . . . . . et al. ~ 1997.~ rS EHS ,~Koller et al. " I 2000,[ aP 2 index S~stem (max PEC/PNEC) IPratt et al. [ 1993|aS
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1 p:preselection, r:relative ranking, a: absolute assessment, S:substances, P:processes 2 according to Pennington and Yue (1999): 1: direct data summation, 2: effect normalization, 3: scoring and ranking approaches, 4: model based approaches 3 1:yes/no condition (based on threshold), 2:3-5 categories, 3:continuous index scale, 4:continuous scale in physical units, i:implicitely 4 1:categories, 2:simple model (equilibrium, Mackay I), 3:detailed model (Mackay III), 4:measured data or detailed site specific model, i:implicitely 5 aquatic and terrestric toxicity assessed using: a:acute data, c:chronic data, b:both, i:implicitely 6 aquatic and terrestric toxicity assessed using: e:effect data, n:no effect data, b:both, i:implicitely 7 primary basis of assessment: I:legal threshold values / categories (e.g. Threshold Level Values), o:other data (e.g. No Observed Effect Level human), b:both 8 1:only structure (and QSARs), 2:Material Safety Data Sheet (MSDS) data without legal classifications (melting point,boiling point, vapour pressure,Kow, solubility, biodegradation halflife,LC50 fish,LD50 oral rat), 3:MSDS with legal classifications, 4: MSDS and other data) 9 1:only one type of data, 2:set of data sources, 3:data hierarchy 10 u:user dependent, w:worst case assumption, b:best case assumption, n:neutral (best available estimation) 11 1:not considered, 2:considered by use of reviewed data, 3:explicite guidelines, 4:presented also in result 12 1:nominal, 2:orclinal, 3:interval, 4:ratio
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results. The exposure part can be modeled in different degrees of detail ranging from simple categories (vapor pressure > lmbar --) substances in air) over models assuming equilibrium between the compartments (Mackay I type), models including transfer and degradation (Mackay III type), to detailed site specific models. Environmental effects can be assessed as condition (substance is degradable), as category (classified as very toxic T+), or on a continuous scale. Continuous scales can have physical units (LD50 value in mg/kg) or index units (0: no danger, 1: high danger). Index scales usually set limits to an aspect (halflife of 1000 days is as bad as halflife of 10000 days). Therefore domination of a single effect during aggregation can be avoided. Physical values allow a model- rather than expert-based aggregation of different effects. Another way of considering environmental effects is to use legal threshold values. Methods using such values developed by a team of experts therefore cover a number of different effects implicitly (marked with i in Table 1). Applying legal data for assessment can be desirable as they represent some kind of societal agreement. One disadvantage however is that they generally do not represent the latest scientific knowledge and that the decision making process might be politically rather than scientifically driven. One detail of effect assessment, which has become subject of intensive discussion recently, is the question of using acute and/or chronic data respectively effect or no effect data as basis for assessment. For details on this discussion and its implications for risk assessment see Koller et al. (2000b). Raw data: The applicability of a method depends on the amount (minimum substance data) and quality of raw data used. Data available in Material Safety Data Sheets represent a base set for assessment during process design. Legal classifications (for instance emission limits) usually are available only for existing and relevant substances. Methods relying on legal classification can therefore not be applied for fine chemical processes as soon as highly complex organic substances are involved. Ideally, methods apply a tiered approach not only relying on a single type of data but proposing a data hierarchy where the best available information is used in a common assessment frame. In a similar way, methods should provide guidelines how to cope with data quality and data ranges. Parameters such as the aquatic LC50 value can vary for several orders of magnitude. Each method should consider this problem in some way (e.g. by taking the mean or the minimum value) as it might influence the result of assessment significantly (Koller et al. 2000b). Combining the quality and uncertainty of input data and presenting it as a final result for the quality of the total assessment has not been attempted so far and remains one of the major challenges when developing new methods. Scale of data: One important and often neglected aspect is the scale of input data as well as of the assessment result. The simplest scale is the nominal scale which can only take two values (yes or no). Methods for selecting substances for priority action use this scale for presenting the result. The ordinal scale ranks objects according to numbers representing the magnitude of an aspect. The difference between these numbers, however, has no relation to the difference in aspects. Only statements like "A is more dangerous than B" can be made. Examples for an ordinal scale are the ratings proposed by the National Fire Protection Agency for fire, health and reactivity. For a quantitative information (A is three times more dangerous than B), at least an interval scale is required. Here, in addition to the order of numbers also the difference between two numbers has a physical meaning and corresponds to the difference in effects. The highest level of scale is the ratio scale where a certain value on the scale (for instance zero) corresponds to a defined physical value, for instance a concentration causing no effects or causing a cancer risk of 1%. All toxicological endpoints are examples for ratio scales. The type of scale used has an important influence on the permissible mathematical and logical operations (Volta and Servida, 1992) and on the final interpretation of results. Combining an ordinal scale with any other scale via multiplication or addition is highly questionable from a methodological viewpoint although it is common praxis for many methods (e.g. Dow Fire & Explosion Index). On the other hand, transferring data from a ratio scale to an ordinal scale (e.g. the regional halflife converted into the persistence score of the WMPT tool (US-EPA, 1998)), and using only the ordinal scale for further assessment and aggregation should be avoided as well, as a lot of information is lost unnecessarily. In each case, the scale of the final result must be identified before applying or interpreting any results of assessment methods.
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4.
Methods Selected for Practical Comparison & Description of Case Study
Using Table 1, the theoretical differences between a number of methods can be identified. As already mentioned, methods for assessing human health and environmental impact of substances and processes were selected for practical comparison that can in principle be applied for early process design (type of method = 2 or 3 in Table 1). Additionally, the methods selected had to display the results at least on an interval scale in order to be combined with the mass of emissions (scale of results >= 3 in Table 1). In order to simplify the interpretation, method comparison was done for environmental impact and human health separately although some methods would allow full aggregation. The seven methods selected were included into the automated assessment tool developed by Koller et al. (1999) so that they could be applied easily on a common set of substance data. If no substance data were available and the method did not mention any default values, the best case assumption was used. Otherwise, the inorganic substances (e.g. nitrogen, carbon dioxide, sodium sulfate) would have dominated the effect scores of many methods, and method comparison would have been impossible. The method of Koller et al. (2000a) was applied without technology factors, assuming that no further treatment of effluents was considered. Thus, all non-product streams were used for assessment. In order to present the results on a common scale, all methods were converted to an exponential scale of four orders of magnitude, resulting in a factor of 10000 for the most dangerous substance in each category (hexachlorobenzene, dieldrin) and a factor of 1 for degradable and non toxic compounds. This equivalence factor was multiplied with the mass per kg of product. The methods were compared using an industrial process from Novartis Pharma AG. From a six stage batch process to 8o~-Amino-2,6-dimethyl-ergoline described by Baenziger et al. (1997), the Curtius Rearrangement converting dihydroisolysergic acid methyl ester to 8ct-amino-6-methylergoline was selected. After transforming the ester into the hydrazide using hydrazine hydrate in 1-pentanol, the intermediate is crystallized from n-heptane and further converted with sulfuric acid and sodium nitrite in aqueous acetic acid. After neutralization with sodium hydroxide and recrystallization from methanol/water, the product is obtained in an overall yield of 78%.
5.
Results Figure 1 shows the different equivalence factors for environmental impact as defined by the methods (see Table 1 for details). Five methods identify hydrazine as the most dangerous compound, whereas the Sw iss emission limits and the KEMI method find equal and much smaller impacts for a number of compounds. 25OO
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Figure 1" Equivalence factors for environm-ental'impact ' assessment of substances in amino-methyl-ergoline production (for method abbreviations see Table 1)
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Figure 2: Assessment of enviromental impact of production of l k g of aminomethyl-ergoline (for legend see Figure 1, for method abbreviations see Table 1)
935
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Fiuure 3: Human health assessment of production of lkg of amino-methyl-ergoline (for method abbreviations see Table 1)
The reason for this is that Swiss emission limits to water are defined for classes of substances and not for single compounds. The results of combining the equivalence factors with the emission inventory are shown in Figure 2. Water is identified as major environmental pollutant by the KEMI method, as it is not degradable and exists at high amounts (9 lkg). This questionable result indicates the general problem of applying existing methods to inorganic substances. Otherwise all methods calculate similar total impacts, although the individual contributions of substances are different. Methods relying on the sum of degradability and toxicity signify a medium danger for non-toxic but non-degradable substances. Methods relying on the product of the two aspects and having a toxicity scale with zero as lower limit result in no danger. Heptane, a moderately toxic, degradable but potentially accumulating substance, which exists at high amounts (15kg), and hydrazine, a highly toxic, not biodegradable and non-accumulating compound emitted at low amounts (0.5kg) are dominating. The different ways in which the methods implicitly weigh these aspects lead to the different contributions for the same compound. Figure 3 compares methods for assessing human health effects. These methods differ slightly in the number of effects considered (carcinogenicity, irritation). For other methodological differences see Table 1. Still, most methods highlight the problem of chronic exposure to hydrazine as a potential carcinogen and the acute effects of sodium hydroxide. The absolute values are in the same range for most methods. Only the Swiss emission limits for air (between 5 and 150rag/m3 corresponding to equivalency factors between only 40 and 1) indicate harmless substances. The compound 1-pentanol is identified as major danger for humans when the PNEC value is used for assessment. This result however is caused by the large safety factor of 100000, as LC50 values for mammalian were the only toxicological endpoints available. These two problems (threshold values not discriminating between different organic substances and large safety factors dominating the result) limit the applicability of these methods for the assessment of chemical processes. 6.
Conclusions & Outlook
A number of different methods for assessing human health or environmental impact were compared on a theoretical and a practical basis. Although developed for the same goal of assessing human health or environmental impact these methods vary significantly in the number of effects considered and the way of processing data. However, the application to a case study resulted in similar total impact except for Swiss emission limits and PNEC values. Neglecting some methodological limitations all methods highlight the same
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substances as hazardous. If these cannot be avoided during process development, a suitable way has to be found to handle them in a safe way. Rather small deviations between methods were found in the contribution of individual substances. Comparably large differences were found in particular for inorganic substances due to the way methods treat non-degradability. When discussing differences in assessment results, it must be kept in mind, however, that compared to a total range of four orders of magnitude, a factor of three does not mean a significant difference between different assessments. The highest differences between assessment methods arise from the default assumptions in case of missing substance data. If Figures 1-3 were shown with the worst case assumption, all methods only relying on toxicological data and not using legal classification systems would highlight the data gaps for inorganic compounds. This fact shows the importance of using all available information ranked according to its quality in automated assessment. Methods only relying on a single type of information often require expert judgement in practical application. The practical comparison of different methods, as presented in this study, was facilitated by using a tool that automatically combines process data with substance databases. Currently, additional case studies (representative set of substances and several processes) are analyzed for a more comprehensive comparison of methods claiming to assess environmental, health and additionally safety aspects of chemical processes during the early design phase. 7.
References M. Baenziger, C.P. Mak, H. Mtihle, F. Nobs, W. Prikoszovich, J.L. Reber and U. Sunay, Org. Proc. Res. Develop., 1 (1997)395. J.A. Cano-Ruiz and G.J. McRae, Annual Review of Energy and the Environment, 23 (1998) 499. European Commission, Technical Guidance Document in Support of Commission Directive 93/67/EEC on Risk Assessment for New Notified Substances and Commission Regulation No 1488/94 on Risk Assessment for Existing Substances, Luxembourg, 1996. B.G. Hansen, A.G.v. Haelst, K.v. Leeuwen and P.v.d. Zandt, Environ. Toxicol. Chem., 18 (1999) 772. G. Pratt, P.E. Gerbec, S.K. Livingston, F. Oliaei, G.L. Bollweg, S. Paterson and D. Mackay, Chemosphere, 27 (1993) 1359. E. Heinzle, D. Weirich, F. Brogli, V.H. Hoffmann, G. Koller, M.A. Verduyn and K. Hungerbtihler, Ind. Eng. Chem. Res., 37 (1998) 3395. E.G. Hertwich, W.S. Pease and C. P.Koshland, Sci. Total Environ., 196 (1997) 13. C.Q. Jia, A.d. Guardo and D. Mackay, Environ. Sci. Technol., 30 (1996) 86. G. Koller, U. Fischer and K. Hungerbtihler, Comp. Chem. Eng., 23 (1999) $63. G. Koller, U. Fischer and K. Hungerbtihler, Ind. Eng. Chem. Res., in press. G. Koller, K. Hungerbtihler and K. Fent, Environ. Sci. Pollut. Res., in press. S.K. Mallick, H. Cabezas, J.C. Bare and S.K. Sikdar, Ind. Eng. Chem. Res., 35 (1996) 4128. D.W. Pennington and P.L. Yue, J. of Cleaner Production, 8 (2000) 1. M.B. Swanson, G.D. Davis and S.L. Jones, Comparative Evaluation of Chemical Risk Ranking and Scoring Methodologies, 87th Annual Meeting & Exhibition, Cincinnati, Ohio, 1994. M.B. Swanson, G.A. Davis, L.E. Kincaid, T.W. Schultz, J.E. Bartmess, S.L. Jones and E.L. George, Environ. Toxicol. Chem., 16 (1997) 372. M.B. Swanson and A.C. Socha, Chemical Ranking and Scoring: Guidelines for Relative Assessments of Chemicals, Sandestin, 1997. Swedish National Chemicals Inspectorate, Selecting Multiproblem Chemicals for Risk Reduction - Sunset Project - KEMI Report, Stockholm, Schweden, 1995. Swiss LRV, Luftreinhalteverordnung, SR 814.318.142.1., Bern, 1985. B.M. Trost, Science, 254 ( 1991) 1471. US-EPA, Waste Minimization Prioritization Tool Spreadsheet Document for the RCRA Waste Minimization PBT Chemical List Docket, Washington DC, 1998. G. Volta and A. Servida, Environmental Indicators and Measurement Scales, in Environmental Impact Assessment, A.G. Colombo (eds), Dordrecht (1992) 181.
European Symposiumon ComputerAidedProcess Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
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An integrated framework of process and environmental models, and EHS constraints for retrofit targeting F. Nourai, D. Rashtchian and J. Shayegan Sharif University of Technology, Tehran, Iran Finding environmentally clean alternatives for retrofitting existing processes has been addressed with various approaches. In this paper, a previous pollution prevention (P2) approach by the same authors is extended to include EHS criteria for targeting waste reduction in chemical processes. The method is applied to an existing facility as a case study and most viable modification are highlighted in order to improve the environmental conditions of the plant with the regulatory constraints in mind. The advantages of the present approach are that the proposed solutions are based on a precise, integrated description (simulated model) of the plant and the environment, practical feasibility of the modifications, and the correspondence of pollution impacts with regulations. The results of this type of analysis are risk-based pollution prevention targets, and the appropriate direction/magnitude of any modifications. 1. INTRODUCTION The chemical process industries (CPI), as a whole, is moving more and more towards improving its environmental impact under high pressures from public opinion and governments on a global basis. Stricter environmental regulations posed on CPI through national, or international agreements, and escalating costs of end-of-pipe treatment, are the main reasons why operators and designers are realizing the benefits of adopting a pollution prevention approach to the problem. Legislators also appreciate the approach [1 ]. It is now possible to say that pollution control technologies are regarded more as complementary tools. Now, the problem is "How to practically modify an existing plant to reduce its environmental impact potential?" This expression has several important implications for process designers: The proposed solutions should be based on a precise description of the plant, the recommended modifications should be realizable in practice, and the pollution impacts should be fully defined in practical terms. Source reduction or pollution prevention (P2), is concemed mainly with modifying process conditions so as to hinder production of pollutants. Several approaches have been proposed in the literature for design and retrofit of chemical processes for source reduction [2]. Some investigators have integrated the approach with quantitative measures to determine the effectiveness of each different route of P2 [3]. Process simulators are found suitable for assessing the efficiency of those modifications [4]. Although helpful, unfortunately these methods do not provide the insight for proposing appropriate direction and magnitude of any modifications and the designer has to do that by inspection. Also, there is a need for quantitative measures to determine how effective each technique is.
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2. A R E V I E W OF PREVIOUS W O R K
2.1. WAR Algorithm In WAste Reduction, or WAR, algorithm [2], 9 potential impact indices are calculated from process material balance data and a relative potential impact score for each chemical. These are used to quantify pollution potential of altemative process flow sheets. Still it does not give the state space of the process operation and the overall impact index is rather uncertain. Nevertheless, by using this approach, in fact an implicit environmental model is included in the problem. In a retrofit project, the process model can be used for rating, and examining the feasibility, efficiency, costs and benefits of any candidate modifications. Combining process simulation model with an environmental model, i.e., macroscale simulation, gives a more comprehensive picture of the problem. 2.2. Process Synthesis
Mass Exchange Network (MEN) synthesis [5] and Total Site Analysis [6] are other solutions to the problem. In the former as extended to waste reduction studies, the network is defined as a system of separators and mass transfer units that can achieve minimal discharge of hazardous waste streams cost-effectively. However, if the final effluents of the process are assumed as the rich streams at the beginning of the design, as described in [7], it will be difficult or impossible to address improvements in the upstream process that has generated the rich (waste) streams; a kind of irreducible structures problem [8]. In this context, it may contribute to cross-media pollution transfer by employing mass separating agents that requires consideration of more precise trade-offs [9]. With a Total Site Analysis technique can predict (target) fumace emissions based on the relation between energy use and pollutants generation [6]. However, a comprehensive method for targeting process-related emissions when they are NOT directly related to energy use is not available and emission targeting is possible only if the emissions can be related to energy use [10]. In this approach, neither the process nor the environment is modeled. 2.3. Life-cycle Assessment
Life-cycle assessment (LCA) is intended for use as a decision support tool in improving environmental performance. It is applied to products and processes. In LCA approach, all activities in production and use of a specific product is considered 'from cradle to grave.' It is believed that in this way, it is possible to determine whether a product or service genuinely causes reduced environmental load, i.e., environmental impacts plus resource depletions, or whether the environmental load is merely transferred from the immediate supplier to other systems [ 11 ]. The applicability of LCA is increasing. However, certain problems still remain to be addressed or resolved in this methodology [11,12], mainly the highly data-intensive nature of LCA, serious difficulties in 'valuation' step of LCA, and its inability to quantify the ultimate limit of performance of the plant.
939 3. EHS RISK CRITERIA
The key advantage of risk assessment is that it offers a systematic framework with scientific foundations to assess and prioritize diverse risks to resources -typically health and safety, natural resources, public goodwill, and financial assets (e.g., equipment and production capacity)- and to make effective use of resources for protecting public health and the environment [13]. Lacking a broad context, there is typically no clear vision of the relative benefits of reducing a particular risk, the alternatives involved, time sensitivity, and potential trade-offs across different facilities and types of risk. The main focus of assessments has been on costs and compliance rather than on strategic advantages that could be derived from a new way of looking at EHS opportunities. To make strategic decisions on environmental policy of a plant, management should know where to optimally invest and to what direction to move. Since EHS have financial impacts on organizations from losses, understanding these risks can have a considerable impact on the company's health, and even survival. The objective is to evaluate risks so that informed decisions can be made about capital investments when different alternatives exist. For risk assessment the probability of occurrence of an event and the probable magnitude of its adverse effects are estimated. A set of scenarios (events) are generated and then their risk is calculated by trying to find: (1) The likelihood of occurrence of each event; and (2) The magnitude of its consequences. Risk can be a complex function of many different variables [13]. The solution is also not necessarily globally applicable, because of many local parameters affecting the assumptions, estimations and solutions. Moreover, it is important to apply EHS risks to normal operation of plants not to plant accidents. 4. PROPOSED APPROACH
First, a mathematical model of the process is built. Because of the level of sophistication and calculation power of today's simulation packages, it will be beneficial to use them for this purpose, whenever possible. In the next step, fate modeling for pollutants, an environmental model is incorporated in the simulation. Since the model should be flexible enough to be customized for each problem, use of a general-purpose spreadsheet program is recommended. Finally, with the use of case study tool within the process simulator and the spreadsheet, one should obtain a state-space of feasible operational modes of the process based on as precise a geometry of the equipment and the environmental conditions as can be reasonably justified. This is more important in retrofit projects. The information obtained in this step can be used for devising a feasible path based on practical, regulatory constraints. Authors have recently proposed a method to address both needs [14] that combines a process and an environmental model to track the pollutants OSBL and defines targets, i.e., the state-space of feasible operating conditions of the process.
940
The present paper extends the method further to evolve into a general framework and to perform the evaluations within more tangible practical engineering context. The updated approach has these new features: 1. Health and safety criteria are added to simultaneously study EHS liabilities 2. Diverse, national and intemational EHS criteria are included 3. The process model, the environmental model, and the EHS criteria are integrated into a common tool using a standard commercial simulation program 4. With a risk-based approach, the costs and benefits of modifications are determined. 5. CASE STUDY In this paper, Pollution Reduction Potential (PRP) in an existing 600-ton/day nitric acid production plant near Shiraz, Iran, is discussed. In this plant, a gaseous mixture containing nitrogen oxides contacts water in a reactive absorption column to form nitric acid. 5.2. Integrated Modeling Reaction mechanisms [15] along with their corresponding kinetic data are developed into the mathematical model of the absorption process and it is implemented in HYSYS, a commercial chemical process simulator with open architecture and extensibility facilities [16]. The process is simulated under different conditions. In this work, fate of pollutants (NOx), is modeled using a Pasquill-Gifford continuous three-dimensional deterministic Gaussian plume model (Figure 1). The parameters of the model are given in the literature [17] according to the stability class of the atmosphere. Holland empirical equation was used to account for buoyancy and momentum of the stack gas. Recorded meteorological data were used for tuning the model. Experimental maximum and average ground level concentration of NOx were used to check the validity of the calculated concentrations within accuracy margin of the dispersion model. EHS criteria are taken from Refs. [ 17] and [ 18]. 5.2. Sensitivity Analysis To optimize the process based on the maximum acceptable concentration of NOx, the process model was linked to the "'"' ..............................." .............. " - ' : i ; , i i ~ i ~ i g h " i i ~ ; ' ~ i U ~ ; t i , ' n " i " c ; i ~ ; . i i a i " S i u d ' y ..................................................... ~ D i s p e r s i o n C a l c u l a t i o n s f o r NO= atmospheric dispersion model. Stack Gas Temperature = 1.~3 "C :i:i:i:i: = 28.1 !:!:!:!:i., R 9 82,06 cruz*tin I glol K :~;~;~:~;" The latter was implemented in a o,,,~y. 067 =,,,, i ! i l i ! i : : ~r~ssiort Rate 9 25.00 k~'t~ ::::::::: Effective Stack Height 9 10919 , ~iii!::il.: spreadsheet (Fig. 1). Linking the : .... c . . . . . . . . . . ,,o..,~o~, .... i!iii!~!i two models made possible the Case 1. =ability Class A iiii::i::ii Cuooi; t~*~llated~t y = 0 m (plvmt ct~erlint} ] r,(m] =,. [ml C =,. {kg,"m3] C=.. [mgkn3) ~!~3~i!?ii study of the relationships :2e3'~ 13.'I9 ~09E.20 ,~09r_,.], ii!iiii:ii ......... 2110 52 ~ 1 29 56 ? DOE.I0 "/60~04 ":':':': between the ultimate ground 300 74 60 4132 2.16F-,08 2.16~-02 :i:!:i:i:i :.:.:.:.:. 350 85.43 6235 4.48E-08 4.$8F_,02 :!:!:!-!: 427 101.17 90.16 $37,~08 J.//g-02 iiiiiiiii level concentration of NOx with 42e 10198 90.55 $37F,08 $ .T'/F.,-02 !i~i!i!!ii 420 102.19 5.17F,,08 .~.77~02 :i:i:!:!: their exit concentrations, the 44O 104.49 9~3~ ~.~,oe ~.~r,,o2 ~iiii!iii Max. ~'ovnd I,,ewlr162162 r procedurel~Id~-: ":':::::: -'. 9 r r . ~ 9 l ~ L.L~-~_j_._':___.L_.L~ i!iii;~ii. ,,~ .................. ~,. . ~.t. ~.....:-.:~-:: : . ~ .:i.~.::f;!: , H i!i!i!i::: geometry of the stack, and the operating conditions of the Tkerefor+..C . . . . 0.0554.6 . 9 / . 3 i,,, ]: i:;: ~: :.~...:,:i... ~... ~... ~. 1 ..... C.,. dkectl~ from tab,l.. 0.05??0 m9,=.3 . . . . i':i':;:;!iii . . . . . =. . . . . . = . . = - " . , . , ' : .... I iii!i~:~i plant. The maximum ground I ;.~i .........
.......v.
9094
tk,
Fig. 1. The Environmental Model as Applied in the Study
941 level concentrations thus obtained were compared to different maximum acceptable levels as set by various legislators in different countries, as well as with local criteria. Sensitivity analysis, graphical representation of improvement trends, and environmental and health analyses were done within the spreadsheet. The results of the sensitivity analysis are summarized in Table 1. The last row of the table shows the extent of change in the three categories of parameters necessary to achieve a certain level of reduction in maximum ground level concentration of NOx. 6. RESULTS AND DISCUSSION
The results of the present investigation show that: 1. For a certain amount of C,,~x reduction, about 11% change in the amount of nitrogen oxides in the stack gas is necessary, compared to more than 14% for other variables (see Table 1). This clearly suggests the suitability of source reduction. 2. As the amount of NOx in stack gas is a function of the operating temperature and pressure of the absorption tower, these variables can be manipulated to find the optimum operating point of the tower. It is noteworthy that in this way the amount of product (nitric acid) produced is also affected in a favorable way. 3. Since nitric oxide (NO) is a pollutant as well as a reactant (feed), its absorption efficiency and the process yield vary in the same direction. In other words, we can have the benefit Table 1. Maximum ground level concentration Plant Parameters Exit Temp. (~ 0.0569 158 0.0551 169 0.0507 198 % Absolute change in plant parameter 25.4
ofNOx (mg/m3) vs. plant parameters Exit Pres. (kg/cm2) 0.875 0.836 0.748
Stack Height (m) 57 60 65
NOx Flow Rate (kg/hr) 25.0 24.5 22.2
14.5
14.0
11.2
of the amount of avoided pollution and the extra amount of useful product produced in this way, simultaneously. Therefore, our first trade-off is between reduced pollution plus higher production rate (favorable), and increased operating costs (non-favorable). 4. Altemative trade-offs can be generated. In this case, since NO2 is more hazardous than nitric oxide (NO), a second trade-off exists between reduced environmental impact and reduced benefits (less useful product). The trade-offs can lead to practical targets for pollution prevention as a retrofit tool.
942
7. CONCLUSIONS The approach used in this study has a number of advantages: 1. Real plant data were used for modeling contrary to a number of other papers. 2. In our approach, manipulation of operating variables can lead to pollution reduction, as opposed to other methods that require additional capital investments. 3. In this case, it makes the plant more profitable by providing extra capacity (nitric acid). 4. In an environmentally critical case, this approach helps plant owners to: - Decide on their environmental strategy (prevention, control, or both). - Gain insights as to how pollution problems can be possibly reduced Explore and implement the most cost-effective modifications within their plant. - (As a result of the previous steps) optimize their investment for modifications. - Obtain a common scientific basis for evaluation of problems with legislators. 5. The interface can be used with other simulation models. REFERENCES
1. S. Shanley, Chem. Eng., Nov. (1993) 30. 2. W.J. Lacy, in Riegel's Hdbk of Industrial Proc., J.A. Kent (ed.), Chapman & Hall, 1997. 3. H. Cabezas et al., Comp. Chem. Eng., in press (1999). 4. M.M. Dantus and K.A. High, Ind. Eng. Chem. Res., 35 (1996) 4566. 5. M.M. E1-Halwagi and V. Manousiouthakis, AIChE J, 35, 8 (1989) 1233. 6. V.R. Dhole and B. Linnhoff, Comp. Chem. Eng., 17 (1993)sl01. 7. K.P. Papalexandri et al., Chem. Eng. Res. Des., 72 (1994) 279. 8. R. Smith, Chemical Process Design, McGraw-Hill, 1995. 9. J. Lowe et al., Total Environmental Control, Pergamon Press, 1982. 10. B. Linnhoff, Chem. Eng. Res. Des., 71 (1993) 503. 11. R. Cliff and A.J. Longley, in Clean Technology and the Environment, Blackie, 1995. 12. B.P. Weidema, Keynote lecture at 2nd National Conf. On LCA, Melbourne, 2000. 13. R.V. Kolluru, Chem. Eng. Prog., June (1995) 44. 14. F. Nourai, D. Rashtchian and J. Shayegan, Proc. of PRES'99, Budapest, 1999. 15. N.J. Suchak et al., AIChE J, 37 (1991) 323. 16. HYSYS.Process Customization Guide, Hyprotech Ltd., Calgary, Canada, 1998. 17. F.P. Lees, Loss Prevention in the Process Industries, Butterworths, 2 nd ed., 1996. 18. AIChE CCPS, Guidelines for Chemical Process Quantitative Risk Analysis, 1989.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScience B.V. All rights reserved.
943
Soft sensor development and experimental application to a wastewater treatment process D. Zyngier ~', O.Q.F. AratSjo b and E.L. Lima ~ ~COPPE, Programa de Engenharia Qufmica, Universidade Federal do Rio de Janeiro Caixa Postal 6850, 21945-970, Rio de Janeiro (RJ), Brazil bEscola de Qufmica, Dept. de Engenharia Qufmica, Universidade Federal do Rio de Janeiro Centro de Tecnologia, B1. E, 21949-900, Rio de Janeiro (RJ), Brazil In this work, two soft sensors are proposed for monitoring concentrations of a few COml)ounds during nitrification of wastewater. One is a hybrid estimator while the other is based on Stacked Neural Networks (SNN), an approach that increases predictor robustness. After simulation, both soft sensors were implemented in an experimental unit with FIX MMI (lntellution, Inc) automation software as an interface between the process and MATLAB 5.1 (The Mathworks Inc.) software. 1. I N T R O D U C T I O N Since the beginning of industrialization era until the 70's, there had never been a great concern with wastewater treatment. As Governmental agencies developed stricter regulations specifying effluent quality, more complex wastewater treatment plants had to be built in order to remove specific nutrients, like nitrogen and phosphorus [1]. In a wastewater treatment unit, however, it may be very difficult to measure directly some process variables, either because there are no physical sensors available, or because these are too expensive. An alternative is to employ soft sensors in such cases to provide online estimates of difficult-to-measure variables through calculations that may involve auxiliary measurable variables. In this work, two soft sensors are proposed and implemented in an experimental process the nitrification of wastewater, which is of great importance during the nitrogen removal phase in biological treatment of wastewater. Due to their relevance attributed by legal rcslrictions in maximum concentrations, the selected variables to be inferred by the soft sensors are the concentrations of nitrate and ammonium ions, and of carbonaceous matter. 2. E S T I M A T O R S State estimators can be based on a process model from which process variables can be inferred. This is known as white-box approach, where physical relationships among process inputs and outputs are well established. Amongst the most widely applied estimator under this denomination is the Kalman filter [2], which accounts for process and measurement noise influences on process variables inferring procedure.
944
When no information about the physical links between process variables is available, the estimators must be based on cause and effect relationships, in a black-box approach. Neural networks (NN) are one of the best known black-box predictors, which, due to their complex structure, are able to represent a wide variety of processes [10]. 2.1. Extended Kalman filter (EKF) Although the Kalman filter has originally been developed for linear systems, a linearization can be conducted on a nonlinear process. The nonlinear model is used in a prediction step, while its linearized version is used when calculating the gain matrix for a correction step. Such an approach is denoted as Extended Kalman Filter (EKF), which has shown good results for many highly nonlinear processes, such as polymerization and bioprocesses [3, 4, 5, 6, 7, 8]. Due to specificities of a system (for example, infrequent or delayed measurements), some variations on the EKF have been developed and successfully implemented by [3, 8, 9]. 2.2. Neural Networks It is known that NN are able to represent a great variety of processes. Nevertheless, their development requires some attention. Because the mathematical correlation obtained lacks physical meaning, the extrapolation capacity of these predictors is generally very limited. Therefore, data used for developing the neural network need to represent the whole operation range of interest, so that it "learns" what the best correlation between the variables is [11]. Depending on the process' nature and its operating scale, however, obtaining large amounts of representative data can be a very difficult or even impossible task [12, 13]. A challenge when developing a new NN is choosing its architecture, for if a different conliguration (number of hidden layers and neurons, NN inputs, activation function) is c:h~+sen, its performance will probably be affected. To select the best architecture, each candidate NN must be tested with a validation data set after the training phase. Developing and validating different types of NN, however, may be very time-consuming. An approach that has shown good results is to combine several individual and architecturally simpler neural networks to provide improved robustness to the stacked neural network. Wolpert [14], who has introduced stacked generalization, states that the purpose of this technique is to achieve greater generalization accuracy as opposed to learning accuracy. This means that, even though the predictor may not have the best performance on training data. it is able to adequately capture process behavior, being thus more robust. 3. OFFLINE APPLICATION: GRAY-BOX SOFT SENSOR A soft sensor was built based on a simplified model of the process, previously developed by Coelho [15]. Since the EKF had a number of successful applications to bioprocesses [3, 6, 7], it was first chosen as soft sensor for this system. The inferred variables were the concentrations of nitrate (NO) and ammonium (NH) ions, and of carbonaceous matter (CM). FIX MMI automation software (Intellution, Inc.) was configured in order to have a "userfliendly" interface in the experimental unit, simplifying process monitoring task. In the studied application, two sets of delayed measurements exists: each ion concentration can be measured offline, with a sampling period of 60 minutes, while carbonaceous matter, which is determined by Chemical Oxygen Demand (COD) method, can only be determined z~t 3 l~our intervals. In order to deal with these delayed, offline measurements, two different alternative soft sensors (SS) were considered:
945 S,~,'I: An iterated EKF, where the variables are reestimated at each CM update. The previous CM values (fi-om the sampling instant to the updating moment) are replaced by the values obtained through the EKF, while the other states are not replaced. Whenever a NH and NO update becomes available, previous values estimated for the three variables are replaced by values obtained through the EKF. ,";$2: A reintegration method, where instead of reestimating values through an EKF (as in SSi), a simple reintegration of the process model is conducted each time a measurelnent becomes available. Experimental data fiom [15] were used to evaluate the performance of each soft sensor. It was verified that SS2 showed superior performance when compared to SSI for NH and NO csti~nation, but none achieved good results for CM estimation. The reason CM inference was not satisfactory is probably because the nitrogen compounds have no mathematical influence on CM in the process model [15]. Fience, CM estimation was next approached as a black-box model which, combined to NO and NH inference, formed a gray-box predictor. A feedforward neural network was employed, which, according to [16], is the most commonly used network in studies with neural networks. Since the three variables have well-defined reaction profiles, and as the process has two distinct time patterns - a filling phase and a batch reaction phase - two networks were trained, both with three inputs (NH and NO from the EKF, and dissolved oxygen, which is an online measurement), six hidden neurons and one output (CM). The activation function used was the hyperbolic tangent function. 4. OFFLINE A P P L I C A T I O N : B L A C K - B O X SOFT S E N S O R As previously mentioned, Stacked Neural Networks (SNN) are built through the combination of several individual neural networks. S(~me authors have recently used Stacked Neural Networks (SNN) as predictors [17, 18, 191]. They seem to agree on the fact that no general rule exists on how to determine the slacking weights in order to combine individual estimators. Least Squares Regression was tise(t in [1O], while [17, 18] recommended using Principal Component Regression or a \vcighe(l average calculated by using the individual NN training error. In this work, four different types of stacking weights were evaluated: I,V/ - "a simple weighed average of the individual NN outputs; I , I / 2 - the weighed inverse of the individual NN prediction error; t'V3 - weighs the sum of all individual NN prediction errors minus the individual NN error; 1 4 / 4 - calculated by Principal Component Regression (PCR) technique. Evolution of the standard deviation of the weights assigned to each NN was analyzed with increasing nulnber of stacked neural networks (NS), for a given data set. It was observed that, ~s NS increases, W2 and W3 standard deviations tend to zero. Assuming that there are enough individual NN, such results lead to the conclusion that it is practically equivalent to adopt W1 ~r ~lle of the proposed weighed averages (W2 and W3), since W1 standard deviation is always zero (all the individual NN are always assigned the same weights in this case). When using W4, no evident pattern was observed. Twenty-five individual NN were trained for carbonaceous matter (CM) prediction. Data from two exl)eriments [15] were separated into training and validation sets. Figure 1 shows the 25 individual NN errors (calculated through normalized residual sum of squares - NRSS) l~l illc training and validation sets.
946
An important aspect of stacking NN is the decision on when to stop the stacking process; that is, when the ratio cost/benefit stops increasing, where cost is related to the parameters and benefit is the reduction in the normalized residual sum of squares (NRSS). The stacking process is pictured in Figure 2, where the NN were stacked in increasing order of architecture complexity. Eighteen individual NN were chosen as the optimal number of NN to stack. An interesting point is that many of the individual NN with complex architecture did not perform adequately in the validation phase. Although they could be expected to diminish SNN performance, only when using W4 such fact was verified. It also can be seen in Figure 2 that, except for the PCR method (W4), there was little difference between the stacking strategies. Although the four types of SNN had similar performances, W l (average) was chosen based on the fact that it is the simplest of them all, making the system easier to build and to maintain. The same procedure was repeated for NO and NH concentrations, where the optimal number of networks to stack were l6 and 14, respectively. T(~ develop new Stacked Neural Networks, the following algorithm is proposed: ,S'let) ]. Split the available process data into training and validation sets, bearing in mind that data should be rich in process information; ,S~le/) 2. Develop approximately 15 different NN structures, trying to make them as simple as possible (with the smallest number of parameters). Creating a minimum number of neural networks is necessary to avoid a local minimum (as can be seen in Figure 2, when only four NN had been stacked); ().()1()() , ................................H...............DNRSStraining ...... ~ ().()()5()
]]
1
2
3
4
5
Vl NRSS validation
6
7
8
9
]l ...................................................................................i........................... I!
~~
~
~
i
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Individual NN Fig. 1 - Individual NN errors ().()()5() ........................................................................................................................................................................................................................... IWl DW2 ~W3 V-1W4 c~ ().()()25 z
I
2
3
4
5
6
7
8
9
10 I I 12 13 14 15 16 17 18 19 20 21 22 23 24 25
NS Fig. 2 - Stacked NN errors
947
,gle t) 3. Train the developed NN and stack them. The stacking process should start with the network with simplest architecture and continue with growing network complexity. s l) 4. Calculate SNN prediction error for each additional NN added to the stack. W h e n e v e r lhe error stabilizes, it is time to stop the stacking process. "Fhe algorithm proposed above intends to minimize the number of NN needed to develop lhe predictor. Furthermore, it has been observed that SNN performance is not affected by stacking individual NN in excess (there is no overfitting). 5. O N L I N E
APPLICATION
TO A BIOPROCESS
Coelho [15] developed an optimal operational strategy for an experimental Sequence Batch Reactor (SBR) bench-scale unit (35 liters). The author verified that there was maximum nitrogen removal when the filling was made by pulses and with aeration, followed by anoxic batch reaction phase. Both process model and experimental data from [15] were used in the development of the soft sensors herein presented. The performance of the two sensors can be seen in Figures 3, 4 and 5. ~l'l~e gray-box estimator is based on a simplified process model to infer NH and NO, and so requires less modeling and maintenance efforts than a complex model does. Besides, whitebox models are usually able to represent the system even during small deviations in process operating conditions. The black-box soft sensor also showed good results for this process. Its development and operation is simpler than the gray-box predictor's, as a process model is not required and there is no need for model parameter adjustments during the reaction. Another advantage is that it does not require updating with offline measurements since its inputs are variables that are available online, thus making it easier to operate. Nevertheless, this sensor should be rctrained each time process operating conditions are altered, since its performance is strongly (tependant on the data set used for training purposes. I()()
cal00 ~ . 9 Experimental Gray-Box Sensor ~, ....... Black-Box Sensor ca50 ,
~-O 5()
"
9
co ()
I,..
[.
.
0
1
.
.
0 ()
I
o
"~
4
tili-e (hours)
5
6
7
Fig. 3 - Sensors for CM Estimation 5()
9 Experimental ~ Gray-Box Sensor ......... Black-Box Sensor
~25 i Z () I
9
.3
4
5
6
- tllile(hours) Fi~. 5 - Sensors for NH Estimation
. A-.-Ik---A ""
..it-
2
.3
" Gray-Box Sensor ......... Black-Box Sensor 4
time (hours)
5
Fig. 4 - Sensors for NO Estimation
6
7
948 6. C O N C L U S I O N S AND D I S C U S S I O N
Two soft sensors were proposed for inferring variables that are difficult to measure online in a wastewater treatment process. The first one was based on the reintegration of a simplified process model for nitrate and ammonium ions, with considerable process mismatch, while carbonaceous matter was inferred by a feedforward neural network. Good results were achieved, suggesting that this soft sensor, besides not having tuning parameters, is robust. Nevertheless, phenomenological models may require greater effort during the development phase. The second soft sensor overcomes such limitation by employing blackbox models. The main disadvantage of using this soft sensor is the same of using any blackbox model: it cannot be used for predictions outside the range covered by training data values. That implies on retraining the predictor each time process conditions are altered. ACKNOWLEDGEMENTS The authors would like to thank CAPES (Funda~:5.o CoordenaqS.o de Aperfeiqoamento de Pessoal de Nfvel Superior), CNPq (Conselho Nacional de Desenvolvimento Cientffico e Tecnol6gico), FUJB (Funda~5.o Universitfiria Jos6 Bonifficio) and FAPERJ (Fundaqg.o de Amparo h Pesquisa no Estado do Rio de Janeiro) for their financial support. REFERENCES
1. K. Gernaey, A. Vanderhasselt, H. Bogaert, P. Vanrolleghem and W. Verstraete, J. Microb. Methods, 32 (1998) 193. 2. D.I. Wilson, M. Agarwal, and D.W.T. Rippin, Comp. Chem. Eng., 22 (1998) 1653. 3. T.J. Crowley and K.Y. Choi, Chem. Engng. Sci., 53 (1998) 2769. 4. D.G. Robertson, J.H. Lee and J.B. Rawlings, AIChE Journal, 42 (1996) 2209. 5. W.W. Woo, S.A. Svoronos, H.O. Sankur, J. Bajaj and S.J.C. Irvine, AIChE Journal, 42 (1996) 1319. 6. L.F.M. Zorzetto and J.A. Wilson, Comp. chem. Engng., 20 (1996) $689. 7. M.A. Myers, S. Kang, and R.H. Luecke, Comp. Chem. Engng., 20 (1996) 585. 8. D.J. Kozub and J.F. MacGregor, Chem. Engng. Sci., 47 (1992) 1047. 9. L.J.S. Lukasse, K.J. Keesman and G. van Straten, J. Proc. Control, 9 (1999) 87. 10. A.J. Morris, G.A. Montague and M.J. Willis, Trans. IChemE., 72 Part A (1994) 3. 11. J. Glassey, G.A. Montague, A.C. Ward and B.V. Kara, Biotech. Bioeng., 44 (1994) 397. 12. B. Schenker and M. Agarwal, Comp. chem. Engng., 20 (1996) 175. 13. Q. Zhang, J.F. Reid, J.B. Litchfield, J. Ren and S.-W. Chang, Biotech. Bioeng., 43 (1994) 483. 14. D.H. Wolpert, Neural Networks, 5 (1992) 241. 15. M.A.Z. Coelho, Ph.D. Thesis - COPPE/UFRJ (1998). 16. G. Montague and J. Morris, TIBITECH, 12 (1994) 312. 17. J. Zhang, E.B. Martin, A.J. Morris and C. Kiparissides, Comp. chem. Engng., 21 (1997) S1025. 18. J. Zhang, E.B. Martin, A.J. Morris and C. Kiparissides, Chem. Engng. Journal, 69 (1998) 135. 19. D.V. Sridhar, R.C. Seagrave, and E.B. Bartlett,'AIChE Journal, 42 (1996) 2529.
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rights reserved.
949
Computer Aided Technique for Pollution Prevention and Treatment Peter M. Harper and Rafiqul Gani* Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark 1
ABSTRACT A framework for the identification of environmentally benign and altemative solvents is presented. The methodology for solvent design and selection contains multiple stages: 1. problem formulation, 2. constraint selection, 3. candidate identification, 4. verification and final selection. The candidate identification step can be performed using a database search method or a Computer Aided Molecular Design (CAMD) approach. A multi level CAMD method capable of generating a selection of candidates meeting the required specification without suffering from combinatorial explosion is presented and the entire framework is illustrated with a case study. The case study identifies alternative solvents for Oleic acid methyl ester as replacements for Ethyl ether and Chloroform. Keywords: Solvent, substitution, CAMD, group contribution, pollution prevention, process synthesis 2
INTRODUCTION One of the principal causes of pollution is the presence of a substance (or substances) in amounts higher than the allowed maximum in one or more streams released to the surroundings. Substitution of the polluting substance (or substances) by another that performs the same function in the process but is environmentally benign is one way of dealing with such environmental problems. Computer aided molecular design, commonly known as CAMD, is well suited to solving this class of environmental problems (pollution prevention and/or treatment) because it can design/find the candidate replacements more efficiently than other search techniques. Current applications of CAMD within this area have addressed environmental concerns and constraints using simple property estimation methods based on the group contribution approach. While this approach is sufficient for some problem formulations, in their present form the group contribution approaches are unable to handle the complex molecular structures of some of the chemicals responsible for causing pollution and are unable to predict the properties of interest with sufficient accuracy. Therefore, it is desirable to employ more appropriate property estimation methods that require a higher level of molecular information (for example, QSPR and QSAR methods) in order to assess environmentally important properties more accurately and "tap-in" to the vast knowledge-base consisting of already developed methods. Also, most CAMD methodologies so far have designed relatively simple compounds by collecting fragments into group vectors (Mavrovouniotis (1998) lists a series of examples with references in his review of CAMD). While the generation of group vectors suits property prediction using group contribution approaches, the generated compound descriptions, however, do not contain the additional structural details needed for the QSPR or QSAR methods, for example, a 3D representation of the molecular structure. It is therefore necessary to generate more detailed molecular descriptions in the CAMD algorithm. * Author to whom correspondence should be addressed
950 However, with an increase in molecular detail there is an associated increase in the size of the search space and computational complexity. It is therefore necessary to take precautions in order not to face a combinatorial explosion leading to unsolvable problems. In this paper, a process systems engineering approach is applied to prevention and/or treatment of pollution through an integrated set of computer aided tools. The proposed computer aided technique integrates molecular-level (microscopic) information with the current group contribution based approaches (handles only macroscopic information) in order to open new horizons of applicability and accuracy of CAMD with emphasis on pollution prevention or treatment. This systems engineering approach has led to the development of a multi-stage, multi-level methodology. 3
SOLUTION METHODOLOGY The method of solution for the compound design and selection problem is an iterative process consisting of multiple stages some of which containing multiple steps. 3.1
Stage 1 In stage one, the causes of pollution are identified together with the polluting substances and their undesirable properties. Once the causes have been identified it is necessary to formulate the strategy for solving (curing) the pollution problem. Pollution Loss of environmentally harmful substance to the environment via waste streams.
Cause Byproduct from process.
Loss of environmentally harmful substance to the environment via waste streams.
Process fluid (e.g. solvent) used is lost through waste streams.
Air emissions from energy production.
High energy use
The production of process fluids used causes pollution. Table 1 Examples of pollution types, causes and cures
Pollution from raw materials.
Cure Control of emission by removal of pollutant prior to discharge (by separation process). Change in operating conditions resulting in minimization/elimination of the generation of pollutant. Identification of replacement solvent having a lower environmental impact and/or lower unavoidable loss to the surroundings. Changes in operating conditions resulting in zero avoidable loss of process fluid. Reclamation of lost process fluid prior to discharge (separation process) Optimize operating conditions to lower energy consumption. Replace process fluids with more energy efficient alternatives. Replace process fluids with alternatives having a less harmful production pathway (without introducing post process pollution).
The routes of information leading to the identification include simulation, engineering knowledge, regulatory requirements, observations of existing practices as well as changes in environmental policy. Examples of possible pollution scenarios are listed in Table 1. C A M D can be used in the instances where the cure either involves the replacement of a process fluid or removal of a pollutant by using a solvent based separation technique. In the case of replacement solvents the general process equipment and operations have already been fixed and the substitute must function in all of them. ff the search is aimed at finding a compound for use in a removal operation there are additional degrees of freedom since the separation method has not been fixed. In such cases different searches can be performed for the various feasible separation techniques. Determining the set of feasible separation techniques to consider is a separate sub-problem involving process design techniques.
951 3.2
Stage 2
In stage two, the pollution prevention or treatment problems are formulated in terms of desirable and undesirable properties for the replacements or solvents. From an environmental point of view it is obvious that the properties of interest include environmentally related properties like: ozone depletion potential, bioconcentration factor, global warming potential, toxicity considerations, biodegradability. It is however also necessary that the compound fulfills its operational role and there are therefore additional specifications that depend on the type(s) of operation(s) the compound is to take part in. A computer based knowledge base is used to assist in the selection of the application-related properties and their values. 3.3
Stage 3
In stage three the identification of compounds possessing the desired properties is performed. Two different approaches are possible: (1) Searching a database of compounds combined with mixture calculation procedures. (2) Generation of compounds, matching the specifications, by assembling building blocks. The evaluation of properties is performed using predictive techniques. If the database approach is used the most reliable results are obtained because identification is primarily based on experimental data. However, the obtainable results are restricted to the number of compounds in the database and the amount of available data for each compound. The limitation of available data is broken if the generation approach is used. By combining fragments to form molecules a wide range of compound can be generated and screened. The limitations of the generation approach lie in the availability and accuracy of the prediction methods. Also, due to the very large number of structural alternatives possible there is always the risk of encountering the so-called "combinatorial explosion" (Joback & Stephanopoulos, 1989) problem associated with structural enumeration, especially when designing molecular representations having a high degree of structural detail. The methodology used for the generation approach is a multi-level method for computer aided molecular structure generation and property prediction. The computational complexity is controlled using two techniques: (a) Partitioning: by subdividing the generation procedure into several successive levels with a screening step between each level (allowing only the most promising candidates to progress to the next level) ensures that computational efficiency is maintained. (b) Feasibility: ensuring that only chemically feasible structures are generated not only improves the quality and ease of interpretation and analysis of the results but also eliminates the computational resources spent on false solutions. The developed method consists of four levels. The first two levels operate on molecular descriptions based on groups while the latter two rely on atomic representations (Harper et al., 1999). In outline form the individual levels has the following characteristics: 3.3.1 Level 1 In the first level, a traditional group contribution approach (generation of group vectors) is used with its corresponding property prediction methods. Group vectors are generated using a set of building blocks as input. The employed approach does not suffer from the so-called "combinatorial explosion" as it is controlled by rules regarding the feasibility of a compound consisting of a given set of groups (Harper et al., 1999). Only the candidate molecules fulfilling all the requirements are allowed to progress onto the next level. 3.3.2 Level 2 At the second level, corrective terms to the property predictions are introduced. These terms (so called second-order groups) are based on identifying substructures in molecules. At this level molecular structures are generated using the output from the first level (first-order description) as a starting point and the second order groups are identified using a pattern matching algorithm developed specifically for this purpose. The generation step of this level
952 is a tree building process where all the possible legal combinations of the groups in each group vector is generated. 3.3.3 Level 3 In the third level, molecular structures from the lower levels are given a microscopic (atomic) representation by expanding the group representations in terms of the atoms each group is made up from. This can generate further structural variations. Furthermore the conversion into an atomic representation (including connectivity) enables the use of QSAR/QSPR methods as well as structural analysis methods. The possibility of using QSAR/QSPR methods and structural analysis significantly increases the applicability of CAMD in environmental applications since many environmental properties are only possible to estimate using such techniques and the available techniques are very specific with respect to the compound types they are applicable to. As an added benefit the structural analysis enables the re-description of the candidate compounds into other group contribution schemes thereby further broadening the range of properties that can be estimated as well as giving the opportunity to estimate the same properties using different methods for comparison. 3.3.4 Level 4 In the fourth level the atomic representations from level three are further refined to include the 3D position of the individual atoms. This conversion gives the opportunity to create further isomer variations (cis/trans and R/S) and is performed in a way that the output is compatible with most molecular modeling applications. Since property prediction using molecular modeling is a task difficult to automate the estimation and screening process is done interactively. Note however that this fourth level is not a necessary step in all applications. Regardless of the approach used (database or design) the result of a successful completion of the algorithm is a list candidate molecules. All the candidates fulfill the property requirements set as design criteria. 3.4
Stage 4
In the fourth and last stage, the final selection from the generated list of feasible candidates is made. The final selection is done after careful analysis of the identified candidate molecules. Even though the results from stage 3 fulfill the property requirements there are properties and criteria that are difficult to handle using automated prediction methods and must be assessed using extemal sources. Examples of such criteria and properties are: Availability, Price, Regulatory restrictions, Long term health effects, Detailed environmental fate, Process-wide behavior. The methods used to assess the additional considerations include external databases as well as other computational tools such as process simulators, environmental fate models and phase behavior calculators. Which tools to use depend to a large extent on the type of application the compound is being designed for and the range of applicable tools available. It is an obvious advantage if the various tools and data sources used are tightly integrated in order to have a easy collection and flow of information. After analyzing the candidate compounds the final candidates must be selected for experimental testing or rigorous simulation. The selection can be performed by formulating an objective function based on compound properties, by using engineering insight and experience or by only considering the candidates that are known to exist on-site and are being used in other processes. Another promising altemative is the integration of the selection into a general computer aided process design problem where the choices of compound can be represented as discrete choices in a MINLP superstructure formulation (Hostrup et al., 1999) Regardless of the approach used for the selection of final candidates, the primary function of CAMD - identifying a set of candidates having the properties needed for a particular application - has been achieved.
953 4
CASE STUDY The fatty acid ester "Oleic acid methyl ester" ((Z)-9-Octadecenoic acid, Methyl ester) is an important compound in a variety of applications, such as: intermediate for detergents, emulsifiers, wetting agents, stabilizers, textile treatment, plasticizers for duplicating inks, rubbers, waxes, biochemical research and as a chromatographic reference standard (NTP, 1999). Reported pure component solvents for Oleic acid methyl ester are: Diethyl ether (NTP, 1999) and Chloroform (CAPEC-Database, 1999) with Diethyl ether being reported as the best solvent. While both of the reported solvents are effective they also have unwanted properties. Diethyl ether is very volatile and flammable (including the risk of formation of explosive peroxides) and Chloroform is a suspected carcinogen. It is therefore desirable to identify alternative solvents that are safer and more environmentally benign than the above mentioned. The actual identification of the candidate solvents is done using the database approach as well as the molecule generation approach. 4.1
Stage 1 Determine a solvent having the following characteristics: (a) Liquid at (ambient) operating conditions. (b) Is non-aromatic and non-acidic (stability of ester). (c) Has low environmental impact and poses limited health and safety problems. (d) Is a good solvent for Oleic acid methyl ester. 4.2
Stage 2 The goals from stage 1 can be formulated as property constraints using the following values: Melting Point (Tin) < 280 K, Boiling Point (Tb) > 340 K. The requirement of low environmental impact can only be addressed in part using property and molecular type constraints (non-aromatic compounds). The true environmental behaviour of a candidate compound must be assessed in stage 4 as part of the analysis of the candidates identified from stage 3. However, it is possible to address some environmental considerations via a property constraint: (a) Compounds must be acyclic and must not contain C1, Br, F, N or S. (b) Octanol/Water Partition coefficient (logP) < 2 (lower is better). The determination of solvent ability towards Oleic acid methyl ester should ideally be calculated using an activity coefficient approach. However, since the solute in question is quite complex and very few predictive methods (e.g. UNIFAC, ASOG) are capable of handling large compound with complex structures the solubility requirement is addressed using a solubility parameter approach. Based on the theory of solubility parameters, a good solvent has a solubility parameter that is close to that of the solute. In the case of Oleic acid methyl ester the solubility parameter is 16.95 (MPa) ~ (CAPEC-Database, 1999). The solubility criteria than then be formulated as: 15.95 (MPa) ~ < Solpar < 17.95 (MPa) ~
4.3
Stage 3
The constraints and design criteria formulated in stage 2 is solved using the database search approach as well as the CAMD approach.
4.3.1
Database approach
Using the specifications from stage 2 and searching the database (containing more than 10000 compounds) two compounds are identified: 2-Heptanone, Diethyl Carbitol.
4.3.2
CAMD approach
Using the formulated CAMD problem with the added constraint of only allowing two functional groups in a compound (prevents generation of very complex and thereby expensive compounds) the following results are obtained: 9 In level 1 of the CAMD procedure 2691 vectors of groups were created. After screening against the constraints 425 representations remained and were passed onto the next levels.
954 9 4593 molecular structures were created in level 2 based on the input from level 2. After screening 1351 candidates were passed on to level 3. 9 No additional isomer forms were generated in level 3and no screening was necessary (all properties had been handled in level 2). 9 The final result from the CAMD approach was a total of 1351 compounds. 9 The total time spent was 45 seconds using an AMD-K6-2 (350MHz) processor with 64 MB (equivalent to a Pentium-II processor at 233MHz). 4.4
Stage 4 Determining which of the 2 compounds, identified using the database approach, is the best compound is difficult since both alternative compounds have EH&S issues. 2-Heptanone can cause liver and kidney damage with prolonged exposure (NJDHSS, 1999) while Diethyl Carbitol can form explosive peroxides (NTP, 1999). The final choice of compound depends on the typical usage environment and the types of operations the solvent will be used in. In order to select the prime candidates from the 1351 alternatives obtained from the CAMD solution an extensive analysis must be performed on the candidates. If only performance considerations are taken into account (i.e. how close the solubility parameter matches that of the solute) the following candidates are the most promising: Formic acid 2,3dimethyl-butyl ester, 3-Ethoxy-2-methyl-butyraldehyde, 2-Ethoxy-3-methyl-butyraldehyde. A more rigorous analysis has been performed but cannot be reproduced here due to the page limitation. The results are obtainable from the authors on request. 5
CONCLUSION The algorithm outlined above provides an opportunity to solve pollution prevention and/or treatment problems in a more rigorous manner since widely used and more accurate property estimation methods can be used without sacrificing efficiency of the method of solution. This enables the user to find solutions that not only protects the environment but also has a high environmental benefit and/or process efficiency. The process systems engineering approach has combined aspects of computational chemistry, property prediction, process design and optimization for the solution of problems of current and future interest. A case study involving replacement solvents with environmentally acceptable substances has been presented. The case study highlight the application of the set of integrated tools needed to solve the environmental problems, the efficiency/flexibility of the multi-level computer aided technique and the analysis/validation of the computed results. REFERENCES NTP, 1999 "Chemical Health & Safety Data", National Toxicology Program, Online database. NJDHSS, 1999, "Right to Know Program", Online database, New Jersey Department of Health and Senior Services. CAPEC-Database, 1999, R. Gani, T. L. Nielsen, P. M. Harper and J. M. Harper. Harper, P. M., R. Gani, T. Ishikawa and P. Kolar, 1999,"Computer Aided Molecular Design with Combined Molecular Modeling and Group Contribution", Fluid Phase Equilibria, vols 158-160, p 337-347. Joback, K. G. and G. Stephanopoulos, 1989, "Designing Molecules Possessing Desired Physical Property Values", FOCAPD '89, Snowmass, CO, p363 Hostrup, M., P. M. Harper and R. Gani, 1999, "Design Of Environmentally Benign Processes: Integration Of Solvent Design And Process Synthesis", Computers and Chemical Engineering, 23, 1395-1414 Mavrovouniotis, M.L., 1998, "Design of chemical compounds", Computers and Chemical Engineering, 22, 713-715
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rightsreserved.
955
Using driving force based separation efficiency curves within an integrated system for process synthesis/design Erik Bek-Pedersen a, Martin Hostrup a, Rafiqul Gani a* aCAPEC, Dep. of Chemical Eng., Tech. Univ. of Denmark, DK-2800 Lyngby, Denmark
Abstract This paper presents an integrated system for process synthesis and design. The integrated system combines three different algorithms, a driving force based separation process synthesis/design algorithm, a thermodynamic insights based process synthesis/design algorithm and an interactive MINLP solution algorithm. The combined integrated system follows a three-stage procedure. The driving force based algorithm and the thermodynamic insights based algorithm help to formulate the problem, generate a superstructure and determine a good (near optimal) initial feasible solution. The interactive MINLP solution algorithm determines the final optimal solution. Two application examples illustrating the main features of the integrated system are also presented. Keywords: Driving force, integrated system, synthesis, design, optimal solution, MINLP 1. Introduction Most separation techniques use driving forces from differences in thermodynamic properties of the mixture compounds and their rates are governed by pure component and mixture properties. By using the insights from the analysis of driving forces, it is possible to make decisions regarding flowsheet design, operational conditions, as well as initialization for the related optimization problem. Synthesis and design of process flowsheets involves generation and identification of feasible flowsheet alternatives, process/solvent design (such as, separation column design, reactor design, solvent design) and structural optimization. The framework used here incorporates tools to handle the above sub-problems in an integrated manner, where it is possible to solve interactively classical synthesis problems as well as retrofit problems. Separation efficiency curves, calculated from the driving forces, provide a useful tool for generation of feasible process flowsheets and design of conditions of operation. Driving forces based on different sets of properties can be related to different separation techniques. In addition to feasibility of a separation technique, the ranges of temperature, pressure and/or composition over which these driving forces are large enough can even indicate the likely limits of operation. For example, at the azeotropic composition or temperature, the driving force based on relative volatility becomes zero indicating separation beyond the azeotropic composition is infeasible by distillation. When the driving force is too small, separation becomes infeasible or difficult, while, as the driving force approaches its maximum value, the separation becomes very easy. From an operational point of view, a process should be designed/selected to operate at the highest possible driving force. Plotting
*Author to whom correspondence should be addressed: Fax: +45 4588 2258, emaih [email protected]
956 the driving forces on 2-dimensional plots of driving force versus composition allows us to visually configure a feasible separation system. The objective of this paper is to present an integrated system for process synthesis/design that uses driving force based separation efficiency diagrams. The integrated system combines the thermodynamic insights based process synthesis/design algorithm, Jaksland et al. (1995) with the idea of driving force based separation efficiency curves and the interactive MINLP solution strategy for structural optimization problem, Hostrup et al. (1999). Two illustrative examples are included to highlight the new features of the interactive system. 2. M E T H O D O L O G Y A three stage algorithm has been developed for the integrated system. In the first stage (Problem Formulation Stage), the different process alternatives are identified together with selection/design of solvents and materials. The driving force based separation efficiency curves together with the extended thermodynamic insights based algorithm for integration of synthesis and design (Hostrup et al., 1999) is used for this step. In the second step (superstructure generation and initial feasible flowsheet), the identified process alternatives are screened through the use of driving force based separation efficiency curves (where applicable) and analysis of mixture properties. The reduced set of process (feasible) alternatives is represented through a superstructure and an initial feasible flowsheet is generated. The results from the second stage form the basis for the mathematical formulation and solution of the problem. Here, the interactive MINLP solver is employed. Gani and Bek-Pedersen (1999), defined the driving force, Fi~, for the binary pair of component i and j, as,
Fi? Yi -
(1)
Xi~ij X i ~-
- Xi
1 + xi (13~j-1) Where, yi and xi are the phase compositions of component i and [30 is the relative separability factor for the binary pair. Note that for vapour-liquid equilibrium, [30 is the relative volatility. Through the above equation, it is possible to model equilibrium as well as rate based separation processes (such as rate-based distillation and pervaporation). The main feature of the driving force methodology is to configure and design separation processes such that the total driving force is at its maximum. Plotting the driving forces on 2-dimensional plots of driving force versus composition allows us to visually configure a feasible separation system. A novel feature of this method is that the determined values of the design variables correspond to optimal (or near optimal) solution with respect to cost of operation, without requiring any rigorous simulation or optimization, Gani and Bek-Pedersen (1999). 2.1.
Distillation Column Design Consider the following problem - given a mixture to be separated into two products in a distillation column with NP plates. What is the optimal (with respect to cost of operation) feed plate location and the corresponding reflux ratio for different product purity specifications? The solution involves the following steps (algorithm 1). 1. Generate or retrieve from a database, the vapor-liquid data for the binary system. For a multicomponent system, select the two key components to define a binary "split". 2. Compute FD1 using Eq. 1 and plot FD1 as a function of xl. 3. Identify the points Dy[maxand Dx[max. 4. Determine NF from NF = (1 - Dx) NP.
957
2.2.
Configuration of distillation columns When configuring a series of separations for a multicomponent mixture, the order of the separations should be done such that the total driving force is at its maximum. The following steps are followed (algorithm 2): 1. Retrieve the necessary vapor-liquid data for the separation techniques to be considered. 2. List all the components in the mixture, NC, according to the size of [3ij. 3. Calculate the driving force diagrams for the adjacent components, preferably all at the same operating pressure or temperature (usually 1 atm.). In total, driving force diagrams for NC-1 components are calculated. Set k = 1. 4. For the split k, select the adjacent pair with the largest driving force value, Dy. 5. Remove the split between the selected adjacent pair from the list. Distribute the components according to [3ij into two products. Set k = k+l. For each product stream, if more than one pair remains, repeat the algorithm from step 4. Otherwise, go to step 6. 6. Add the separation technique(s) for the selected components to complete the flowsheet. 7. For each distillation column in the flowsheet, apply the single column design algorithm. In case the mixture is not to be separated into pure components, but the mixture is to be separated into fractions of more than one component, then there are less than NC-1 neighboring adjacent key components, there is a non-sharp separation and
2.3.
Integration aspects The combination of the driving-forces based synthesis/design algorithms with the extended Jaksland algorithm for thermodynamic insights based synthesis of process flowsheets (Hostrup et al. 1999), makes it possible to generate the list of feasible alternatives that form the basis for a comprehensive superstructure. The next step in the integrated system is to use the generated superstructure to determine the optimal process flowsheet through formulating and solving a structural optimization problem. For structural optimization problems, a framework for applying MINLP solution methods in an interactive manner has been incorporated into the integrated system. Figure 1 shows the information flow diagram of the interactive algorithm. Here, the NLP sub-problem is solved by the process optimization feature of the integrated system. Through an user-interface which the designer can control, the MILP master problem is set-up and solved in an outer-loop through a MILP solver. In this way, any MINLP problem can be set-up in the integrated system and solved interactively. The advantage is that the user does not need to supply the process model equations, they come directly from the simulation engine. Therefore, rigorous models can be considered in the steps related to the solution of the final synthesis/design problem. Note that, the final mathematical problem being solved may only require solutions of NLP (Non Linear Programming) models for given sets of binary variables or solution of the reduced MINLP (Mixed Integer Non Linear Programming) problem. The resulting method is therefore an interactive method where it is possible to go backwards in the generation and then forward again to determine a new solution. 3. APPLICATIONS The integrated system is illustrated through two application examples. The first example highlights the use of the driving force based technique near optimal solutions while the second example formulates and solves a structural optimization problem.
958 3.1.
Sequencing of distillation columns in an ethylene plant separation section This problem involves the separation of a multicomponent, non-azeotropic mixture in the production of ethylene. Only distillation columns are considered as the method of separation. The data for the effluent stream are actual plant data given by Hoch and Eliceche (1999), it consists of 10 hydrocarbons at 270 K and 10 bar. Hoch and Eliceche (1999) have formulated a NLP problem to minimize the distillate and reflux flow rates, both with the feed plate location in distillation columns as fixed, and with the feed plate location as a continuous variable. Hoch and Eliceche (1999) have first determined optimal reflux and distillate flow rates for the original feed locations with respect to the performance specifications see. Secondly, they have optimized the operating conditions, including the feed locations as variables in each column. The feed locations obtained from this optimization study are given in table 1. With the integrated system, the main problem is to decide on the sequence of distillation columns that leads to the most energy efficient design. According to algorithm 2, the sequence in figure 2 is the near optimal sequence in terms of energy consumption and matches with that proposed by Hoch and Eliceche (1999). The distillation columns are identified through the following numbers: Deethanizer (T3101), Ethylene-Ethane splitter (T3801), Depropanizer (T5001), Propylene-Propane splitter (T5601), Debutanizer (T6001). Note that, as Hoch and Eliceche (1999) operate with actual plant data, the results obtained by the sequencing algorithm are therefore validated. With the identified Dx values, the optimum feed location are also calculated (with algorithm 1). The calculations are done with the actual number of stages in the columns, as listed in table 1. Also given in table 1 is a list of the largest driving force values for each split, the corresponding optimum feed locations and the total number of stages. Table 1 compares the actual plant data and the optimization results of Hoch and Eliceche (1999) with the calculated values of the proposed algorithms. The optimized feed locations refer to the results of the NLP problem with the feed location as a variable carried out by Hoch and Eliceche (1999). *Org. feed Optimized feed Dy Energy TOptimal Dev. of optimal Nv from Column N location, Nr location, Nv Saving [%] Nv the optimized loc. T3101 40 24 18 0.47 3.12 21 7% T3801 90 60 68 0.065 0.82 52 18 % T5001 29 10 14 0.38 57.24 15 3% T5601 134 80 68 0.066 2.37 75 4% T6001 22 9 13 0.33 5.04 13 0% Table 1. Comparison of the optimized Nv as given by Hoch and Eliceche (1999) with a comparison of the calculated Nv in this paper. (*Hoch and Eliceche (1999); tAlgorithm 1)
Linearized _ I~ASSi~ m o d e ~ t ! p ~ C ~ Si~,a!~J j~
a i n a ~
~ 0~'~~
Contineousvariables
Figure 1. Inf. flow in the interactive MINLP alg.
~
Ethylene,99.92%
"
Ethane I. I, Propylene
.-
T5601
Propane ;Butane T6001
%~anve:rn"
Figure 2. Near optimal seq. accord, to algorithm 2.
959 It can noted that a combination of algorithms 1 and 2 provide further energy savings. Note, however, that these results were obtained without rigorous simulations. 3.2.
Production of Cyclohexane: For the production of cyclohexane, a high conversion reactor with conversion >95% is usually available. In this paper, production of cyclohexane with reactors giving conversions of 70% and 80% respectively, are considered. Due to this reason the recovery of product (cyclohexane), which could be carried out with a flash unit (for conversion >95%) becomes more difficult and at least an extra distillation column is neccessary. Since benzene and cyclohexane are close boiling components that also forms a binary azeotrope, other separation alternatives also needs to be investigated. In the algorithm for generating separation superstructures Hostrup et al. (1999), the first step is to list the separation techniques to be considered. The second step involves search for external mediums, leading to poly (vinyl chloride)-graft-poly(butyl methacrylate), Yoshikawa and Tsubouchi (1999), that can be used in pervaporation. The binary mixture property based analysis, first validates the presence of a low-boiling azeotrope. Investigating the pressure dependence reveals that the benzene content at the azeotropic point increases with increasing pressure. However the driving force becomes too low at pressures >3 atm to exploit this pressure dependence in pressure swing distillation. The next step is to generate solvent altematives to be used in extractive/azeotropic distillation. Vega et al. (1997) have found 5 high boiling solvents to be used for this task, with n-methyl2-pyrrolidone as the most suitable for obtaining pure cyclohexane. These can also be found using CAMD, Harper et al. (1999). Although with CAMD low boiling solvents can also be found, they are not considered further since preliminary analysis ruled them out. With this information three altematives for separating cyclohexane from benzene can set up in a superstructure (see figure 3), including one distillation column and recycling a stream with the azeotropic composition, one distillation column with a membrane based separation for "breaking" the azeotrope, and an extractive distillation configuration with n-methyl-2pyrrolidone as the solvent. The complete superstructure is given in figure 4.
y
y
e-
The object function minimizes cost: FOBJ -- Min {CT y + f(x)} = Min{- Cc clohexane*Fc-clohexan Che~t *Heat produced in the reactor + Cheat * (Heat required in dist. + Preheating before reactor) + Ccooling * (Cooling required in dist. + Cooling before flash) + Cbenzene * Fbenzene-kChydrogene * Fhydrogene + Cpurge*Fpurge + Yi * capital cost reactori + Yi * capital cost separationi } s.t. Composition of cyclohexane in product ___0.99 9 < Hydrogen/Benzene ratio in stream to reactor _< 14 Recycle
Benzene
~___
Reaction product
Purge
roduct
'
R2
Y ~
.~Product: xane
yyclohe
emb
Figure 3. The superstructure for the Figure 4. The complete superstructure is obtained by combining the separation section, separation superstructure with the two alternatives for the reactor.
960
Iteration Binary variables NLP-solution FoBJ[$/hr] 1. Inner Y2, Y5 (70% conv, membrane) FH2 = 714.08; PFinner loop = 0.05 380.20 1. Outer yl, y4 ( 8 0 % conv, 1 dist column) MILP Evaluation 741.74 2. Inner Yl, Y4( 8 0 % conv, 1 dist column) Fm= 701.41; PFinnerloop = 0.05 705.07 2. Outer yl, y4 ( 8 0 % conv, 1 dist column) MILP Evaluation *732.68 Table 2. Solutionsummaryfor cyclohexaneproductionusing the InteractiveMINLP. FH2- Total flowrate of hydrogen feed; PFinnerloop= Purge fraction in inner loop; - RelaxedMILPsolution. The mass and energy balances for individual units and overall flowsheet are included in the model equations through the process simulator. The solution summary is given in table 2 and the optimal flowsheet employs reactor 2 (with 80% conversion) and separation scheme 1 (with one distillation column). 4. CONCLUSIONS The addition of new features for separation process synthesis/design based on driving force and thermodynamic insights help to formulate and solve the structural optimisation problems more efficiently. The interactive MINLP solver, employs the integrated simulation engine to solve the NLP sub-problem. As a result, rigorous process models are used in the MINLP problem formulation and solution. The simulation engine also provides the derivative information that is needed to generate the linearised model for the MILP master problem. The algorithms based on driving force and thermodynamic insights, together, provide near optimal solutions as initial feasible solutions and significantly reduce the size of the MINLP problem. The application examples confirm that the integrated system is able to solve non-trivial problems and that it can be a very useful tool for process engineers. Current work is extending the integrated system to handling more complex process integration problems, to applications in batch operations and preparation of industrial case studies. References
Gani, R. and E. Bek-Pedersen, 1999, AIChE annual meeting, 221d, Dallas, October 1520, 1999 Harper, P. M., R. Gani, T. Ishikawa and P. Kolar, 1999, "Computer Aided Molecular Design with Combined Molecular Modeling and Group Contribution", Fluid Phase Equilibria, 158-160, p 337-347 Hoch, P. and A. M. Eliceche, 1999, "Energy Savings in the Operation of Distillation Columns", PRES'99 "2 nd Conference on process Integration, Modeling and optimization for Energy saving and Pollution Reduction", 335-340 Hostrup, M., P. M. Harper and R. Gani, 1999, "Design Of Environmentally Benign Processes: Integration Of Solvent Design And Process Synthesis", Computers and Chemical Engineering, 23, 1395-1414 Jaksland, C. A., R. Gani, K. Lien, "Separation process design and synthesis based on thermodynamic insights", Chem Eng Sci, 50, 511 (1995). Vega A., F. Diez, R. Esteban and J. Coca, 1997, "Solvent Selection for CyclohexaneCyclohexene-Benzene Separation by Extractive Distillation Using Non-Steady-State Gas Chromatography", Ind. Eng. Chem. Res., 36, 803-807 Yoshikawa M. and K. Tsubouchi, 1999, "Specialty polymeric membranes. 9. Separation of benzene/cyclohexane mixtures through poly(vinyl chloride)-graft-poly(butyl methacrylate)", Journal of Membrane Science, 158, 269-276
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
961
Pairing Criteria and Tuning Methods Towards Integration for Heat Exchanger Networks S.Chetty and T.K. Zhelev Chemical Engineering Department, University of Durban-Westville, Private BagX54001, Durban 4000, South Africa. The paper addresses the problems of heat exchanger network control synthesis with perception of simultaneous process and control system synthesis. Using algebraic and transient heat exchanger modeling, a framework for heat exchanger networks control synthesis is proposed, incorporating a bump matrix, which bears similarities to the relative gain array though involving less computational effort and network distortion. The performance of the control system using the described method compared well to results reported in literature. 1. INTRODUCTION The design of heat exchanger networks (HEN) involves the pairing of hot and cold streams, and calculation of heat exchanger areas for decided matches. The matches are subject to cost and energy considerations. The first stage of control system synthesis involves the pairing of manipulated variables and target temperatures. Thereafter, controller tuning occurs where controller constants such as gain and rest time are determined. Hence HEN's topology and control system synthesis occur as two distinct steps. Various methods of HEN control synthesis were reported [1-3], but these methods could not be included in the synthesis process since they involved either static optimization or dynamic controllability criterion which involved extensive computation to be included in synthesis implicitly. A framework for the synthesis of control systems for heat exchanger networks is submitted here where the first stage of control synthesis can be used within HEN design, due to its simple computation. Methods of controller tuning employed yield insights, which are useful during HEN design. The objective was to implement methods of control synthesis to aid the design engineer, thereby providing a platform to integration of control and HEN synthesis. Although the framework was used on a retrofit case, merging with synthesis is an envisaged goal of future work.
2. BUMP MATRIX When energy is introduced into a system or the system's parameters are changed, the system must redistribute its energy intemally to negate the initial change via manipulated variables. Bypass over heat exchangers was chosen as manipulated variables, and ideally bypasses used to control target temperatures of which they have strong influence over. The bump matrix described in this work identified such strong influence. In order to judge the systems response to energy inputs, the heat capacity flowrates of hot and cold streams were individually increased and increased as a step change. The corresponding
962 uncontrolled responses of all target temperatures were studied and a bump matrix was created using the ratio of target temperature fluctuation to step change of heat capacity flowrate for each entry, (ATI/ACp). The dimensions of the bump matrix generated, were n control targets, and p possible control inputs in the form of heat capacity flowrates. Initially dynamic simulation of the system was used to generate the bump matrix; however, the process was time consuming, and difficult to include in syntheis. Since the matrix could be generated from steady state values, with a maximum reported error of 3%, algebraic equations [2] were used to accomplish this task on a spreadsheet. However, dynamic simulation was used at a later stage when the control system structure was determined. A one-percent change in each heat capacity flowrate was induced and the systems response was used to generate the bump matrix. Bump matrices for increases in heat capacity (positive case) and decreases in heat capacity (negative case) were generated. Where necessary the row elements were normalized by dividing by the highest row value. 3. STREAM-TARGET TEMPERATURE PAIRING The Bump matrix yielded useful insights into the system. A non-zero element guaranteed structural controllability. The stream with highest influence over a target (greatest magnitude in column) was chosen as manipulator or bypass stream to that target provided other row elements were lower in magnitude to reduce interaction. The final step in control pairing was choosing the area to be bypassed in each control loop. 4. BYPASS-HEAT EXCHANGER PAIRING The choice of pairing of a bypass placement with a particular control objective or target temperature would influence the area increase (capital cost) and change in utilities (operating cost). Where a chosen stream passes through more than one heat exchanger, this step became a necessary one. The heat exchanger with the lowest efficiency [2] was bypassed since the increased area would increase its efficiency and improve energy recovery for the system, and ability to absorb disturbances. Algebraic Model O
Bump Matrix
Stream-Target Pairing
Bypass-HE Pairing
AdditionalArea
Dynamic Simulation ~
Bottlenecking 1 I
~Goal
Seek
~[
Controller selection and tuning
Fig. 1. Suggested Framework for Control System Synthesis 5. GOAL SEEK The desired goal was to change the system parameters in the form of nominal bypasses and additional area to function during system variations. By using low nominal bypass ratios it meant that the bypass could be further increased or decreased to absorb disturbances. Bottlenecked heat exchangers (low efficiency) identified sutiable sources of area increase. The worst case scenario was first isolated by grouping disturbances and/or system variations that contributed towards maximum target tempertature variation. The best case was associated
963 with minimum target temperature variation as a result of system variations. Both cases were simulated with the steady state model on a spreadsheet to ensure swift results. Using Excel Goalseek Macro, with the worst case employed and bypasses set to zero, the area of the heat exchanger with the lowest efficiency was recalculated such that the target tempertaure associated it, was maintained at its steady state value. The macro involved varying the designated area until the target temperature associated with it, was maintained at its desired value. The remaining target temperatures were maintained by either varying the nominal bypass used to control the target temperature or increasing the area of the associated heat exchanger as described. The bypasses set to zero ensured that a greater bypass range was available during operation, based on the area used. This method was used to debottleneck a low efficiency point and hence increase the operating space allowing greater variation and disturbances to be absorbed by the retrofitted system. 6. DYNAMIC S I M U L A T I O N A hybrid dynamic model was used for evaluating transient responses and tuning of the control specifications, subject to the following assumptions. 9 Constant heat capacity 9 No phase change 9 A delay of 10 seconds per baffle (or per cell in dynamic model) Each hot stream in a heat exchanger was modeled as a row of CSTR's that interchanged heat with a row of CSTR's simulating the behavior of the cold stream. Differential energy balances were used to develop temperature gradients for each cell. Log mean temperature difference driving force was used and the model order was set equal to the number of baffles. Hence each heat exchanger was modeled as m hot cells exchanging heat with m cold cells. The formula for the temperature gradient in each cell was determined as equation (A) and (B).
~m-1
Th, m/e
T ~ i ~ i - 1
m
~. 1
......
Tc, i
Th,~
,,
1
m
Tc, 1
Th, i
o .....
Tc,j- 1
Tc,j
~ - I ~ T c , m/e
Tc,m-1
In general for the i th cold cell
dTc,
m
d(tmr)
UAce n Cpc
(A) In
Zhm-i
--
rc
i
Th(m+l_i) - Tc~_ 1
In general forthe j,h cold cell dTc j d(t-
r)
UA cen (Th m_j - Tc j ) - (Th (m+I_j) - Tc j-I ) . + Thj_~ - Thj Cp~
In
Thin_ j - Tc J
(B)
Th (m+l_j) - Tc j _1
Hot Bypass Mixer:
Tho-- (1-yh) The + YhThin
(C)
964 Cold Bypass Mixer:
(D)
Tco = (1-yc)Tce + YcTctn
Hence there would be 2m differential equations to solve for each heat exchanger. The model was programmed into Matlab/Simulink since excellent response monitoring was available. Fully object-orientated modules for the heat exchanger, the mixer-tank and control loops were developed and tested hence facilitating the building of the system by instantiation of the desired components. 7. C O N T R O L L E R SELECTION AND TUNING PI controllers and pure integral controllers were nominated as principal choices employing negative feedback. The choice depended on transient oscillatory responses to step changes in input parameters. Each controller had obvious advantages and disadvantages, which were quantitatively analyzed. The retrofitted system was dynamically simulated with the decided manipulated variable and target temperature pairings. Tuning of the loops was then accomplished, furnishing the controller specification namely proportional gain, K and reset time, Tr. The Ziegler Nichols reaction curve method [9] was employed for direct bypass control and the ultimate method [8] used for other cases. The latter method was swifter, realized lower steady state error, and gave better insight as to whether PI or pure integral action was to be used. Hence, the ultimate method was generally used. However, recommended gain and reset time settings were not necessarily employed. Though critical gain and ultimate period of oscillation was established using reported procedures in quoted literature. 8. CASE STUDY The fouling problem from [4] was adopted as a case study and served as an application example for the described framework. The network was subject to variation 0.12 _> U_> 0.081 kW/m2.K and disturbance input Th2,i, with Thl,0 and TCl,0 (outlet temperatures) as control objectives or target temperatures. Table 1. Stream Data Stream Supply Target Heat Capacity H1 H2 Temp., ~ Temp.,~ Flowrate,kW/K C1
20
175
20
C2
20
175
50
H1
155
90
60
H2
200
115
60
C1
Table 2. Heat Exchanger Parameters C2
Fig. 2. Schematic Representation of case study
HE
A nom
Unom
efficiency
1
305
x
0.807
2
600
0.225
0.773
3
63
0.15
0.357
4
173
0.36
0.581
965 Using the algebraic model, the bump matrix was generated. From the matrix it is clear that on a structural controllability basis control variable C1 cannot be paired with manipulated variables associated with C2 (bypass in this case). Table 3. Bump Matrix for CASE STUDY - positive case A Tcl
ATe2
A TM
A Th2
ACpd
-2.65
-1.15
-0.1
-0.6
ACpc2
0
-2.15
-0.85
-2.5
ACphl
-0.02
0.1
0.82
0.25
ACph2
0.12
0.63
0
0.75
Comparing the magnitude of entries in each control target column, it is clear that stream C1 is best suited to control target temperatureTcl and C2 is suited to control target Th2. From the efficiencies in Table2, it is clear that HE3 is a bottleneck due to a low efficiency value. This result was later confirmed by a single heat exchanger dynamic response.
The additional area was determined from goal seek during worst case (i.e. when heat exchanger 1 was completely fouled, U1 = O.081 kW/m2.K). The new area for heat exchanger 3 was calculated as 253.62 m 2. For the best case (U1 = 0.12 kW/m2.K) the required area was determined as 172.34 m 2 but this area did not de-bottleneck the heat exchanger during worst case, hence an area of 263.62 m 2 from heat exchanger 3 was successfully used. At this stage, the bypass and target pairings were known i.e. cold bypass over HE3 to control TCl and cold bypass over HE4 to control Thl. The ultimate method was used to tune each Table4. Recommended Control Specifications control loop and critical gain and ultimate Proportional Gain Reset Time periods were reached. However, using recommended K = 0.45* critical gain and KlO'c/) -3.6 139.17seconds Tr = ultimate period~1.2 [8] yielded an unstable system. Both control loops yielded g2(Yc/) -16 55.56 seconds highly oscillatory responses even at gains below the critical gain. In the first loop lower temporal violations were sacrificed for a more stable system by using a gain o f - 3 rather than the recommended -3.6 and the second loop lower response time was sacrificed and pure integral action was used stabilizing the system. Direct temporal Violation
Response to Step Change In U
v
A 1
2000
0.5
1500
0 -16
-14
-12
-10
-8
-6
-4
1000
-2
ne
-16
-14
-12
-10
Gain, K2 Response -9
-6
.
. -5
. -4
.
. -3
-6
-4
-2
Gain, K2
Direct temporal Violation
.
-8
11 -2
,, -12 -1
~6 m~ I ca.
~,
~
to Step Change In U
, ~
3000
,,,,, 9
2000 1000
-6
Gain, K1
Fig. 3. Control Performance for Different Control Settings
-5
-4
-3 Gain, K1
-2
-1
966 Yr 4
/--
0.6 rx f..- ~ xJ
0.4 ~
It\
0.2 ' 0
I
\ 1
~ f -
0
1000
2000
3000
4000
Fig. 4. Bypass response to Th2 input disturbance of +5~ The developed control system with area increase on heat exchanger 3 to 253.62 m 2 and bypasses Ycl 3 and YC24 used to control TCl,o and Tht,o respectively shows good system stability and response times. As confirmation, U1 was stepped down from 0.15 kW/m2K (best case) to 0.12 kW/m2K and the responses for various controller settings tabulated. Although this test was extreme, all configurations unlimited by physical constraints reached desired target with zero steady state error. The values of bypasses at the worst case was Yc13=0.000 and Yc24=0.0406 and at the best case when U =0.12, YCl3= 0.4962 Yc24= 0.0406. 9. CONCLUSIONS AND RECOMMENDATIONS With respect to HEN control synthesis, the bump-matrix was successfully used to create manipulated variable and target temperature pairings. Controllability aspects namely shortterm system stability, response times and safe transition from one operating point to another were entirely dependent on the controller constants subject to physical constraints. Since physical constraints were removed by installation of additional area, returning the system to desired operating point was accomplished by choosing suitable control settings. It was determined that low gains produced improved short-term stability operability and pure integral control provided ideal transition at the expense of lower response times and higher temporal violations. REFERENCES 1. C.Boyaci, D.Uzturk, A.E.S.Konukman and U.Akman, Dynamics and Optimal Control of Flexible Heat Exchanger Networks, Comp.& Chem. Engng, 20 (1996), $775. 2. A.Aguilera and J.L.Marchetti, Optimizing and Controlling the Operation of Heat-Exchanger Networks, A1ChE J., 44 (1998), 19. 3. K.W.Mathisen, S.Skogestad and E.A.Wolf, Bypass Selection for Control of Heat Exchanger Networks". Paper presented at ESCAPE-l, (1992), Elsinore, Denmark. 4. K.P.Papalexandri and E.N.Pistikopoulos, Synthesis and Retrofit Design of Operable Heat Exchanger Networks.2., Ind. Eng. Chem. Res., 33 (1994), 1738. 5. E.A.Wolf, K.W.Mathisen and S.Skogestad, Dynamics and Controllability of Heat Exchanger Networks, Computer-Oriented Process Engineering, Elsevier., Amsterdam, 1991, 117. 6. B.Glemmestad, K.W.Mathisen and T.Gundersen, Optimal Operation of Heat Exchanger Networks based on Structural Information, Compt.& Chem. Eng., 20 (1996), $823. 7. S.Papastratos, A.Isambert and D.Depeyre, Computerised Optimum Design and Dynamic Simulation of Heat Exchanger Networks, ESCAPE-2, 1993. 8. P.Harriot, Process Control, Robert E. Krieger Publishing Company, Malabar, 1964. 9. G.F.Franklin, J.D.Powel and A.Emami-Nacini., Feedback Control of Dynamic systems, AddisonWesley Publishing Company, Reading, Massachusetts, 1986.
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rightsreserved.
967
A Conceptual Programming Approach for the Design of Flexible HENs L. Tantimuratha*, G. Asteris, D. K. Antonopoulos, and A. C. Kokossis t Department of Process Integration, UMIST, PO Box 88, Manchester M60 1QD UK. A conceptual tool is presented to address the flexibility and operability objectives for heat exchanger networks. Previous work addressed the variations in the process parameters (stream temperatures and flow-rates) either on the basis of a fixed structure or with modelling considerations in the synthesis superstructure. The approach presents a screening model to accommodate considerations ahead of design. The new development is essentially a targeting tool based on the Area Target Model that is a part of the Hypertarget methodology of Briones and Kokossis (1999a, b, c). The targets accommodate for an early pre-processing of matches and assess implications due to parameter variations. The feasibility targets provide ahead of design information on the costs incurred due to the variations. Flexibility targets assess the ability of the system to handle variations for different driving forces. The potential of the matches to handle variations can be further exploited with superstructure schemes. However, the optimisation of these schemes (as NLP problems) would still require extensive and difficult formulations to embed the process variations. Such a crude approach would not be practical or useful to apply to industrial size problems. The paper instead proposes a systematic algorithmic procedure that converges to flexible networks at the expense of only a minor set of additional constraints. Examples are presented to illustrate the approach, explain its ability to assess the impact of variations and prove the ease of its use in real-size problems. 1. I N T R O D U C T I O N Heat exchanger networks are usually developed under the assumption of fixed design parameters that represent standard values at nominal operating conditions. However, the nominal conditions are expected to change as different feedstocks are processed, the heat transfer deteriorates (fouling), and the flow conditions are requested to follow different sets of process specifications. With frequent operational changes a rather common feature of modern sites the emphasis should be widened to consider their own impact on the design. Screening and targeting tools should subsequently be extended to address expected or predicted changes ahead of detailed considerations. The first effort to embed design uncertainty as part of a mathematical formulation should be attributed to Halemane and Grossmann (1983). In the steps of this work, a systematic framework has been presented by Swaney and Grossmann (1985a, b) to address flexibility. The flexibility analysis comprised two similar test problems: one to assess the ability of the design to remain feasible for a given set of variations; another one to assess the range of variations the design can afford (flexibility index). The tests could function on specific layouts and required an exhaustive enumeration of vertex points.
E-mail: [email protected] ; E-mail: [email protected]
968 Grossmann and Floudas (1987) contributed with a novel MIP formulation to prevent the exhaustive search. The literature of HEN problems is apparently much more extensive. It includes methodologies based on thermodynamics and extends to mathematical programming techniques and formulations of several types and purposes. Most notably, one should refer to the MILP transhipment model by Papoulias and Grossmann (1983) and the NLP superstructure by Floudas et. al. (1986). The former contributed with a conceptual model able to identify matches of process streams; the latter extended these developments with superstructure schemes one could employ to automatically develop energy integrated networks. However, the merits of Pinch Technology had enabled the method to continue its ascendancy over mathematical programming in the vast majority of industrial applications. In due recognition of the power of thermodynamics, a new generation of applications illustrated venues to regain all the merits of the Pinch Method within a mathematical programming framework. Hypertargets (Briones and Kokossis, 1999a, b, c) combine the merits of thermodynamics and mathematical programming and introduce a systematic framework for HEN synthesis and retrofit applications. Hypertargets can be used to provide targets ahead of design in the form of a solution stream of design options available at different driving forces. A library of targeting models is introduced (ATM, TAME, HEAT) one is able to launch with minimal effort either to enable the development of targets or to set up superstructures for layout optimisation. The screening models combine information about the number of matches, the level of energy recovery and the potential savings on capital investment (heat transfer area, re-piping, modifications). 2. M O T I V A T I O N AND THE OUTLINE OF THE APPROACH 2.1. Motivation The application of the ATM (Briones and Kokossis, 1999a) designates HEN layouts that are attractive from an economic viewpoint. The experience with the use of the model indicates that there exist numerous combinations of matches that feature similar- if not identicalperformance. The select pool of the design candidates comprises the primal set of the ATM. The detailed optimisation of the candidates eventually determines configurations that, although of similar cost, perform quite differently in their ability to adjust to process variations and parameter changes. The idea to expand the ATM model has been motivated by the large size of the primal sets in the majority of industrial applications. Rather than processing all candidates, it would be more efficient to consider only the ones with an additional potential to handle variations. 2.2. Outline of developments The new developments include: (i) conceptual models to screen layouts for grassroots design; (ii) conceptual models to screen modifications for retrofit applications; and (iii) a design procedures for the development of layouts of networks with the ability to handle variations. Models in (i) and (ii) apply prior to the development of a HEN network and are formulated as MILP models to measure the ability of the primal set to handle variations at different levels of ATtain. The results can be plotted as .flexibility targets alongside the Hypertargets. Good designs are located in areas of large flexibility targets and low total cost. The selected designs from (i) and (ii) do not represent networks yet. They are simply
969 combinations of matches with a promising potential to account for assumed variations in the design parameters. To exploit the potential of these matches, the mathematical models need be expanded with a significant amount of additional modelling information. Instead the paper outlines an iterative procedure that requires only incremental modifications in the model and converges in practice within very few iterations. 3. SCREENING M O D E L FOR TARGETING
The new developments are based on the ATM and TAME models (Briones and Kokossis, 1999a, b, c) which consist of targeting terms for the heat transfer area. They contain energy balances for different temperature intervals and involve residuals that reduce to the minimum heat duties required. The participation of residuals in re-formulations with an additional consideration of flexibility (Swaney and Grossman, 1985a, b) results in MILP problems with a large number of variables. In the new formulation: (i) all constraints that contain residuals are relaxed; (ii) an additional set of inequality constraints accounts for physical bounds on the available energy to recover; and (iii) a set of inequality constraints accounts for possible violations of the exchanger minimum approach temperature (EMAT). Constraints (ii) and (iii) result from the relaxation of the residuals. The reformulated ATM/TAME model is optimised over the primal set to determine acceptable combinations. Plotting the optimisation objective against different values of ATtain determines the flexibility targets. Flexibility targets that assume a minimum number of units, Umin, give rise to wave-like shapes (flexibility waves) whose peaks and lows relate to favourable trade-offs. Alternatively, flexibility waves can be developed for Umin+l and Umin+2 cases. 3.1. Example 1 The data of this small problem is shown in Table 1. Variations relate to the CP values of process streams hl and c l. The primal set contains four primal solutions at ATmin 10 ~ listed Table 2 with the summarised results from the application of the new models. The results explain that only two of the four options (i.e. S1 and $2) are acceptable; the other two are unable to afford the assumed variations. The meaning of the feasibility and flexibility (Table 2) strictly follows the propositions by Swaney and Grossman (1985a, b). The difference is that the results from this model are only "promises" for a performance and do not contain guarantees the actual, final design will be featuring these qualifications. There is no real drawback, however, in having targets and solutions deviate at this stage. One should bear in mind that all primal solutions correspond to good designs and that additional considerations for flexibility account for an extra "bonus" on the solution.
Table 1 Stream data Stream H1 H2 H3 Cl C2
for example 1 Nominal CP (kW/~ 1.7 1.5 0.5 1.25 0.8
Variation of CP (kW/~ +0.5 +0.45 -
Tin (~ 400 370 410 200 250
Tout (~ 280 190 388 370 400
970 Table 2 Feasibility Solution St $2 $3 $4
and Flexibility targets of each options for example 1 Matches Feasibility target h I -c 1,h 1-c2,h2-c 1,h2-c2,h2-cu,h3-c 1,hu-c2 - 1 t.0 !11-c1,h 1-c2,h2-c 1,h2-c2,h2-cu,h3-c2,hu-c2 - 11.0 h 1-c l,h2-c 1,h2-c2,h2-cu,h3-c2,hu-c 1,hu-c2 0.7 h 1-c I ,h2-c 1,h2-c2,h2-cu,h3-c 1,hu-c I ,hu-c2 2.5
Flexibility target 1.85 1.76 0.99 0.92
The consideration of time-varying effects (i.e. fouling) can also be addressed with the proposed models. For example, given a time-variant profile for the heat transfer coefficients, the fouling resistance can be added into the targeting expressions of the ATM/TAME models. Accordingly, the synthesis objective has to be upgraded in the form of a time-weighted average of the targeted heat transfer area. Similar arguments can be made for the flow rates, the temperatures and the availability of utilities. 3.2. Example 2 Consider a larger problem of 7 hot and 3 cold process streams shown in Table 3 with • variations in the process flow capacities. The availability of options at different ATmin (U=Umin) is illustrated in Figure 1. The Figure explains ranges of ATmin (30-35~ where no flexible solutions are available which . The results obtained for Umin have motivated the development of additional curves for Umin+l units. Figure l a presents the Hypertargets (annual costs) against ATmin. The primal set contains matches for Umin and Umin+l. The shaded region accounts for flexible solutions; the remaining area relates to matches unable to handle variations. Figure l b concentrates on the flexible designs. The shaded area of Figure 1b relates to matches with Umin number of matches. The remaining region is accessible with Umin+l units. For ATmin > 25~ flexible designs require Umin+l units. Figure l c explains the flexibility targets against ATmin. Figure l d plots the cardinality of the primal set. It is interesting to see that for small ATmin, the deviation from the minimum- number-of-units rule (Pinch Analysis). 4. NETWORK OPTIMISATION As one is required to translate the selected matches to detailed networks, superstructure models (Floudas et al., 1986) need to be formulated at all vertex points. The number of constraints faces a dramatic increase with the parameters whose variations are considered. However, the benefit in processing a suitable match can be exploited through an iterative procedure that proceeds along the following steps: (i) solve the NLP optimisation with the parameters at the nominal point; use a single superstructure model So; obtain an optimal network Li ; (ii) check Li at each vertex point to confirm feasibility; terminate in cases all tests are successful; and (iii) for a vertex point i that relates to an infeasible variation include an additional superstructure model Si; augment the mathematical formulation; iterate with Step (i). Step (i) usually results in inappropriate layouts. Although the initial combination of matches has a potential to account for variations, the mathematical formulation is driving the search towards cost-effective configurations. Step (ii) is trivial to complete and identifies violations. Step (iii) requires further explanation on the type of constraints that need be included in the formulation. The additional constraints ensure that all superstructure models S/ converge to the same layout.
971 Table 3 Stream data for example 2 Stream Nominal CP (kW/~ H1 470.00 H2 825.00 H3 42.42 H4 100.00 H5 357.14 H6 50.00 H7 136.36 CI 826.09 C2 500.00 C3 363.64
Tin (~ 140 160 210 260 280 350 380 270 130 20
Tout (~ 40 120 45 60 210 170 160 385 270 130
Figure 1 Hypertargets, Fi-exibility targets, and number Of solutions for Example 2
hl h2
h4 hS
()
[
(
(
()
O
h7 cl ~, O ___
(
T [
c2~ - < 2 c3
(a) Design A (inflexible) Figure 2 Network solutions for Example 3
c~~ c l - A ~r77~
c3 9
(b) Design B (flexible)
972 4.1. Example 3 Let us consider the problem discussed in Example 2. At ATmin=30 ~ a combination is selected from the optimal set featuring 12 units. The flexibility target for the combination is 1.30 and indicates an appropriate candidate to account for the parameter variations of that problem. A superstructure set up and optimised on the basis of these matches yields the network of Figure 2a. The cost of the network is 6690 k$/yr but it fails to handle the expected variations in the parameters. The results from the iterative scheme that is proposed iterative earlier yields the network of Figure 2b. The procedure converges after the first iteration. The network (of the same combination) features a different structure from the network A and can afford all of the variations with a very small penalty in the total cost (1.9%).
5. CONCLUSION The paper outlines a conceptual tool that is able to address flexibility objectives for heat exchanger networks. The work extends the screening models (ATM/TAME) of the Hypertarget methodology to enable variations in the process parameters. The new models retain the screening nature of the early developments in their ability to identify options ahead of detailed calculations. Flexibility targets are calculated alongside the Hypertargets and are useful to assess trade-offs between the design flexibility and economics. Moreover, the optimisation approach is enabled with an iterative scheme to materialise the targets of the screening stage. The procedure optimises the network layout using superstructure developments. Although it addresses a much larger problem, the additional synthesis effort is very limited: in most cases, the scheme converges within two iterations. REFERENCES
Briones, V. and Kokossis, A. (1999a), Hypertargets: a Conceptual Programming approach for the optimisation of industrial heat exchanger networks-I. Grassroots design and network complexity, Chem. Engng. Sci. 54, 519-539. Briones, V. and Kokossis, A. (1999b), Hypertargets: a Conceptual Programming approach for the optimisation of industrial heat exchanger networks-II. Retrofit design, Chem. Engng. Sci. 54, 541-561. Briones, V. and Kokossis, A. (1999c), Hypertargets: a Conceptual Programming approach for the optimisation of industrial heat exchanger networks-Ill. Industrial applications, Chem. Engng. Sci. 54, 685-706. Floudas, V.A.; A.R. Ciric; and I.E. Grossmann (1986), Automatic Synthesis of Optimum Heat Exchanger Network Configurations, AIChE Journal, 32, 2, 276-290. Grossmann, I.E. and C.A. Floudas (1987), Active Constraint Strategy for Flexibility Analysis in Chemical Processes, Comput. Chem. Engng., 11, 6, 675-693. Halemane, K.P. and I.E. Grossmann (1983), Optimal process Design under Uncertainty, AIChE Journal, 29, 3, 425-433. Papoulias, S.A. and I.E. Grossmann (1983), A Structural Optimization Approach in Process Analysis-II: Heat Recovery Networks, Comput. Chem. Engng., 7, 6, 707-721. Swaney, R.E. and I.E. Grossmann (1985a), An Index for Operational Flexibility in Chemical Process Design: Part I-Formulation and Theory, AIChE Journal, 31, 4, 621-630. Swaney, R.E. and I.E. Grossmann (1985b), An Index for Operational Flexibility in Chemical Process Design: Part II-Computational Algorithms, AIChE Journal, 31, 4, 631-641.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
973
Mass Exchange Network Synthesis by Coupling a Genetic Algorithm and a SQP Procedure S. Shafiei, A. Davin, L. Pibouleau, S. Domenech, P. Floquet Laboratoire de G6nie Chimique - UMR-CNRS 5503 - ENSIGC. INPT 18, Chemin de la Loge - 31078 Toulouse Cedex 04- France The design of mass exchange networks (MEN), where a number of process streams rich in terms of some components (for example pollutants) have to be matched with lean or utility streams in order to meet given specifications on their final compositions, is addressed in this paper. These networks are typically composed of mixers, splitters and mass transfer units that carry out the discharge of hazardous waste, according to an economical objective function.. This paper describes a novel approach for the synthesis of MEN by coupling a genetic algorithm with a classical nonlinear programming method. The encoding procedure for the genetic algorithm describes a wide search space, considering all possible combinations within a superstructure mass exchange network. The procedure is illustrated by an example provided from the literature in which the total annual cost is significantly improved. 1. INTRODUCTION Mass-exchange operations have a wide range of applications in chemical process industries. Mass exchange networks (MENs) are used to reduce the waste generated by a plant to an acceptable level at the cheapest cost. The potential use of mass exchange networks arises in situations as diverse as feed preparation, product recovery and waste minimization. MEN synthesis has been tackled in a variety of ways. The first attempts at algorithmic methods tried to solve the problem of minimizing the amount of mass separation used. El-halwagi and Manousiouthakis (1989,1990a) applied a pinch method similar to that of HEN to MENs and developed a two stage approach based on linear programming (LP) and mixed-integer linear programming. This method was extended to include regeneration (El-halwagi and Manousiouthakis 1990b) and mass transfer with reaction (El-halwagi and Manousiouthakis 1992). The major limitation in such an approach is that the sequential synthesis procedure does not offer overall cost optimality of the final network, since cost parameters are not simultaneously optimized. Furthermore, no systematic method is provided for the derivation of the network configuration. More recently, the problem has been extended to include full capital costing as well as operating costs, resulting in a nonlinear problem. These have been solved using a
9/4 superstructure approach (Papalexandri and Pistikopoulos, 1994), state space techniques (Gupta and Manoustakis, 1993; Bagajewicz and Manousiouthakis, 1992) and the Process graph synthesis procedure (Lee and Park, 1996). Introducing superstructure parameters to the problem results in adding integer variables which makes the problem a Mixed Integer Nonlinear Programming Problem(MINLP). So all limitations associated with a MINLP problem still remains. In this paper we propose a new approach that makes use of a combination of genetic algorithm and classic solution of linear or nonlinear programming problem. The general strategy for designing the optimal process is a two level approach. The fundamental difference between the proposed procedure and the well-known MINLP approach is that a stochastic method, namely a Genetic Algorithm (GA), is included to solve the integer master problem, related to the structural optimization of the network, instead of a classical Mixed Integer Linear Programming(MINLP). Compared to the deterministic method, the main advantage of this stochastic procedure lies in the fact that it may avoid to be blocked on local optima and the supply of a family of good solutions at the end of the search. 2. P R O B L E M S T A T E M E N T
The MEN synthesis problem can be stated as follows, by considering : 9 A set of rich process streams, their flow rates, their initial compositions, with respect to the component that must be removed, and their desired outlet compositions (imposed either by environmental regulations or economical considerations); 9 A set of lean streams (or Mass Separating Agents) which may be process streams or auxiliary external lean streams, their initial compositions, an upper bound or outlet compositions, and their costs; 9 An equilibrium relation for the distribution of the transferable component between rich stream and lean stream; The objective is to synthesize a mass exchange network in terms of stream matches, lean stream composition, network topology and investment cost, that can satisfy the specifications for the rich and lean streams. The following assumptions are made : I. The mass flow rate of each stream remains constant throughout the network. II. The equilibrium of the considered component does not depend on the other solutes. III. The mass exchangers considered are only of counter-current type. IV. No mass exchange between rich streams is allowed. V. Heat integration between streams is not allowed. VI. The network operates under constant pressure. The first assumption is realistic when small composition changes takes place. The second assumption refers to linear equilibrium relation, where the equilibrium constants are mutually independent for the various components. This assumption can be easily removed by defining any nonlinear equilibrium relation. The two last assumptions refer to the applicability of one equilibrium relation throughout the network.
975 3. G E N E T I C A L G O R I T H M Genetic algorithms are stochastic optimization methods based on the biological principles of natural selection (Beasly et al 1993; Goldberg, 1989;Holland, 1975; Michalewicz, 1994). The basic idea of GA is to place the parameters of the problem to be optimized within what is referred to as a chromosome (or individual) which consists of 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 operators including crossover and mutation. Members on which to operate are chosen from the population using a fitness based selection method. One can use any of a large variety of operators and selection methods. The underlying feature of all 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. In optimization studies the fitness is quite often the value of the objective function for the given parameter or it may be the solution to an LP or NLP that is generated from the chromosome. A genetic algorithm terminates when a user specified criterion is met. Typically, this is some expected fitness, a number of iterations (known as generations), or some threshold on the diversity in the population. There are several advantages to the use of genetic algorithms: 9 They perform a global search and are less likely to get trapped in local optima. 9 Multiple solutions are available at the end of the search. For these reasons we have implemented a genetic algorithm for the optimization of the structure of mass exchange networks (master problem). 4. MASS E X C H A N G E N E T W O R K S U P E R S T R U C T U R E The superstructure is defined so that all possible matches are considered between rich and the available lean streams, in all possible network configurations. Adapting the definition of superstructure by Papalexandri et al. (1994), involves the following basic features. 9 Each potential match between a rich and a lean stream corresponds to a potential mass exchanger (one-to-one correspondence). 9 Each stream entering the network is split towards its potential mass exchange units. 9 The possibility of multiple mass exchanges between two streams is considered. 9 Prior to each potential mass exchanger a mixer is considered for each stream, where the flow from the initial splitter and bypass flows from all the other potential exchangers of the stream are merged into a flow towards the exchanger. 9 After each mass exchange unit a splitter is considered for each stream, from which potential flows are driven towards the final mixer of the stream and the other exchangers of the stream. Once structure is defined then the mass exchange network synthesis problem can be formulated as a nonlinear programming problem (slave problem) coupled with a genetic
976 algorithm (master problem), where the total annualized cost expressed in terms of operating and capital investment cost is minimized. For each feasible structure the NLP is executed and total cost obtained becomes the fitness of the chromosome that represents the structure. 5. I M P L E M E N T A T I O N In the master part of the procedure an initial population is first generated. Each chromosome is introduced to the NLP part to evaluate the objective function and thus fitness of the chromosome. Before entering NLP procedure all chromosomes are checked to represent a feasible structure. With an initial population in hand, the simple manipulations of GA such as random selection crossover and mutation are performed for each generation until the pre-specified number of generations performed. The procedure can be best represented as the following flow chart: begin initialize population evaluate solution population (NLP procedure) while (termination condition not met) iterate update generation counter select parents for the next generation (based on fitness) reproduce, recombine and mutate parents to form offspring evaluate offspring (NLP procedure) end iteration end 6. ILLUSTRATING EXAMPLE The mass exchange network synthesis concept is illustrated with an example from copper Technology (E1-Halwagi and Manousiouthakis 1990b) Etching of copper is achieved through ammoniacal solution and etching efficiency is higher for copper concentrations in the ammoniacal solution between 10 and 13w/w%. To maintain the desired copper concentration in the solution, copper must be continuously removed. Copper must also be removed from rinse water, with which the etched plates are washed out, for environmental and economical reasons. Mass flow rate data and specifications on the concentrations of the two copper-rich streams are given in table 1. Table 1. Rich streams of copper recovery problem Streamno R1 R2
Descrip_t_!on_.........G._.(Kg/s ) ys amm. Solution 0.25 0.13 rinsewater 0.1 0.06
yt 0.1 .02
Two extractants are proposed for copper recovery LIX63 (an aliphatic cxhydroxyoxime, S1) and P1 (an aromatic [~- hydroxyoxime,S2). Data on copper concentrations and cost of the two available lean streams are given in table 2.
977 Table 2. Lean streams of copper recovery problem Stream'n0 Descpt'.' Xs .......X't......... Cost ($/Kg) Ann. Cost ($s/KgYr) $1 LIX63 0.25 .13 0.01 58,680 $2 Pl 0.1 .06 .12 704,160 Within the ranges of copper concentrations of interest, the copper transfer between the given rich and lean streams is governed by the following linear equilibrium relations : R1,S1 : yl=0.734xl+0.001 R2,S1 : y2=0.734xl+0.001 R1,S2 : yl=0.11 lx2+0.008 R2,$2 : y2=0.148x2+0.013 Two types of contactors are considered : a perforated plate column for S1 (LIX63) and a packed tower for S2(P1). The annualized investment cost of a plate-column is based on the number of plates Nst which is determined through kremser equation. The cost of a packed tower is based on the overall height of the column (HRNz) ; annualized investment costs are given in table 3. Table 3. Capital cost data for copper recovery problem Cost of Plate column 4,552 Nst $/Yr Cost of Packed column 4,245H$/Yr Following the sequential synthesis approach, the proposed solution by E1-Halwagi and Manousiouthakis(1990b), based on minimum extractant consumption features an annualized cost of $27,758/yr, whereas the investment cost of the required equipment is $95,695/yr. The proposed synthesis model is employed to determine the total annualized cost optimal mass exchange network for copper recovery. Operating and investment cost are optimized simultaneously with the structure. The program is written under Microsoft Fortran power station software and uses a SQP (Successive Quadratic Programming) library routine to optimize the NLP part of the problem. The problem consists of 28 integer and maximum of 52 real variables, maximum of 36 equality and 8 inequality constraints. The obtained network is shown in the following figure.
R2(0.1Kg/s) ys=0.06 R1(.25Kg/s) ys=0.13
0.2783Kg/s SI rxs=0.03 ~ ~ ~~yt=.l
$2" .01929Kg/s lxs=0.001 yt=0.02 xt=.02
978
It features annualized cost of 48,1345 out of which 29,9145 is the cost of extractants. Note that the total annualized cost obtained with the proposed approach has been reduced from $123,453/yr to $48,134/yr (61% reduction). The operating cost has been increased from $27,758/yr to $29,914/yr (7.8% increase), whereas the equipment cost has been reduced from $95,695/yr (80% reduction). 7. C O N C L U S I O N S A new approach to the mass exchange network problem was proposed. The procedure does not rely on the decomposition of the problem since it accounts simultaneously for the trade-off between stream cost and equipment installation cost. While reducing the MINLP problem to NLP one simplifies the problem and avoids all difficulties concerned with MINLP; finally the use of GA likely prevents the solution to be trapped in local optima. Notation
G H Nst Ri Sj
Flow of rich stream (Kg/sec) Height of each theoretical equilibrium stage in packed column(m) Numberof theoretical plates in extraction column Rich stream number i Lean stream number j
x~
Xs Xt Yj Ys Yt
Concentration of lean stream i Supplied concentration of lean stream Target or upper limit concentration of lean stream Concentration of rich streamj Supplied concentration of rich stream Target or desired concentration of rich stream
REFERENCES
1. 2. 3. 4. 5.
Bagajewicz, M. and Manousiousthakis, V., AIChE Journal, Vol. 38, No.11, pp. 1769, 1992 Beasly, D. Bull, D.R. and M.R., Univ. Comput. 15(2), pp58-69, 1993 El-halwagi M., M. Manousiouthakis V., AIChE Journal, Vol. 35, No.8, pp. 1233, 1989 El-halwagi M., M. Manousiouthakis V., AIChE Journal, Vol. 36, No.8, pp. 1209, 1990a El-halwagi M., M. Manousiouthakis V., Chemical . Engineering Science, Vol. 45, No. 9 pp.2813-2831, 1990b 6. El-halwagi M., M. Manousiouthakis V., Chemical Engineering Science, Vol. 47, No. 8 pp.2113-2119, 1992 7. Goldberg, D.E., 'Genetic Algorithms in Search, Optimization and Machine Learning' Addison-Wesley, Reading, MA, 1989 8. Gupta A. Manousiouthakis V., Ind. Eng. Chem. Res., Vol. 32, PP. 1937-1950, 1993 9. Holland, J.H. 'Adaptation in Natural and Artificial Systems', MIT Press, Cambridge, M.A., 1975 10.Lee S., and Park S., Computers and chemical. Engineering. Vol. 20, Suppl., pp. $201-s205, 1996 ll.Michalewicz, Z., 'Genetic Algorithm + Data Structures = Evolution Programs', Springer, New York, 1994. 12.Papalexandri K. P., Pistikopoulos E.N., Floudas C., Trans IChemE, Vol. 72, PP. 279-194, 1994
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
979
SYNTHESIS OF REACTOR NETWORKS IN OVERALL PROCESS FLOWSHEETS WITHIN MULTILEVEL MINLP APPROACH
B. Pahor I and Z. Kravanja ~ 2 1. EVACO, Ltd., Vodovodna 28, Maribor, S I - 2000 Slovenia Fax: ++ 386 62 31 83 35, e-mail: [email protected] 2. Faculty of Chemistry and Chemical Engineering, University of Maribor, P.O.Box 219, Maribor, SI - 2000 Slovenia, Fax: ++ 386 62 22 77 74, e-mail: [email protected]
ABSTRACT: This paper describes the superstructure-based mixed-integer non-linear programming (MINLP) approach to the synthesis of reactor networks in an equation-oriented environment. The approach is an extension to the MINLP synthesis procedure of plug flow reactor (PFR) networks (Pahor and Kravanja, 1995) upgraded with the multilevel-hierarchical MINLP approach (Kravanja and Grossmann, 1997). The model comprises a general superstructure in which the exact formulation of the recycle reactor ( R R ) - a train of differential PFRs with a common recycle stream, and a continuous stirring tank reactor (CSTR) are embedded and can easily be extended for N elements so as to enable a different feeding (cross flow, side stream), recycling and bypassing. The reactor arrangement is capable of representing several reactor systems such as a pure CSTR, pure PFR, pure RR and their combinations. Further, it is possible to represent a cross flow reactor (CFR). The superstructure is a suitable either for isothermal or non-isothermal, simple or complex reactor network. Since it is simple, it can by means of heat integration simultaneously be incorporated into an overall process synthesis, which favours simple results. With the multilevelhierarchical strategy, it is possible to postulate the superstructure at different levels of representation of flowsheet alternatives. Therefore, the superstructure is optimized more effectively and reliably. The approach has been applied to a complex non-isothermal reaction p r o b l e m - an industrial example of a production of allyl chloride. 1. I N T R O D U C T I O N Even though the reaction system and the reactor design determine the character of the flowsheet (the types, sizes and operating conditions of the reactor and other process units), there does not seem to be a general procedure for the synthesis of non-isothermal reactor networks which could be efficiently integrated into an overall process scheme. Previous approaches for addressing this problem can be classified as either superstructure based methods or targeting methods. In the former approach, the plug flow reactors (PFRs) were approximated by a series of equal- sized sub- continuous stirring tank reactors (CSTRs), while integer variables were used to represent the existence of the reactor units. The resulting o To whom correspondence should be addressed.
980 formulation was a large-scale, complex, non-convex MINLP problem. The limitation of such an approach is that the solution obtained is only as rich as the proposed superstructure. Increasing of richness comes at the cost of increasing of complexity of the model. Recently, the properties of the attainable region (Feinberg and Hildebrandt, 1997) were considered in the way the smaller and simpler MINLP formulations of reactor network synthesis are developed (Lakshmanan and Biegler, 1996; Schweiger and Floudas, 1999). Still, the formulation is too complex to efficiently perform simultaneous synthesis of reactor network and process flowsheet.
2. PROBLEM STATEMENT The main idea in our contribution is to overcome the mentioned drawbacks and to improve the superstructure-based approach to the synthesis of reactor networks as follows: 9 PFRs/RRs are described with accurate differential algebraic equation (DAE) models that can be simultaneously solved within the MINLP synthesis procedure. A train of differential PFRs represents RR with a common single recycle stream (Figure 1). i
i+1
i+
i+
i+4
Figure 1: a RR m o d e l - a train of differential PFRs Each train is further accompanied by a side stream and a component separator, which from the view of the whole superstructure, could be understood as a compact approximation of a cross flow reactor (CFR) with intermediate separation of products. A simplified, yet fairly general superstructure of reactor networks is proposed which can cope well with the majority of reaction systems, and can successfully be incorporated into the multilevel-hierarchical MINLP approach (Kravanja and Grossmann, 1997) to perform a simultaneous synthesis of reactor networks, heat integration and the overall process synthesis/optimization at different levels of superstructure details and model aggregation. Motivated by the mentioned disadvantages of either the superstructure or the targeting approach, the goal of our contribution is to derive such a model and to postulate such a reactor network superstructure which would combine both approaches. On the one hand, the resulting reactor network superstructure should be sufficiently rich to resemble most alternatives, while on the other hand, it should be compact enough to favour simple results in the overall process synthesis. Also, it should be adapted to the multilevel synthesis (Z. Kravanja and I.E. Grossmann, 1997), which is performed hierarchically from reactor network synthesis (MINLP 1) to separation (MINLP 2) and HEN synthesis via task identification to task targeting and to task integration. Since the goal of our research was oriented towards a detailed synthesis of the optimal reactor networks, we thus limited our study to MINLP 1 only, which is from the point of view of the reactor network synthesis the most detailed one. The approach has been implemented by the multilevel-hierarchical MINLP computer synthesizer MIPSYN (a successor of Prosyn/MINLP by Kravanja and Grossmann, 1994).
981 3. R E A C T O R N E T W O R K S U P E R S T R U C T U R E The generation of a reactor network superstructure is subjected to the multilevel MINLP process synthesis. The attainable region theory suggests that for one and two dimensional problems, the only reactor types that are required to achieve all possible compositions for a given reaction are PFR, CSTR, whilst for higher dimensions a CFR is introduced as well. The general model formulation of PFRs and RRs is an extension to the ideas presented by Pahor and Kravanja (1995). Since the motivated example, the industrial problem of allyl chloride production, is complex and high dimensional, the fundamental reactor units considered are the CSTR and the RR (a train of differential PFRs).
FEED-2
t
.~
L
__
by products
products~
LEGEND FOR PROCESS UNITS
LEGEND FOR PROCESS UNITS
PFR
[]
S,NGLE CHOICE STREAM SPLITTER
IS
intermediate separation
0
MULTIPLECHOICESTREAMMIXER
FS
final separation
I----~----I
MULTIPLE CHOICE STREAM SPLITTER
Figure 2: Proposed reactor network and simplified separation superstructure (MINLP 1) The resulting superstructure for MINLP1 (Figure2) comprises a series of basic substructure elements (1 to 5). In each element a RR and a CSTR (optional) are embedded in parallel arrangement so as to enable a different feeding, recycling and bypassing. It consists of a detailed reactor superstructure and simplified alternatives for feeding and recycle purification. Further, Duran' s model for the simultaneous heat integration is used in MINLP1. The results are therefore much more reliable and represents the identification of reactors present in the optimal structure (type, number and size of reactors, each of them determined by optimal inlet and outlet conditions). In order to achieve the convergence of non-trivial problems, the recycle ratio, R, in RR can be discretized. The initial reactor network superstructure for the MINLP 1 therefore consists only of the RRs along with mixers, splitters, heaters and coolers. It is therefore suitable either for isothermal or non-isothermal reactor network synthesis. In cases when in MINLP 1 a CSTR is identified by a high recycle ratio of the RR, the algebraic model of the CSTR substitutes the RR model in MINLP 2. When side streams and intermediate separation(s) are identified together with several RRs, the resulting superstructure in principle represents a CFR. 4. M O D E L L I N G THE R E C Y C L E R E A C T O R For the modelling of the RR the assumptions made are steady one-dimensional flow, instantaneous mixing and no axial diffusion. Constant density and ideal gas assumptions were used to fully specify the model. The method of Orthogonal Collocation on fixed Finite
982
Elements (OCFE) is used and extended by an appropriate MINLP formulation to simultaneously perform MINLP synthesis and the solution of DAE. In our case each finite element (FE) resembles one of the differential non-isothermal PFRs in the reactor train, RR. A multiple choice mixer and splitter with a recycle stream are added to each train in order to implicitly determine the optimal recycle. Instead of determining the optimal reactor outlet as a linear combination of FE outlets (targeting approach), we used the following procedure: a) discrete decision: the MINLP is used to determine an optimal FE in each existing train, b) continuous decision: the parallel Lagrange polynomial is used to determine an optimal point within the optimal FE which then becomes the optimal PFR outlet. It should be mentioned that in order to avoid problems with Lagrange multipliers in the MILP step, the optimal reactor outlet is determined by the last collocation coefficients of the optimal FE rather than by the linearizations of the parallel Lagrange polynomials. The superstructure can be modelled by the generalized disjunctive programming representation as follows: m a x Z : f(x)-~_,c r -~__~c; i
Global constraints
s.t.
i
h(x) = 0 g(x) <_o
r~ hi(x ) =0
Disjunctions for reactors
Disjunctions for optimal finite elements Disjunctions for recycle ratios Disjunctions for intermediate separation
ri (x) < O r r Ci -- ]/i
h i%(X)(x) . =<
LBij x = o
[il 0Iv Ril =
lRi2 =
F islr is, his (x) = 0
where
r
]
Lci --o
j
LRi3 =
_BiS x
[ris (X) ~_ O [
.Lc; -
llV
J ~ DFE
j
=0
(D-MINLP)
L cs - o
ic= DRR,xC= Rn,c >O,Y~_ {True, False}
The design variable - reactor v o l u m e - is obtained by summing up the increment volumes dV from the beginning of the reactor train up to the optimal outlet point. Since the heat balance of each differential PFR train has been extended for possible internal cooling/heating, the RR model could resemble the non-isothermal conditions in the reactor. The advantage of our approach is that the NLP is carried out only for the existing PFRs (all up to the optimal one) and outlet conditions are evaluated only in the optimal FE. The NLP problem is consequently smaller and more robust. Furthermore, since many differential
983 PFRs are embedded in the train, the corresponding flow/concentration relations have to be given only at the entrance and outlet of RR, which additionally decreases the non-linearity of the model. Proposed disjunctive model (D-MINLP) has been solved by the Modelling and Decomposition (M/D) strategy which is a special disjunctive programming scheme for process flowsheet, and the Boolean variables have been replaced by binary variables in order to carry out the optimization by modified OA algorithm (Kravanja and Grossmann, 1994). 5. E X A M P L E P R O B L E M
An industrial case of allyl chloride manufacturing is used. Allyl chloride is manufactured by means of non-catalytic chlorination of propylene in the vapour phase. In general, the reactions could best be represented by the non-isothermal Van de Vusse reaction scheme that involves four species: kl k2 k3 Consecutive reaction: A ~ B --~ C and parallel reaction: 2 A --~ D.
(4)
where reaction rates are kl = 1.5 9 10 -6 s -1, k2 - 4.4.108 s -1, k3 = 100 1.mol-~.s-1, and E1 = 662761 J/mol, E2 = 99410 J/mol, E3 = -33140 J/mol. The specific heat capacities, Cp, and entalpies for reactions, ArM i w e r e taken for the allyl chloride manufacturing problem. The reaction rate vector for components A, B, C, D respectively is:
R(x) = [- klCA
-
-
2 kl CA -k2Cb , k2Cb , k3 C2 ] k3CA,
(5)
The objective is the maximization of the yield of intermediate species B. Please note that the industrial example above involves additional reactant and a by-product. The objective of the optimization is to maximize the profit at the fixed production rate of allyl chloride at 7.56 mol/s. 5.1. Solution of MINLPI: reactor network synthesis The optimal structure of heat integrated allyl chloride problem comprises a set of three reactors and is shown in Figure 3: a) The first one is a preheated isothermal internally cooled RR (R = 3), determined by one finite element and followed by an additional cooling and an intermediate separation. b) The second reactor is an isothermal internally cooled RR (R = 3) determined by one finite element, fed by a preheated side stream and followed by the intermediate separation. c) The last one is again an isothermal internally cooled RR (R = 3) determined by one finite element and accompanied by a side stream feed and a successive cooler unit. The simultaneous heat integration yields only the consumption of the cooling utility of 0.662 MW. The optimal solution yields the profit of 18.961 M$/yr which is significantly better than just one preheated isothermal PFR in the industrial case (16.349 M$/yr). The solution was found by MIPSYN in the 12 th major MINLP iteration using the modified OA/ER algorithm (Kravanja and Grossmann, 1994). The size of the problem at the optimal solution was 934 equations with 913 variables at NLP step and 2593 equations with 4892 variables, 23 being binary ones at MILP step.
984
FEED-2
~_~
T /c:0':00
T Qlc=0'4~O0
]
I Qc=0'437 M W ~ ~ , " A / ~ _
by
E,~---~product s HEAT INTEGRATED PROCESS BENEFIT:18,961M$/yr Q = 0,0 MW Q = 0,662 MW
Figure 3" Optimal structure of the reactor network at MINLP 1 The optimal solution indicates that significant benefit could be gained if temperatures of the reactor system were carefully controlled and reaction could be accompanied by intermediate separation. 6. CONCLUSIONS AND SIGNIFICANCE The more general DAE model for PFRs and the compact superstructure for reactor networks have been proposed that together enable the simultaneous synthesis of reactor networks and the synthesis/optimization of the remaining process flowsheets. Their effectiveness has been studied and confirmed within the first level (MINLP 1) of the multilevel - hierarchical MINLP approach applied to the non-trivial industrial case study. The optimization models are much smaller and more compact, the results are more realistic and the selected optimal structures much simpler. Further investigation of the use of a hybrid model that would efficiently handle isothermal and non-isothermal process streams for the simultaneous heat integration within MINLP 2 is under way. REFERENCES: 1. M.A. Duran and I.E. Grossmann, Simultaneous Optimization and Heat Integration of Chemical Processes. AIChe J. 32, 123 - 138 (1986). 2. M. Feinberg and D. Hildebrandt, Optimal Reactor Design from a Geometric ViewpointI. Universal Properties of the Attainable Region, Chem. Engng. Sci., 52, 1637-1665 (1997). 3. Z. Kravanja and I.E. Grossmann, Multilevel-hierarchical MINLP Synthesis of Process Flowsheets, Computers chem. Engng, 21, $421-$426 (1997). 4. A. Lakshmanan and L.T. Biegler, Synthesis of Optimal Chemical Reactor Networks, Ind. Eng. Chem. Res., 35, 1344 (1996). 5. B. Pahor and Z. Kravanja, Simultaneous Solution and MINLP Synthesis of DAE Process Problems: PFR Networks in Overall Processes, Computers chem. Engng, 19, S181-S188 (1995). 6. C.A. Schweiger and C.A. Floudas, Synthesis of Optimal Reactor Networks, Computers chem. Engng, 23, $47-$50 (1999).
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rights reserved.
985
Synthesis of reactive and extractive Distillation Units Using Distillation Line Diagrams L. Jim6nez a, O. M. Wanhschafft b, V. Julka b aChemical Engineering and Metallurgy Department, University of Barcelona, Martf i Franqu6s, n~ 08028 Barcelona, Spain* bAspen Technology, Inc., Ten Canal Park, Cambridge, Massachusetts, USA 1. INTRODUCTION For any reactor, the composition of reaction mixture is limited by chemical equilibrium. This limitation can be overpassed only by selective product removal from the reaction zone. Nowadays membrane reactors, recycling systems (reactor and distillation column joint by recycle) and combined reaction-mass-exchange systems (reactive distillation) have been successfully implemented in industrial processes. Currently, the low throughput of available membranes makes impracticable the applicability to large-scale chemical processes. In this concern the challenge is to develop technology based on high capacity mass-exchange process for product removal from the reaction zone. Therefore, reactive distillation has received an increasing attention as a promising alternative to the classical approach that considers separately and consecutively reaction and separation. However, reactive distillation is not advantageous in every case, and there is need for a systematic investigation concerning the phenomena (Gmehling et al, 1993). The design of reactive distillation is currently based on expensive and time consuming sequences of laboratory and pilot plant experiments. Distillation lines diagrams and residue curve maps (RCM) are build based solely on the system physical properties: vapor-liquid equilibrium (VLE), liquid-liquid equilibrium and solubility data. Although missing model parameters can be estimated from several group predictive methods, physical properties' model accuracy (Carlson, 1996) had to be checked, especially when the specifications for high purity products are based on maximum impurities. In a non-reactive mixture the temperature always increases along a residue curve line, and the singular points are either nodes (stables or unstables), or saddles. This information allows us to assign, without any computing, the RCM topology for the whole composition space, making RCM a promising technique in the early phase of development of any project (i.e. solvent screening, preliminary design) and increasing the engineer productivity. Several authors have transferred the RCM concept to reactive distillation by overlaying a chemical reaction. Barbosa and Doherty (1988) assume chemical equilibrium, Solokhin et al.
Present address: Chemical Engineering Department, University Rovira i Virgili, Carretera de Salou s/n, 43006 Tarragona, Spain. Tel.: 34-977-559617; fax: 34-977-559667/21; e-mail: [email protected]; One of the authors (L. Jim6nez) thanks CIRIT (Generalitatde Catalunya, Spain) for the financial support.
986 (1990) apply rate equations while Venimadhavan kinetics expressions.
et al.
(1994) use homogeneously catalyzed
3. E X T R A C T I V E D I S T I L L A T I O N Solvent selection is the key factor in extractive distillation due to the high influence of the solvent recovery system and recycles in the global process (Momoh, 1991). The huge amount of possible solvents had lead to previous selections based upon heuristics (effect in VLE, chemical stability, difference in boiling point). Usually, there is any solvent that matches all these characteristics and thereby, compromise solutions, using cost criteria and additional constraints (melting point, form a characteristic azeotrope, viscosity) are required. The parameter most widely accepted to rank a set of feasible entrainers is selectivity. To compare among different solvents, it is a common practice to consider the situation of infinite dilution (Si[). The higher the selectivity, the better the solvent.
3.1. Propylene oxide solvent selection The purification of propylene oxide (PO) can be simplified to three major separations. Crude propylene oxide is fed to the first column where the light components are removed. In the second one, PO is withdrawn as top product, while the third separation includes the solvent recovery system. Physical properties for the multicomponent system (PO, acetaldehyde, methanol, npropionaldehyde, acetone, water, propylene, cis-l,3-pentadiene and 2-methyl-pentane) was estimated using NRTL-Redlich Kwong equation of state with all parameters retrieved from ASPENPLUS database. The results have a similar behavior for all the separations involved, and the solvents rating is: n-alquil > alquil-benzenes > chlorine-benzenes. We can conclude that selectivity is a useful technique to cluster the solvents in groups, but to go further additional criteria should be involved (i.e. toxicity, volatile organic carbon restriction).
TM
4.- R E A C T I V E D I S T I L L A T I O N Reactive RCM show that the reaction modifies the diagram topology. However, for reactive systems we can not make any definitive conclusion about the temperature change along a residue curve line. In this work we will restrict our analysis to systems in which a liquid and a vapor phase are at equilibrium and where a set of series and parallel reversible reactions occur in the liquid phase. If we suppose ideal vapor phase behavior, the chemical equilibrium constant (Keq) in the liquid phase can be written as
Keq = (I-I Y ~' ) (I-I x~' ) =ky "kx
(1) A set of transformed mole compositions, Xi and Yi (Ung and Doherty, 1995) defined as follows will be used.
)-i
X i = I_I)ToT(I)Re f --l)i x i (1)Ref)_1XRefXRef(i = 1..... C-R)
(2)
987 where l)i T is the row vector of the stoichiometric coefficients for component i in each of the R reactions, and XRefis the vector of mole fractions of the R reference components in the liquid phase. These new variables behave in a similar way than mole fractions (~iCl RX
i = 1
)in non-
reactive mixtures, and can be thought as reaction-invariant compositions. We can represent multicomponent systems in a lower dimensional composition space (C-R-1 degrees of freedom). For example, in a ternary mixture with a chemical reaction, all residue curve lines collapse in just one.
4.1. Re-esterification of methyl acetate with butanol One of the by-products in the Poly-(vinyl alcohol) (PVA) production is an azeotropic mixture of MeOH + MeAc. In the past, this mixture was sold as paint solvent, but Volatile Organic Compounds legislation has cut down drastically this market. A reactive extractive distillation process was designed (Jim6nez, 1997) to produce butyl acetate and recycle the methanol. Physical properties were estimated using NRTL model with Hayden-O'Connell for vaporphase non idealities. All parameters used were based on Jim6nez (1997). The temperature dependence of the molar fraction equilibrium constant, Kx is as follows. 1265 lnK x = 3 . 5 1 8 - - (3) T/K 1.0
MeOH
-
1.0
,g
0.8
:~. 0.6
0.6
0.4
,,6 0.4 ,5 0.2
0.2
~ o.o
:~ 0.0
0.2 0.4 0.6 0.8 Transformed mole composition for BuAc, XBuAc
1.0
Fig. 1. RCM in molar transformed composition for re-esterification of MeAc with BuOH and o-xylene at 101.3 KPa.
0.0 [ ...............................................................................'2iiiiU22?22.::=-= ............................. 0.0 0.2 0.4 0.6 0.8 1.0 MTBE Mole fraction of reactant isobutene, XiB IB
Fig. 2. Non reactive RCM of MTBE + MeOH + IB at 101.3 KPa.
An accurate analysis of the quaternary non-reactive RCM diagram reveals that there is just one distillation region. The two ternary distillation boundaries go from pure MeOH and MeAc to BuOH + BuAc azeotrope. The MeOH + MeAc azeotrope acts as an unstable node. These two aspects make necessary the use of an entrainer for a proper product separation. Solvent selection analysis, based on selectivity at infinite dilution was performed (Jim6nez, 1997), and o-xylene was selected as a promising alternative. To compute the reactive RCM (Figure 1), the reference component for the transformed molar composition was BuOH. The reactive boundary generates two different regions, but fortunately, the working conditions, even during start-up and shut down, lie far and there is no need for any boundary-crossing strategy.
988
4.2. Synthesis of MTBE About 20 million tons of MTBE (Methyl tertiary butyl ether) are produced every year by the liquid-phase reaction of isobutene and methanol. Liquid activity coefficients ~/i for the MTBE example can be computed using UNIQUACRedlich Kwong (for component related parameters see ASPENPLUS TM Database). Thermodynamic chemical equilibrium as a function of temperature for the heterogeneous catalysed MTBE-synthesis was reported by Nijhuis et al. (1993). k eq 284.exp f (T / K) =
Z(T/K) = -1493.(T -1 -To-1)-77.4"log Too + 1.1 l'10-6"(T 3 -To3)-6.28"lO-l~
+0"508"(r-r~
-r~
4 -To 4)
where To is 298.15K. An interesting note about this separation concerns the fact that the presence of a non-reactive MeOH + MTBE azeotrope plays an essential role in the ability to recover high-purity MTBE from a chemical equilibrium amount of MTBE + MeOH + isobutene (IB). The RCM given in Figure 2 shows that this azeotrope and the high purity IB azeotrope are linked by a distillation boundary and, consequently divides the non-reactive triangle in two regions. Assuming that the chemical equilibrium is achieved, it is possible to pass the non-reactive distillation boundary, just by effect of the reaction in the topology. Hence, one stable point at region of MTBE/IB vertex disappears, and all residue curve lines collapse and end at pure MeOH. As example, if n-pentane is selected as the entrainer, the non-reactive RCM is shown in Figure 4. A distillation separatrice, linking the three binary azeotropes acts as an additional constraint.
Fig. 3. RCM for MTBE synthesis using npentane as entrainer at 101.3 KPa.
Fig. 4. RCM in transformed molar compositions for MTBE synthesis and npentane as solvent.
To compute the transformed molar composition shown in Figure 5, MTBE was the reference component. The shape of the solution space is a triangle where pure IB, n-pentane and MTBE are each located at a vertex. MTBE does not form a vertex since it can not exist as a pure component due to the equilibrium reaction. The hypotenuse is the reactive edge of IB + MeOH + MTBE. The other two edges are the corresponding non-reactive binary systems. On
989 the n-pentane + MeOH edge, the system exhibits an intermediate boiling point azeotrope that divides the diagram in two distinct regions. This azeotrope acts as an additional constraint in the product purification, and thereby, n-pentane is not a good entrainer for this separation.
4.4. Synthesis of formaldehyde VLE data in aqueous and methanolic formaldehyde mixtures are needed for the design of equipment to absorb formaldehyde (FA), a very common unit operation in the FA production for recovery as well as for enviromental purposes. Figure 6 shows the physico-chemical model assumptions (Hasse and Mauer, 1991), including a set of ten reactions (consecutive and competitive). For practical purposes, it is possible to neglect the small amounts of monomeric FA in the liquid phase (=0.4% in a ternary equimolar mixture) but not in the vapor phase. The majority species in the liquid phase are hemiformal (HF) and methylene glycol (MG). Formaldehyde polymers of higher chain length (polyoxymethylene glycols and hemiformals) are virtually not present in the vapor phase. All the reactions are reversible and, in order to get a composition profile in a lower dimension, total water, FA and MeOH mole fraction were used (Figure 7). The minimum boiling point binary reactive azeotrope found in the water + FA system acts as an unstable node. When the pressure is decreased (Figure 7), the binary reactive azeotrope "opens down" (i.e. disappears), and the residue curves do not exactly go through this point. If the feed process is close to the line that links the MeOH vertex with the binary azeotrope, operability problems may be expected related to changes in the feed composition. For smilar feed composition lying in different zones, both top and bottom products could be similar. However, for the feed with less MeOH, the composition profile would approximately be a distillation line which runs close to the water axis and then to the MeOH axis to reach the azeotrope, while for the feed composition rich in MeOH, the column profile will change dramatically, following the FA axis and then the MeOH axis. Moreover, in the contest of distillation, this azeotrope acts as a tangent pinch, because there is a high curvature in regions close to the FA + water edge. +
FA FA FA
+ W
W MeOH
MG
~ ~
MeOH
Vapor
MG HF
<
"~
HF
1• 0
FA ~
o.8
" ~
.........3.s a,m
~
- - - ' - - 2.5 otto - - 1.5 a r m
~
.......
0.6
FA
W
MG
MeOH
LiauM
HF
FA MGn. ]
+ +
W W
~ ~
MG MG n+
W
FA HFn. 1
+ +
MeOH HF
4-+ *--+
HF HF n
MeOH
---
'1.0 atm ,o._______,,tm ~s
~ 0.4 E
0.2
~ " '' ...........
0.0 +
I
Fig. 5. VLE and reactions in the formaldehyde + water + methanol system. 5. CONCLUSIONS
'4.5 arm
~
~
0.0 MeOH
0.2
0.4
0.6
0.8
T o t a l m o l e f r a c t i o n of w a t e r , XH20
1.0 H=0
Fig 6. Azeotrope dependence with pressure for FA + MeOH + water.
990 It seems safe to state that apply the RCM to non-ideal reactive and extractive systems require a substantial effort. The new graphical tools for the synthesis and design dramatically change the way complex distillation processes, including reactive systems, can be analysed, either for retrofit (solvent selection for MTBE), troubleshooting (operational problems in formaldehyde process) or new designs (BuAc re-esterification). For extractive distillation, choose solvents just on the basis of selectivity tends to emphasise the cost of the extraction column, whereas the cost in the solvent recovery system is sometimes significant and may control the cost of the global process. RCM show that the entrainer presence produces important changes in the activity coefficients but we can neglect these influences in equilibrium constant. The in-situ removal of product form the reaction zone causes equilibrium-limited reactions to be shifted forward, thus allowing high conversion. Moreover, the chemical reaction can change the topology of the non-reactive RCM, and therefore, we can create or eliminate distillation boundaries and avoid azeotropes as products, offering new options for the separation. Influence of operating pressure is shown in the FA and MTBE examples. In the first one, at an adequate low pressure, the reactive FA + water azeotrope disappears, while in the second one, very different composition profiles are obtained. For extractive and reactive distillation, accurate analysis of the non-reactive RCM is needed. Moreover, in order to visualise the solution, transformed molar composition (Ung and Doherty) provides information on the feasibility and column sequencing. This is done by determining the stables and unstables nodes, detecting any reactive azeotrope and reactive distillation boundaries, and targetting them as potential bottoms and distillate products, respectively. Examples of benefits are cutting down iterations (increasing the quality and efficiency of process modelling), quickly finding suitable entrainers, improving process designs for new plants and significant capacity increases through changes in process sequences of existing plants. REFERENCES Aspen Technology Inc., 1998, "AspenPlus TM Reference Manual", Cambridge, MA, USA. Aspen Technology Inc., 1998, "Aspen Split TM Reference Manual", Cambridge, MA, USA. Barbosa, D., Doherty M. F., 1988, Chem. Eng. Sci., 43, 541-550. Carlson, E. C., 1996, Chem. Eng. Prog., 35-46. Doherty, M. F., Perkins, J. D., 1988, Chem. Eng. Sci., 43, 541-550. Gmehling, J., Li, J., Schiller, M., 1993, Ind. Eng. Chem. Res., 32, 178-193. Hasse, H., Maurer, G., 1991, Fluid Phase Equilibria, 64, 185-199. Jim6nez Esteller L., 1997, PhD Thesis, University of Barcelona, Spain. Momoh, S. O., 1991, Sep. Sci. & Tech., 26 (5), 729-742. Nijhuis, S. A., Kerkhof F. P., Mark, A. N. S., 1993, Ind. Eng. Chem. Res, 32 (11), 2767-2774. Solokhin, A. V, Blagov, S. A., Serafimov, L. A., 1990, Theor. Found. Chem. Engng., 24, 103109. Ung, S., Doherty M. F., 1995, Ind. Eng. Chem. Res., 34, 2555-2565. Venimadhavan, G., Buzad, G., Doherty M. F., Malone, M. F., 1994, AIChE. J., 41) (11), 18141824.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
991
Separation System Synthesis of Fractional Crystallisation Processes With Heat Integration L. A. Cisternas a, C. P. Guerrero a and R. E. Swaneyb* aDepartment of Chemical Engineering, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile bDepartment of Chemical Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, WI 53706, USA A methodology is presented for the synthesis of fractional crystallisation process with heat integration. The methodology is based on the construction of three networks. The first network is based on the identification of feasible thermodynamic states. Using equilibrium data for a candidate set of potential operation point temperatures, a network flow model is constructed to represent the set of potential separation flowsheet structures that can result. The second network is used to represent the variety of tasks that can be performed at each multiple saturation point. Multiple saturation nodes can be used for different tasks depending on the characteristic of the input and output streams. These tasks include cooling crystallisation, evaporative crystallisation, reactive crystallisation, dissolution, and leaching, This multiple task condition for each equilibrium state is modelled using disjunctive programming and then converted into mixed integer programming. Heat integration is included using a heat exchanger network which can be regarded as a transshipment problem. The method is illustrated through the design of a salt separation example. 1. INTRODUCTION In the last seven years several articles have been published on the synthesis of crystallisation-based separation. There are two major approaches for the design of the flowsheet configuration and its operating conditions. In one approach, the phase equilibrium diagram is used for the identification of separation schemes [1-3]. This procedure can be mixed with heuristic rules to determine the flowsheet. While this procedure is easy to understand, it is relatively simple to implement only for simple cases. For more complex systems, such as multicomponent systems, multiple temperature operation, or systems that form double salts, the procedure is very difficult to implement because the graphical representation is complex and because there are many alternatives to study. The second strategy is based on simultaneous optimisation using mathematical programming based on a network flow model between feasible thermodynamic states [4-5].
*The authors will like to thank the support received for this project from CONICYT (Chile), through the project Fondecyt 1990956.
992 Although in terms of energy requirements crystallisation requires much less energy for separation than do distillation or other commonly used methods of purification, energy cost can have an impact on process economics and process design [6]. However, there is not literature on process synthesis of these systems with heat integration. In this work a methodology is presented for the synthesis of fractional crystallisation separation schemes with heat integration. 2. MODEL DEVELOPMENT 2.1. Networks for Fractional Crystallisation with Heat Integration The model proposed in this paper is composed of three major networks that will be described in this section. These networks are: (1) the thermodynamic state network, (2) the task network, and (3) the heat integration network. Based on these networks a mathematical model will be generated to solve the process synthesis problem. The first network is based on the identification of feasible thermodynamic states. Using equilibrium data for a candidate set of potential operating point temperatures, a thermodynamic state network flow model is constructed to represent the set of potential separation flowsheet structures that can result. This representation was introduced by Cistemas and Swaney [4-5]. The crystallisation, dissolution, leaching, evaporation, and dilution steps can be represented in terms of material flows between particular thermodynamic equilibrium states. Therefore, knowing the phase, compositions, and temperatures pertaining to each state, a network model can be constructed. Figure 1 shows the thermodynamic state network representation for a two solute system at two temperatures with double salt formation (intermediate product) at one temperature. The second network is the task network (See Figure 2). Each multiple saturation state can be used for different task depending on the characteristic of the input and output stream. For example, if solvent is added to an equilibrium state the task can be: (1) a leaching step if the feed is solid; (2) a cooling crystallisation step if the feed is a solution with a higher temperature; or (3) a reactive crystallisation step if the feed is a crystalline material that decomposes at this temperature or in the solution fed to this state (examples are the decomposition of schonite to form potassium sulphate or the conversion ofNa2SO4.10 H20 to Na2SO4). After the tasks are identified, it is necessary to determine what type of equipment can perform each task. In this work a single equipment unit is assigned at each task. Several alternatives exist for leaching, which include in situ leaching, vat leaching, agitation leaching, etc. [7]. Potential crystallisers can be chosen so that it meets the requirements for (a) product size, (b) product quality, and (c) scale of operations [8]. The third network, a heat exchanger network, can be regarded as a transshipment problem [9]. This transshipment problem can be formulated as a linear programming problem. In this representation hot streams (source nodes) and cold streams (destination nodes) corresponds to the arc in the thermodynamic state network. 2.2. Mathematical Formulation Having derived the networks for the separation problem, a mathematical programming formulation is presented for each network to select the optimum flowsheet alternative of the separation sequence.
993
.-~-. Iz
x
/
X
i
x\ \
/
....
/
\x~),--
--\x/y'
I
~,/
/ ~" ~
"~.
,, "~. ,,..
, . - . , -." ..~,.,, ; ,s,
\ I'"
I
~' '~ .,''~, ,. s / : .......... ,, C K_K-2,'
<,
- - ........... "..
lI '"
-:
\~
M I. "".3! ".~_." jAk"
s .~ .." I
Fig. 2 Task network. Fig. 1. Thermodynamic state network F=feed, S=solvent, M=multiple saturation point, I=intermediate product, CC=cooling crystallisation, EC-evaporative crystallisation, L-leaching
P=product,
9 The thermodynamic state network The mathematical formulation for this network is the same developed by Cisternas [5] which gives a LP problem if coprecipitation is not allowed and NLP problem if coprecipitation is an alternative. Here a brief description is given. Only the sets that are needed to formulate the mathematical problems of the other networks will be defined. First, the set of thermodynamic state nodes will be defined as: S={s, all nodes in the system}. This includes feeds, products, multiple saturation points or operation points, and intermediate solute products. The components, solutes and solvents, will be denoted by the set I= {i}. The arcs, which denote streams between nodes, will be denoted by L={l}. Each stream l is associated with the positive variable mass flow rate wl and with the parameter composition of each component Xl,~. Having defined the sets, parameters, and variables that describe the overall network for the separation problem, the constraints that apply are in general form as follows" h(w) = 0
(1)
where w is a vector of mass flow rates and h(w) is a vector of equality constraint functions. These constrains includes: mass balances for multiple saturation nodes and intermediate product nodes, mass balance for each component in multiple saturation nodes and intermediate product nodes, specification for feeds flow rates, and specification for product flow rates [5]. 9 The heat integration network In order to consider heat integration, the approach followed by Papoulias and Grossmann [9] will be used in this work. We consider the case without constrained matches for simplicity in this work, but constrained matches can be easy included. First, it is considered that there is a set K={k} of temperature intervals that are based on the inlet temperatures of the process streams, highest and lowest stream temperatures, and of the intermediate utilities whose inlet
994 temperatures fall within the range of temperatures of the process streams. The following index set are defined: Hk={ l / l ~ L, hot stream that supplies heat to interval k E K}, Ck={ l / l E L, cold stream that demands heat from interval k E K}, Vk={ m /hot utility that supplies heat to interval k ~ K}, and Wk={ n / cold utility that demands heat from interval k E K}. Figure 3 considers a given temperature interval k, where Qm v , Q W and Rk are positive variables that represent heat load of hot utility m, heat load of cold utility n, and heat residual exiting interval k respectivily. (CpAT)Hlk and (CpAT)Clk are known parameter that represent the heat content per unit mass of hot stream l ~Hk and cold stream l ~Ck in interval k. The only constraints that apply are heat balances around each temperature interval k: Rk - R k - 1 -
ZQVm + ZQWn : m~V k
n~W k
Z
Wl ( C p A T ) g -
l~H k
Z
Wl ( C p A T ) C
k s K
(2)
I~C k
where R0 and RK are equal to zero. 9 The task network A task network is constructed for each multiple saturation point node s. Therefore, a subset is defined as SM={S / s eS, multiple saturation nodes}. Let T(s)={t} define the set of conditional tasks in the multiple saturation point node s ~ SM. This network requires the use of discrete variables, yts to represent the choices of task t within each node s ~ SM, with which the model becomes a mixed integer linear program. The following subsets are defined: Sin(s) = {l / l E L, is an inlet to node s, s ~ SM}, s~ = {l / l ~ L, is an outlet from node s, s E SM}, S~ = {l / l e L, is the solid outlet from node s, s ~ SM}, and S~ = {l / l ~ L, is the solvent outlet from node s, s E SM}, The variables are defined as follows: Ginlt are the internal mass flow rates from stream l to task t, G~ are the internal mass flow rates from task t to stream l, Qtsc are the heat loads of crystallisation or dissolution in each task t, Qtss are the heat loads of evaporation in each task t, VCts and FCts are the variable and fixed costs incurred for the equipment associated to the task t. The parameters are defined as follows: HQts is the crystallisation/dissolution heat per unit mass product in each task t of node s ~SM, liSts is the latent heat of evaporation in task t of node s ~SM, ats and/~s are the fixed and variable cost coefficients of task t of node s ~SM. At equilibrium the heat of crystallisation is equal and opposite in sign to the heat of solution. Using the heat of solution at infinite dilution as equal but opposite in sign to the heat of crystallisation is the equivalent, however, of neglecting the heat of dilution. With many chemicals the heat of dilution is small in comparison with the heat of crystallisation and the approximation is acceptable. The formulation, using disjuntive programming (a set of constraints of which at least one must be valid), is given by:
wl =
~ Gltit1 t~T(s)
l
E S in ( s )
,
s ~ SM
995
Wl =
out ~-' "-"tl t~T(s) in
Z
l s S in (s)
clt
:
l ~ S out (s)
,
~ out
Y~
l s S ~ (s)
s~S M
t~T(s)
"'tl
,
(3)
_
Yts
I
F C ts = a ts VC ts = fits
in
~
=
HQts
~out "-"tl
Q s = H S ts ~"~tl~
l~~
~"d
(s)
l ~ ~.~ ,~ ~ (s)
]
-0[
vlVC,
G lt
I~S in (s)
QC
~Yts
=o /
t ~ T(s),
s ~ S M(s)
Lo :o I
_
g(Yts ) = True g(yts) represents the logic relations between Boolean variables to select the task based on
input/output streams properties [10]. The model of equation 3 must be transformed into a MILP form [11]. 9
The objective f u n c t i o n
The objective function is to minimise the venture cost. The following equation can be used as an objective function, min
c
c
s
s
v
-
~ ~ ( FCts + VCts + Cts Qts + Cts Qts ) + ~ cmQm + s s S m t~T(s) m~V new
-_ycO
(4)
equation 4 represents the total cost given by the investment and utility cost. In this way, the objective function in equation 4, subject to constraints in equations 1 to 3 defines a mixed integer linear programming problem. The variables to be optimised are mass flow rates, Wl ; heat load of hot utility m, Qm v ; heat load of cold utility n, QnW; heat residual exiting interval k, Rk ; the internal mass flow rates from stream l to task t, Gin# ; the internal mass flow rates from task t to stream l, G~ ; heat loads of crystallisation or dissolution in each task t, Qts c ; the heat loads of evaporation in each task t, Qts s ; variable and fixed cost, VCts a n d FCts ; and the binary variables, yt~. The numerical solution to the MILP problem can be obtained with standard algorithms. 3. ILLUSTRATIVE EXAMPLE This example considers the production of KC1 from 100,000 ton/year of sylvinite (47.7% KC1, 52.3% NaC1). The data needed was taken from [5] and [6]. The optimal solution (see Figure 4) divides the feed into two parts, and both NaC1 and KC1 are obtained by leaching. Therefore, the KC1 produced can have insoluble impurities. Further solid operation, for example recrystalisation, can be needed. If the cost of KC1 leaching increases because of these solid operations, the model will select cooling crystalisation to produce KC1.
996 sylvinite
Rk_l C'W
Z
Z wz (CpAT)c
wl (CpAT)~
leH k
leC k
~1 9 Interval
ZI "7
Leaching
k
meV k
\
[ Leach g
n~Wk
Rk KC1 Fig. 3. Heat flows in temperature interval k
NaC1
Fig. 4. Solution to example.
4. CONCLUSIONS AND COMMENTS The objective of this paper has been to present a method for determining the desired process flowsheet for fractional crystallisation processes. To achieve this goal, a systematic model was introduced consisting of three networks; the thermodynamic state network, the heat integration network, and the task network. Once the representation is specified, the problem is modelled as a MILP problem. An example was included to illustrate the method. The size, shape, and size distribution of particles in a particulate product are important for several reasons. Therefore, it is important to include solid processing operation such as centrifugation, washing and drying. These operations are not included in this model, but further development is underway. REFERENCES
1. 2. 3. 4. 5. 6. 7.
L.A. Cistemas and D.F. Rudd, Ind. Eng. Chem. Res., 32 (1993) 1993. D.A. Berry and K.M. Ng, AIChE J., 42 (1996) 2162. D.A. Berry, S.R. Dye and K.M. Ng, AIChE J., 43 (1997) 91. L.A. Cisternas and R.E. Swaney, Ind. Eng. Chem. Res., 37 (1998) 2761. L.A. Cisternas, AIChE J., 45 (1999) 1477. S. Rajagopal, K. M. Ng and J.M. Douglas, Ind. Eng. Chem. Res., 27 (1988) 2071. R.K. Prabhudesai, in Schweitzer P.A., Handbook of Separation Techniques for Chemical Engineers, second edition, McGraw-Hill Book, (1988). 8. R.C. Bennett, in Myerson A., Handbook of Industrial Crystallization, ButterworthHeinemann, (1993). 9. S.A. Papoulias and I.E. Grossmann, Comp. & Chem. Engng., 707 (1983). 10. R. Raman and I.E. Grossmann, Comp. & Chem. Engng, 15 (1991) 73. 11. M. Turkay and I.E. Grossmann, Ind. Eng. Chem. Res., 35(1996) 2611.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
997
Optimization of bleed streams in evaporation systems based on pinch analysis" new approach D. L. Westphalen a and M. R. Wolf Maciel b aDepartment of Chemical and Food Engineering - Maufi Institute of Technology, Estr. das Lfigrimas, 2035 - Silo Caetano do S u l - SP, 09850-900, Brazil. E-mail: [email protected] bFaculty of Chemical Engineering- Campinas State University, CP 6066 - Campinas- SP, 13081-970, Brazil. E-mail: [email protected] Evaporation systems are separation processes widely used in the chemical industry and food processing operations. Several alternatives for energy savings have been proposed for this type of equipment. However, new paradigms for process design have been developed where unit operations can not be optimized in isolation, but as part of an overall process. In this way, evaporation systems can be integrated with a background process by bleed streams. In this work, a new algorithm based on Pinch Analysis was developed to optimize bleed streams of evaporation systems. 1. INTRODUCTION Evaporation systems are separation processes widely used at chemical industries. There is a lack of established methodologies for process design of this unit operation because of the large number of possibilities for number of effects, effects configuration (frontal, reverse or mixed) and inclusion of accessories (mechanical compressors, thermocompressors, heat exchangers, flash cooler, bleed streams, and condensate recovery systems). Some guidelines can be found in the literature for the process integration of multiple effect evaporators [1,2]. These guidelines are based on a heat cascade representation of the evaporator in a temperature - enthalpy diagram, against the grand composite curve of a background process. Westphalen and Wolf Maciel [3] proposed a new rigorous temperature - enthalpy diagram for evaporators, called heat curves, where some aspects are not neglected, as: boiling point rise, effect of pressure on latent heat of water, sensible heat of liquid streams, heat of mixing, effects configuration and inclusion of accessories. Figure 1 shows some examples of heat curves of evaporation systems, where each effect is represented by a trapezoid. The optimization of evaporators have been traditionally performed as a stand-alone unit operation. This optimization includes basically the optimum number of effects in a multipleeffect configuration and the use of recompression systems. Smith and Linnhoff [4] established some general principles for the optimization of separation processes in the context of a global process. From the same principles, Smith and Jones [5] developed an algorithm for the heat integration of evaporation systems based on the temperature- enthalpy diagrams. These authors assume that heat transfer coefficients have the same value in all effects, and this assumption leads to a conclusion that the equipment minimum capital cost will be obtained if all effects operate using the same temperature difference. However, it is observed that heat transfer coefficients are strongly influenced by temperature and solids concentration [6] and therefore their conclusions are not suitable for real cases. Indeed, the algorithm proposed by Smith and
998 Jones is based on shifting effects pressure, however, in same cases as food processing, shifting pressures is limited because of restrictions as undesirable reactions. In this work, it was developed a new methodology for process integration of evaporation systems. This tool is based on the rigorous heat profile of a evaporation system and its placement against the grand composite curve of the background process. Steam (a)
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Enthalpy flow (a) frontal feed; (b) reverse feed; (c) mixed feed Fig. 1. Heat curves of evaporation systems 2. USE OF BLEED S T R E A M S IN M U L T I P L E - E F F E C T E V A P O R A T O R S Vapor bleeding is a strategy for reducing energy consumption in industries where a evaporator is used as a separation process. In this technique, a fraction of evaporated water from each effect is used as a heating medium for a process stream, instead of been used exclusively at the next effect in a multiple-effect configuration. Schwartzberg [7] shows an example of a beet sugar evaporator with vapor withdrawal. Adapted data from this author is presented in Figure 2 and simulations of this equipment were performed using the software Evsim [8]. When no steam is extracted from any effect, it was calculated a steam consumption of 22712 kg/h, while, with bleed streams shown in Figure 2 the calculated steam consumption was 58304 kg/h. It has to be stressed that 52900 kg/h total steam is extracted from the evaporator and for this purpose, only 58304 - 22712 = 35592 kg/h of additional steam were spent. The inclusion of bleed streams causes a redistribution of vapor flows through the effects, leading to a smaller amount of vapor sent to condenser, as shown in Figure 3. Therefore, the vapor withdrawal allows a better usage of evaporated water from
999 each effect. It can be concluded that bleed streams in multiple-effect evaporators is a feasible scheme for energy savings in the overall process context.
27026kg/h
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(a) No steam is extracted; (b) Steam is extracted by bleed streams Fig. 3. Heat profiles of beet sugar evaporator Although bleed streams are employed for a long time, it can not be found in the literature a general methodology for design and optimization of these streams. Itahara and Stiel [9] proposed an algorithm for optimization of bleed streams in a configuration where extracted vapor is used to preheat the evaporator feed stream. Therefore, only stand alone evaporators can be optimized using this algorithm and in a new paradigm, any unit operation should be designed as an integral part of the overall process. Leal et al. [ 10] shows the results of employing vapor bleeding in a cane sugar plant, and it is clear that all proposed process modifications were based on engineers experience and intuition. 3. A L G O R I T H M F O R O P T I M I Z A T I O N OF B L E E D S T R E A M S In this work, it was developed an algorithm for optimization of bleed streams based on Pinch Analysis principles. The basic concept of this methodology is to fit the evaporator heat curve to the process grand composite curve. A graphical representation called evaporator placement diagram was created and is exemplified in a case study. The algorithm, was implemented in a evaporation systems simulator [8] and can be described by the following steps: Step 1) From hot and cold streams of the background process, and a specified process minimum temperature difference, calculate the hot and cold utility targets and locate pinch temperature, using the "Problem Table" algorithm.
1000
Step 2) From the evaporation system previously described by the user (number of effects, liquid flow pattern, accessories, feed stream information, product concentration and operating pressures of all effects), identify the effect operating with the smallest pressure, that is, connected to the condenser. Step 3) Verify if there is a bleed stream connected to the effect. Step 4) Calculate temperature of the effect and shift this value by-ATtain/2. Step 5) Compare the effect shifted temperature and process pinch temperature. If the effect shifted temperature is lower than process pinch temperature, zero is assigned to the bleed stream flow rate and the algorithm search for the next effect and goes back to step 3. Step 6) Calculate the heat flow from the grand composite curve for the effect shifted temperature. Step 7) Verify if this heat flow is located inside an envelope in grand composite curve. This verification is done comparing the heat flow calculated in step 6 with the hot utility target. If this heat flow is greater than the hot utility target, the heat duty of hot utility target is assigned to the bleed heat duty of the analyzed effect. Otherwise, the heat duty calculated in step 6 is directly assigned as the bleed heat duty. Step 8) Calculate the bleed stream flow rate as the bleed heat duty divided by the latent heat of water at effect pressure. Step 9) Verify if the bleed stream flow rate is greater than the maximum bleed stream flow rate specified by the user. When heat duties existing in a evaporator are too small when compared to process heat duties, an infeasible design may be developed. In case the calculated flow is greater than the specified value, this specified value is assigned to the effect bleed stream flow rate. Step 10) Search for the next effect in the multiple-effect structure and goes back to step 3. The procedure is repeated until all effects, from the lowest to greatest pressure, are analyzed. 4. CASE STUDY The proposed algorithm for optimization of bleed streams will be illustrated in a case study: the production of crystal glucose plant, using data presented by Klemes et al. [ 11]. The industrial process for production of glucose consists basically of the hydrolysis of corn starch, followed by product purification [12]. Stream data for this process were adapted from data presented by Klemes et al. [11 ] and are shown in Table 1. Evaporator data is not included in Table 1, as in the original paper, because the integration of this equipment will be performed using the algorithm developed in this work. Hot and cold utility targets and pinch position were calculated using the problem table method, resulting in 2718 kW, 634 kW and 56~ respectively, for a minimum temperature difference of 8~ In this process, there is a triple effect evaporator with frontal feed configuration. This equipment was simulated using Evsim [8] from data presented in Table 2, and a steam consumption equal to 1781 kg/h was calculated. In this process, two steam levels are available: 110~ (144 kPa, low pressure level) and 150~ (477 kPa, medium pressure level). From the grand composite curve, it can be calculated that the hot utility target can be divided in these two steam levels: 2399 kW and 319 kW, or 3883 kg/h and 550 kg/h for the levels 110~ and 150~ respectively.
1001 Table 1. Stream data for crystal glucose plant Correntes
Ti (~
1A - Starch juice 1B - Starch juice 3 - Filtr. juice 4 - Thin juice 5 - Thick juice 6 - Water 7 - Air 8 -Cooking 12 - Hydr. juice 13 - Cryst. juice 14 - Cooking vapors 18 - Crystallis. Table 2. Evaporator data Feed stream: Flow rate Temperature Concentration Product stream concentration Steam pressure Pressure of effect # 1 Pressure of effect #2 Pressure of effect #3
25 50 60 50 68 38 10 70 95 70 60 47
Tf (~ 50 145 75 96 80 70 55 70 60 47 60 32
CP (kW/~ 7.2 7.6 8.9 9.6 5.3 18.7 4.4 9.3 4.3 3.7
AH (kW) 180.0 722.0 133.5 441.6 63.6 598.4 198.0 1410.0 -325.5 -98.9 -1184.0 -55.5
9400 kg/h 96~ 32% 56% 140 kPa 82 kPa 40 kPa 25 kPa
The algorithm developed in this work was used to optimize the integration of the evaporator and the background process, using 3000 kg/h as the maximum bleed stream flow rate. Figure 4 shows the evaporator placement diagram, and Table 3 summarizes the results. The integration of the evaporator and the global process allows an economy of 3435 kg/h of steam at low pressure level. This value is obtained from the sum of all bleed flow rates. For the non integrated process, total low pressure steam consumption is 5664 kg/h: 3883 kg/h from process streams and 1781 kg/h from the evaporator. For the optimized process, total low pressure steam consumption is 4343 kg/h: 448 kg/h from process streams and 3895 kg/h from the evaporator. This economy represents savings of 23% of this hot utility. Considering the total low pressure level steam consumption and also the medium pressure level steam consumption, it can be calculated a total hot utility target of 2894 kW for the integrated process. Klemes et al. [ 11] using traditional Pinch Analysis tools to the same process obtained a total hot utility target of 3435 kW. The value obtained in this work is 16% smaller than the value obtained by those authors. Table 3. Optimization results Steam flow rate to Bleed stream from Bleed stream from Bleed stream from
the evaporator effect # 1 effect #2 effect #3
3895 2724 522 189
kg/h kg/h kg/h kg/h
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Fig. 4. Evaporator placement diagram for the crystal glucose plant 5. CONCLUDING REMARKS A new algorithm for integration of evaporation systems with a global process was developed, using Pinch Analysis principles. This new tool is based on the use of bleed streams and it was implemented in a computer software. It can be concluded from this work that the integration of evaporation systems and a background process is economically viable. It has to be stressed that the optimization method developed in this work does not make use of simplifications as neglecting boiling point rise of solutions, sensible heat of liquid streams and differences on latent heat of water. This algorithm can be used in any kind of evaporator configuration, even when some complex accessories are used as vapor recompression, flash coolers, flash condensate systems, and others. REFERENCES 1. I. C. Kemp, J. Separ. Proc. Technol. 7, (1986), 9. 2. E. MacDonald, Process Engineering, November, (1986), 25 3. D. L Westphalen and M. R. Wolf Maciel, Submitted for publication to Brazilian Journal of Chemical Engineering (1999). 4. R. Smith and B. Linnhoff, Chem. Eng. Res. Des., May, 195 (1988), 66. 5. R. Smith and P. S. Jones, Heat recovery systems & CHP, 10 (1990), 341. 6. C. R. F. Pacheco, C. A. Cdzar and T. W. Song, Chem. Eng. and Proc. 38 (1999), 109. 7. H. G. Schwartzberg, Food Properties and Computer-Aided Engineering in Food Processing Systems, edited by R. P. Singh and A. G. Medina, Kluwer Academic Publishers, Dordrecht, 1988. 8. D. L. Westphalen, Ph.D. Thesis, Campinas State University, 1999. 9. S. Itahara and L. I. Stiel, Ind. Eng. Chem. Proc. Des. Dev. 7 (1968), 6. 10. D. Leal, P. Friedman and A. Vald6s, Int. Sugar J., 88 (1986), 205. 11. J. Klemes, G. Kimenov, N. Nenov and A. Nedanova, 13th International Congress of Chemical and Process Engineering - CHISA '98, F6.6, Prag, Czech Republic (1998). 12. F. W. Shenck, Glucose and Glucose-Containing Syrups. Ullmann's Encyclopedia of Industrial Chemistry, VCH Verlagsgesellschaft mbH, Weinheim, 1989.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1003
Knowledge Based Models for the Analysis of Complex Separation Processes Piyush B. Shah* and Antonis C. Kokossis Department of Process Integration, University of Manchester Institute of Science and Technology, P.O.Box 88, Manchester M60 1QD, United Kingdom. Abstract The paper describes a conceptual formulation to screen, scope and review design options of complex separation systems without the need for extensive regression experiments. It employs a supertask representation with new synthesis formulation that exploits thermodynamics and engineering insights of primary separation. The thermodynamic and separation efficiency is systematically assessed with the prepositions of general terms, "conceptual losses". These terms explicitly account for the competing design drives of the problem and enables trade-offs to become clear in the optimal solution. In all cases, the approach guarantees simple mathematical models that one can solve to global optimality. Furthermore, the approach provides venues to interpret the results and explain the layouts selected by the optimisation. The approach is illustrated with the example and results are explained by reviewing important conceptual terms. Indeed, it provides a unique opportunity to understand and build confidence in the selected options. 1. INTRODUCTION The development of chemical process flowsheets embraces a large amount of conceptual knowledge, a result of the lifelong experience the engineers have with the process. In an effort to assess trade-offs and make design decisions, concepts are combined with thermodynamic insights, physical insights, and a variety of increasingly available modelling tools. Synthesis, design and optimisation approaches based on mathematical programming largely acknowledge the importance of engineering knowledge (Daichendt, and Grossmann, 1994; Raman and Grossmann, 1991), but widely ignore its involvement in the models. Engineering insights are occasionally used as additional logical constraints (i.e. heuristics turned into mathematical constraints, Raman and Grossmann, 1991), to simplify superstructures (Bauer and Stichlmair, 1997), and to devise decomposition schemes (hierarchical approaches: Douglas, 1988; problem decompositions Daichendt, and Grossmann, 1994;). These decomposition schemes are often useful, but they naturally disregard the strong interactions between the individual subsystems. As concepts and models are kept separate, the use of engineering knowledge often raises more limitations than it solves. The synthesis of complex separation systems entails a variety of challenges and difficulties due to the large number of design alternatives, the complex design models and the diverse economical implications. The objective of this work is to absorb the process knowledge to understand the dominant trade-offs and employ conceptual models with the rigorous optimisation framework to screen and scope all design options with basic process information ahead of detailed simulation.
*Present Address: AEA Technology Engineering Software (Hyprotech) Suite 800, 7078th Ave. SW, Calgary AB T2P 1H5 CANADA. Phone: (403) 520 6659, Fax: (403) 520 6060; Email: [email protected]
1004 The work addresses the separation synthesis problem with a different approach that postulates discrete representation and defines conceptually rich performance models that make simultaneous use of engineering insights and mathematical programming. It extends discrete representation of Hendry and Hughes (1972) to accommodate complex column configurations with the development of hybrid tasks and transformations. The performance models build on the developments of Shah and Kokossis (1997) to extend the application of "conceptual losses" to complex distillation systems. The models are possible to set up ahead of simulation and the optimisation models are mixed-integer linear programming problems that are easy to solve and trivial to adjust as feed conditions and specifications vary. This paper first outlines the conceptual background and defines conceptual terms to formulate the optimisation problem. It illustrates the approach with an example to demonstrate its strengths in explaining and understanding selected design options. 2. SYNTHESIS REPRESENTATION A task based representation for complex column configurations is accomplished by defining hybrid tasks and their transformations. A hybrid is defined as an ordered combination of simple distillation tasks. The second order hybrid is formed by combining two simple tasks and it features 2 splits, 2 heavy keys, 2 light keys and 3 products. Figure 1 illustrates a hybrid (AB/CD/EF) formed by simple tasks AB/CDEF and CD/EF. Hybrid splits can be used to classify components into groups. For a second order hybrid, the groups consists of a light components, LC, a heavy components, HC and an intermediate components, IC (see Figure 1). A second order hybrid features two associate sequences, one related to a direct sequence (direct associate) and another one related to an indirect one (indirect associate). The hybrid B/C/D of Figure 2 is either defined as a combination of B/CD and C/D (direct associate) or BC/D and B/C (indirect associate). Transformations of a hybrid are all the complex arrangements represented by the hybrid. For the second order hybrid following transformations are considered: * side-column arrangements (viz. side-rectifier (sr) and side-stripper (ss)) 9 prefractionator arrangements (viz. prefractionator (pJ) and Petlyuk columns (pc)) Figure 2 illustrates various these transformations for hybrid B/C/D. All simple tasks and hybrid transformations are integrated together to develop the supertask scheme. The detailed discussion on the supertask can be found in Shah (1999). 3. CONCEPTUAL BACKGROUND The analysis of the reversible multicomponent distillation is useful to set targets and review design alternatives. Fonyo (1974) and more recently by Koehler et at. (1992) and Bauer and Stichlmair (1997) have demonstrated systematic applications of thermodynamic insights in distillation sequencing problems. These ideas are now extended to address complex distillation problems with the introduction of the primary separation to develop conceptual terms.
3.1 Primary separation: The primary separation is a concept required to assess thermodynamic efficiency of potential distillation layouts. The hypothetical fractionation of the feed stream of each hybrid task under near reversible conditions denotes a primary separation. It is modelled to completely separate LC from HC and let IC distribute with minimum reflux (see Figure
1005 3a). The distribution of IC identifies a natural affinity amongst components and provides useful insights. Following terms are introduced to quantify these insights. A
tight
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Fig 1: Definition of hybrid: an aggregate of simple tasks ~ ~ ~ U,~:'o t
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Fig 2: Supertask representation
3.2 Definition of basic conceptual parameters The intermediate distillate recovery, 9t a, is defined as the ratio of the molar flowrates of all intermediates in the distillate to that in the feed. The importance of fractionation and the significance of sloppy splits are related to the intermediate recoveries. As the recoveries reach their natural bounds (0 or 1) the intermediates are selectively shifted to the bottoms and distillate products. The deviation of the separation from the sharp distillation is accounted by the sloppy factors @lp,a and @tp,b for the distillate and bottom product. The sloppy factors monitor the position of the operating line (D-F-B) of Figure 3b and capture the influence of relative volatilities. For example, if the light split is much easier than the heavy split then ICs will accumulate with HC and Q~zp,a will be close to zero and Q/tp,b will approach unity.
Recovery Potential, ~ r~,~, assesses the uniformity of the distribution of intermediates and is defined as the absolute deviation of 9t a from the equal distribution. The smaller the value of ~ecv the more uniform the distribution of intermediate components in the distillate and the bottoms. Fractionation potential, ~ac, values the purpose of fractionation and is defined as r =2.x~/l+x~. The value of g/rat increases monotonically with X/Yc. The higher the value of r the greater the potential for distributing intermediate components. Pressure correction factor, Qpres assesses the impact of pressures on the efficiency of the thermal coupling. If associate tasks feature operating pressures that differ widely, the pressure correction factors assume low values and imply diminishing benefits from thermal coupling.
1006 4. DESIGN CALCULATIONS Each complex configuration is decomposed into thermodynamically equivalent configurations of simple columns. The shortcut methods based on Underwood-FenskeGilliland are applied to each simple column of the equivalent arrangement and the basic calculations determine design parameters and the vapour-liquid traffic in the different sections of the units. The details of this model and calculation procedure are adopted from Triantafyllou and Smith (1992). The pressures are set to the minimum levels (above atmospheric) to enable the use of cooling water in the coldest condenser. The relative volatilities are estimated from the bubble and dew point calculations at the specified pressure. 5. CONCEPTUAL LOSSES AND MATHEMATICAL FORMULATION The conceptual losses for the simple columns were defined in Shah and Kokossis (1997). These ideas are now extended for the complex distillation configurations represented by hybrids and transformations. For a hybrid the conceptual loss is a function of the individual losses of its associate tasks. For transformations with simple columns the function is the simple summation of the losses. For complex transformations (sidecolumns and prefractionators) the losses are reduced due to thermodynamic benefits associated with each integrated scheme. These reductions are accordingly defined as conceptual gains. Let, H ={hth is a hybrid task} and C = {clc is a transformation of hybrid task}. CLh,c and CGh,c respectively denote the conceptual loss and gain of c~C, h~H. The conceptual gains, CGhx, for each transformation are developed from design parameter associated with the separation efficiency and the cost effectiveness. The gains consists of (i) conceptual duties, Z; (ii) weight factors, ~t; and (iii) pressure correction factor, Qore~.The Overall conceptual gain is given by
CGhc, = { ~ Z h,c k "[dh,c k }.Qpres h,c
V h~H
(1)
The conceptual duties bring about one of the thermodynamic and separation potential introduced earlier. The weight factors, ~h.c ~ [0,1] assign a relative significance to the efficiencies and describe the extent to which these terms appear significant. The pressure correction factor, Qpr**,accounts for adverse effects of the thermal coupling due to higher operating pressures. Following section briefly describes the conceptual duties and weight factors for different complex column configurations. The detailed explanation and definitions can be found in Shah (1999).
5.1 Side-column arrangement The side-columns are efficient when ~{d in the primary separation shifts to its bounds (i.e either close to 0 or 1). The recovery potential, ~cv, reviews this and favours the sidecolumn arrangements at its maximum value. Hence the first conceptual duty, Z Ah,c, is defined using qtrer The first weight factor, ~tAh,o, is defined using the concentration of intermediate components in the products of the primary separation. Low concentration of intermediate components requires less separation effort in either top or the bottom end hence it favours the side-column arrangements. The separation load at the top or bottom end of the primary separation decreases when the fractionation potential, ~/~, is low hence it is used to define the second conceptual duty, ZBh,c. The second weight factor, ~Bh,c, is defined to absorb the effect of relative difficulty
1007 of separation using sloppy factors. Side column arrangements are favoured by low value of one of the sloppy factors. 5.2 Prefraetionator arrangements
The balanced distribution of intermediate components in the products of the primary separation spreads the downstream separation load uniformly and delivers best results from prefractionator arrangements. The recovery potential, wre, reviews the component distribution and favours prefractionator arrangements for its lower values. Therefore, it is used to define the first conceptual duty, ZAh,c. The higher concentration of intermediate components results in efficient prefractionator arrangements. Hence, the first weight a factor, ~tA~r is defined using x IC,h. The higher values of fractionation potential, q~pac ensure a comparable downstream separation load at both the ends. Hence ~rrar favours the prefractionator arrangements at its maximum value and defines the second conceptual duty, Z Bh,c. Higher values for both of the sloppy factors correspond to symmetric splits and favour the prefractionator arrangements. Therefore, the weight factor is defined using sloppy factors accordingly. 5.3 Mathematical formulationThe synthesis model (hybrids and transformations) is formulated as an optimisation problem. The objective of the synthesis problem is to minimise overall conceptual loss. The formulation includes following synthesis constraints: 9 Mass balances around tasks and hybrids 9 Expressions for the conceptual losses and gains 9 Logical constraints (to control the complexity of the sequences) 9 Contingency constraints.
In all cases the formulation is a mixed integer linear programming (MILP) problem that is solved using OSL/GAMS. The formulation guarantees the global optimality in all cases as it involves only linear expressions of the continuous variables. 6. ILLUSTRATIVE EXAMPLE: SEPARATION OF LIGHT PARAFFINS The mixture of light paraffins (Rathore et al. 1974) is separated to obtain five pure component products (A: propane 5%; B: i-butane 15%; C: n-butane 25%; D: i-pentane 20% and E: n-pentane 35%). Figure 4 summarises the selected designs. Design I features two simple columns for ABC/DE and D/E and a side-stripper for the separation of products A, B and C. Design H favours the use of a side-stripper for AB/C/DE and pursues the downstream separation using simple columns. Design III combines the two simple columns of Design I to arrive at the side-rectifier configuration for separating ABC, D and E. The best simple column sequence reserves the difficult splits B/C and D/E for the end and favours a balanced split in the beginning. In this example, ~ for the hybrids AB/C/DE, ABC/D~ and A/B/C is higher (as they involve non-symmetric splits) so the side-column configurations are used extensively. The less integrated design (Design I) performs better than the more complex layout of Design III due to the adverse effects of pressure constraints (i.e lower ~pres). The penalty on the reflux due to higher pressure outweighs the benefits of thermal coupling. A compromise in the extent of thermal coupling is subsequently deserved. Conceptual losses of the first three designs are similar and they represent competitive options. These designs promise 15-20% saving as against the best simple sequence (see Figure 4). The
1008 complete analysis of all the conceptual terms and the detailed simulation results for the selected options are presented in Shah (1999). The optimisation required only 40 CPU seconds on P 166 personal computer. [---~
A. F.I"2-" A
Best simple sequence (100%) Fig 4: Promising designs for the Illustrative Example 7. CONCLUSION The results demonstrate the ability of the approach to screen designs with minimum effort and provide reliable and efficient layouts ahead of simulations. It explains how alternative, task representations can outsmart superstructure developments, simplify the synthesis effort and elevate mathematical programming as the technology to support novelty and innovation at the early design stage. Design I (81%)
Design II (81%)
Design III (85%)
REFERENCES
Bauer, M.H. and Stichlmair, J. "'Superstructurers for the mixed integer optimisation of nonideal and azeotropic distillation processes," Comput. Chem. Eng., 20, $25-$30 (1997). Daichendt, M.M. and Grossmann, I.E., "Preliminary Screening Procedure for the MINLP Synthesis of Process Systems-I. Aggregation and Decomposition," Comput. Chem. Eng., 18(8), 663-677 (1994). Douglas, J.M., Conceptual design of chemical processes, McGraw Hill Inc. (1988). Fonyo, Z., "Thermodynamic analysis of rectification I Reversible model of rectification," International Chem. Engng., 14(1), 18-27 (1974). Hendry, J.E. and Hughes, R.R., "Generating Separation Process Flowsheets," Chemical Engineering Progress, 68(6), 71-76 (1972). Koehler, J., Aguirre, P. and Blass, E., "Evolutionary thermodynamic synthesis of zeotropic distillation sequences," Gas Sep. and Pur!f, 6,153-167 (1992). Raman, R. and Grossmann, I.E., "'Relation between MILP modelling and Logical Interface for Chemical Process Synthesis," Comput. Chem. Eng., 15, 73-84 (1991). Rathore, R.N., Van Wormer, S.K.A. and Powers, G.J., "Synthesis of Distillation Systems with Energy Integration," AIChE 2., 20, 940-950 (1974). Shah, P.B. and Kokossis, A.C. Design targets of separator and reactor-separator systems using Conceptual Programming. Computers chem. Engng. 1997, 21, S 1013-S 1018. Shah, P.B. Ph.D. thesis, Conceptual Programming: A new approachfor the optimisation, analysis" and novel development of complex separation systems, UMIST, U.K. (1999). Triantafyllou, C. and Smith, R. The Design and Optimisation of Fully Thermally Coupled Distillation Columns. Trans IchemE, 1992, 70, 118-131
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1009
Synthesis of Solvent Extraction Separation Schemes in Hydrometallurgy L. A. Cisternasa and E.D. G~lvezb aChemical Engineering Department, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile* bDepartment of Metallurgical Engineering, Universidad Cat61ica del Norte, Casilla 1280, Antofagasta, Chile In this paper, we present a method for the synthesis of solvent extraction separation schemes in hydrometallurgy. Using equilibrium data the potential number of extraction and stripping stages are determined, then a superstructure for each stream is postulated. The superstructure contains those stream interconnections among the unit that can potentially define all configurations with no stream splitting, with stream splitting, with stream mixing, and with bypass streams. The result can include reduction in the number of stages, cross-flow, countercurrent, and bypass systems. Then a general mathematical model is formulated to solve the problem. Solution of the mathematical model shows the optimal stream pattern between the candidate extraction and stripping stages. The technique is illustrated with one example. 1. INTRODUCTION Process synthesis has been widely studied and used by the organic chemical industry, however, these advances cannot be easily applied to the synthesis of extractive metallurgy and inorganic chemical processes [1]. In the last years, several works have been developed to reduce the scarcity of methodologies that can be applied to inorganic chemical industry and extractive metallurgy [2-5]. The extraction of metals from ores and/or concentrates is carried out either by pyrometallurgy or by hydrometallurgy. Hydrometallurgy is essentially concerned with methods whereby metals, metal salts, or other metal compounds are produced by means of chemical reactions involving aqueous and organic solutions. The alternative hydrometallurgical route involves leaching of the ore to provide a leachate that can be fed to the extraction circuit, and a winning step to obtain pure product. Leaching or dissolution is the first prerequisite or the front-end of any hydrometallurgical processes. The dowstream of the operation involves treatment of the leach liquor by some separation process for the recovery of metals, metal salts or metal compounds. The various front-end hydrometallurgical processes produce either a dilute process stream or concentrate liquor. Various methods for treatment of both dilute process streams and concentrated liquors are illustrated in Table 1.
*Financial support from CONICYT (FONDECYT 1990956) is gratefully acknowledged
1010 Solvent extraction is the more common operation for purification and concentration of solution before the final metals can be recovered effectively. The use of solvent extraction as a unit operation in hydrometallurgy extends to a wide range of metals from a variety of feed materials including low-grade ores, scrap and waste, and dilute aqueous solutions. Solvent extraction can have potential application in the recovery of several metals from complex ores or deep-sea nodules. The nodules, apart from manganese and iron, the major components, contain appreciable quantities of Ni, Co, Cu, Mo, Zn and V. Table 1. Methods for treatment of diluted and concentrated leach liquors Method Function Examples Electrochemical reduction Recovery of metal Au, Ag, Cu, Cd Ionic precipitation Recovery of metal compounds Fe, U, Th, Be, Cu, Co Crystallisation Recovery of metal compounds A1, Cu, Ni, Mo, W Reduction with gas Recovery of metal or metal Cu, Ni, Ag, Mo, U compounds Electrolytic reduction Recovery as metal Au, Ag, Cu, Ni, Co, Zn Solvent extraction Purification and concentration Ag, Cu, Ni, Co, Zn, Zr, Hf, Nb, Ta, V, Mo, W, Re, Th, Pu Ion Exchange Purification and concentration U Carbon Adsorption Purification and concentration Au In this work, we present a method for the synthesis of solvent extraction separation schemes in hydrometallurgy. First a superstructure is constructed for each stream to represent all the potential stream interconnections. Then, based on material balance, equilibrium equations, and operation conditions a mathematical model is developed. The solution of this mathematical formulation gives the flow rates and concentration of each stream in the superstructure. The model is formulated such that it can be used for different applications. 2. SUPERSTRUCTURE CONSTRUCTION The methodology assumes that, as a result of laboratory studies, a suitable solvent system has been identified that will extract the desired component selectively from the less-desired components, that suitable physical and chemical conditions for carrying out the extraction have been found, that the maximum number of extraction and stripping stages are known, and that the rates of extraction of the desired components have been determined. Based on the maximum number of extraction and stripping stages two superstructures are constructed: one for the aqueous streams and other for the organic streams. In Figure 1 it is shown the aqueous stream superstructure for a 2-extraction 2-stripping circuit. Mixing and division of streams are included to represent several alternatives. The organic stream superstructure is shown in Figure 2. In this figure, extraction and stripping stages are the same that in the aqueous stream superstructure. Therefore, aqueous stream AS4 is mixed with organic stream OS4 in the extraction stage 1 to give aqueous stream AS6 and organic stream OS6. Streams bypass is included, but organic stream bypass between extraction and stripping stages are not included in Figure 2 for simplicity, but it can be added to the superstructure.
1011
Ore or concentrate u
C
EX-
I~.
1
O86
084 EX - 1
EX - 2 EX - 2
sw-'0 jl sw2
'~ Metal Fig. 1. Superstructure for aqueous streams. Fig. 2 Superstructure for organic streams. L=leaching, EX=extraction, ST=stripping, EW=Electrowinning The superstructures in Figures 1 and 2 represent 2-extraction/2-stripping, 2-extraction/Istripping, 1-extraction/2-stripping, and 1-extraction/l-stripping circuits. It also represents several streams topologies as cross-flow, counter-current, and bypass. Counter-current flows can give higher extraction efficiencies, cross-flow can give more capacity, and bypass can reduce the flow rates in the circuit [6]. The superstructures can be extended to an arbitrary number of extraction and stripping stages. The superstructures for the aqueous and organic streams can be developed according with the following scheme: (1) initial split where the streams are directed to all the stages in that superstructure. (2) outlet of the stages is split and mixed with the inlets of other stages and with final mixing point. 3. MATHEMATICAL MODEL With the superstructures the formulation can now be presented. The notation is as follows. Process streams are divided into two sets, set AS for aqueous streams, represented by index i, and OS for organic streams, represented by index j. Index k is used to denote the extraction and stripping stages given by the set XT. Mixers and splitters are divided into MA for aqueous mixers and splitters (represented by index m), and MO for aqueous mixers and splitters (represented by index n). Also the following sets, parameters and variables are used in the formulation:
1012 S e t s : sAinm and SA~ m input and output streams to aqueous mixers and splitters respectively, soin n and SO~ n input and output streams to organic mixers and splitters respectively, sAink and SA~ input and output aqueous streams to extraction and stripping stages. SOmk and SO~ input and output organic streams to extraction and stripping stages. Parameters: l~/~x_ai and MJNAi maximum and minimum concentration in aqueous stream i. MAXOj and MINOj maximum and minimum concentration in organic stream j. MAXOAk and MINOAk maximum and minimum organic aqueous ratio in extraction and stripping stages. M production rate for adsorbed specie. Variables: L and G aqueous and organic flow rates, x and y metal concentration in aqueous and organic streams. With the above definitions, the formulation can now be presented:
Mass conservation equations: 1. Mass balance for aqueous/organic mixer and splitter
ZLi=O V m ~ M A
ZLii~SA
~Gj- ~Gj:O V n ~ M O
;
,~/[ out iE_._ m
in
9 jESO
in n
jE
(1)
SOOUt
2. Mass balance for adsorbed specie in aqueous/organic mixer and splitter. ZL i x i i ~ SA im n
ZL i x i = 0
V m ~ MA
l 9E ~,-,A ,out m
Z
Gj y j j~SOin n
(2)
Gj y j =0
V n ~ MO
j~SO ~
3. Mass balance in leaching and electrowinning stages
(3)
Linput =L output 4. Mass balance for adsorbed specie in leaching and electrowinning stages
(4)
Linput Xinput + M =L output Xoutput
5. Aqueous/organic mass balance in extraction and stripping stages E
Li -
i~SA~n
Z
Li : 0
V k ~ XT;
. ,., , o u t IE,3A k
Z . JESO
Gj
-
in k
E
jeouk
Gj
= 0
_ o,-~out
VkeXT
(5)
6. Adsorbed specie mass balance in extraction and stripping stages Z
Lix i-
i~SAikn
Z
Li x i+
,-, , o u t t 9~ ,3Ak
Z j ~ SOi~"
Gj y j -
~ ] 9~ S O ko u t
Gj y j
0
VkeXT
(6)
7. Equilibrium equations
yj = f(xi)
V j ~ SO~ ut , i ~ SA~ ut , k ~ X T
(7)
1013 Operational equations
8. Maximum and minimum concentration in streams MINA i < x i < MAXA i ;
AJAXO j
(8)
V j ~ SO~ n ,i ~ SA~n , k ~ X T
(9)
MINO j < y j
<
9. Maximum and minimum organic aqueous ratio MINOA k L i < G j < MAXOA k L i
10. Same composition for inlet and outlet streams in splitters and nonnegativity conditions on flow rates and concentrations x i =x i
,
yj =yj
,
L i>_O
,
Gj >0
(10)
Objective Function
Several objective functions can be formulated based on the purpose of the problem. For design problem the objective function is to minimise the total cost. This includes the fixed and variable cost. Integer variables can be included for the fixed cost. For increase the capacity of the plant, the objective function can be to maximise the production rate for adsorbed specie, M. In this case the incremental capacity for flowrates and concentration must be included in the model. The optimisation problem defined by the objective function subject to the constraints in equations 1 to 10 corresponds to a nonlinear programming problem that has as variables the flowrates and the concentrations. Those flowrates that take a value of zero will then "delete" the streams that are not required in the superstructure. The likelihood of multiple local optima is this problem is somewhat reduced because of the operational specifications. This NLP can be solved with a large-scale gradient method as MINOS. 3. EXAMPLE: COPPER EXTRACTION CIRCUIT The production of copper by hydrometallurgy has increased in the last ten years. Usually two extractions and two or one stripping steps are used. In this example a simple minimumflow objective function was used. The data needed are given in Table 2 for a copper production rate of 4500 Kg/h. The superstructures are the same that in Figure 1 and Figure 2. The result solvent extraction separation scheme is shown in Figure 3. In this figure dash lines represent organic streams and solid lines represent aqueous streams. 5. CONCLUSIONS Metallurgical and inorganic chemical process synthesis have received relatively little attention in the process design literature. A method for the synthesis of solvent extraction separation schemes in hydrometallurgy was presented. Two stream superstructures were used to represent the streams interconnections among the equilibrium units. These stream interconnections can potentially define all configurations with no stream splitting, with stream splitting, with stream mixing, and with bypass streams. A general mathematical model was formulated to solve the problem. A simple example was used to illustrate the method.
1014
~LCopper ore
A!
I.. 1-~ ~1 EX-1 ~I~.... ]
ST- 1
r-
~
1
i
',
~~~,~ I ST 2 i~ --tll~
It.........
"1
I ~' Copper
Fig. 3. Result for copper solvent extraction circuit. ~ A q u e o u s streams. ---Organic streams. This study needs to be extended to include proper objective functions and to include more examples. This is important because the study of several examples can give general guidelines about when to use cross-flow, bypass, and counter-current systems. Most of these efforts are now underway. Table 2. Data for copper extraction circuit Maximum number of extraction/stripping stages Maximum and minimum PLS concentration Maximum and minimum reffinate concentration Maximum and minimum loaded organic concentration Maximum and minimum stripped organic concentration Maximum and minimum strong electrolyte concentration Maximum and minimum spent electrolyte concentration Maximum and minimum phase ratio (O/A) in extraction stages Maximum and minimum phase ratio (O/A) in stripping stages Equilibrium condition in extraction stages Equilibrium condition in stripping stages
2/2 9.5 - 9.5 Kg/m 3 2 - 6 Kg/m 3 7 - 11.5 Kg/m 3 5.5 - 7.5 Kg/m 3 34-58 Kg/m 3 37.5-37.5 Kg/m 3 0.5 - 2.4 0 . 2 - 1.0 y = 5.1296 + 1.32592 x 0.07037 x 2 y = 4.777 + 0.0377 x
REFERENCES 1. L.A. Cisternas, Minerals Engineering, 12 (1999) 15. 2. L.A. Cisternas, AIChE J., 45 (1999) 1477. 3. A.F. Connolly and R.G.H. Prince, XVIII Interamerican Congress of Chemical Engineering, San Juan, Puerto Rico, December 6-10, (1998). 4. M.A. Reuter, S. Sudh61ter, J. Krfiger, S. Koller, Mineral Engineering, 8 (1995) 201. 5. K.P. Thomsen, P. Rasmussen, R. Gani, Chem. Eng. Sci., 53 (1998) 1551. 6. D.A.White, Hydrometallurgy, 21 (1988) 145.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
SYNTHESIS OF SEPARATION REASONING
PROCESSES
1015
BY USING CASE-BASED
Elina Pajula, Timo Seuranen, Tuomas Koiranen, Markku Hurme Helsinki University of Technology, Laboratory of Chemical Engineering and Plant Design, P.O. Box 6100, FIN-02015 HUT, Finland The paper presents a new approach to process synthesis, case-based reasoning (CBR). CBR is based on the reuse of proven process solutions, which are used for solving new problems. As an example a conceptual reasoning of a separation system is given. The phases of reasoning in this application are: the search for specific separation methods for a single separation phase, search for creative new solutions by using analogies, using negative cases to exclude some solutions, adaptation by engineering formulas and comparison of cases. Two quality factors, maturity and performance, can be used to evaluate the quality of the cases 1. INTRODUCTION Typical task in process design is to determine the configuration of a separation sequence e.g. Barnicki and Fair (1990). It can been seen from the studies that the synthesis problem is difficult to handle by rules. Also creating rules (generalisations) causes information losses. Most of these methods are suitable only for limited types of separation processes. Therefore it is obvious that more specific information is required in process synthesis. The objective of this paper is to introduce a new method for finding feasible separation processes and process structures by case based reasoning (CBR). This means finding the most alike existing processes and applying the knowledge of their separation capacity and design for solving new problems in the early phases of process design. The method does not try to replace any rigorous simulations in the process design, but gives a few feasible ways to split the given feed into products. In this way it limits the number of processes that need to be considered and gives a systematic way of utilising earlier designs in new problems. 2. C B R - GENERAL FEATURES
CBR-applications consist of four parts, retrieval, adaptation, revising and retaining. The current problem is given as a query by giving the essential parameters e.g. feed components and product purity requirements. Based on these the similarity is calculated and a user-defined number of the most similar cases are retrieved. The user can select a case and launch adaptation routines, e.g. scale-up calculations. The more similar cases can be found, the fewer accurate simulations are needed, because a large part of the design can utilise data already available in the existing cases. The use of a database also forces to a systematic documentation practice. Documentation in a standard form can be used in the database as "cases" and no additional work is needed to maintain the database.
1016 3. CBR APPLIED IN PROCESS SYNTHESIS Hurme and Heikkil~i (1999) have applied CBR to evaluate the value of safety index used as an objective function in process synthesis. To our knowledge, no complete CBR based process design methodology has been presented earlier. 3.1 Hierarchy Dividing the process into different levels can be used for creating the hierarchy. Levels are for example concept level (e.g. solvent recovery process), process systems level (e.g. a distillation column with all its auxiliaries) and equipment level (such as selection of filter type). The use of design levels makes the use of analogies more feasible. Such hierarchy includes complex domains in which cases have different structures. Process design is far too complex field to be presented by using a treelike structure alone. For instance linking equipment to special separation method and describing the related equipment properties in a feasible way is not simple. Our solution to this problem is the use of relational attributes. Relational attributes represent a direct binary relation between two objects. 3.2 Similarity The nearest case is selected by comparing different characteristics between the user input and the existing cases. These characteristics are the phases present, kind of substances to be separated (e.g. hydrocarbons), chemical properties (e.g. relative volatility), and product purity requirements. The "design quality factors" have to be taken into account to make sure that old mistakes won't be repeated. These factors consist of technical maturity and performance as discussed in section 4.1. When the best case is selected, the new design is created based on adaptation rules and heuristic knowledge included in the adaptation part of this application. 3.3 Target An important point to be considered is how to describe the target of the separation process. The target could be e.g. removal of the solvent as waste, when the amount of water in the solvent is far less important than purity of water leaving the separation section. If we are recovering dilute valuable component to be recycled, also the purity of the recovered can be very important. Therefore, the primary target of separation has to be defined. 3.4 Creativity Another important point in process synthesis is creativity. The system should not only be capable of modifying old designs included in the database but also capable of creating new designs. One possible way of including creativity into synthesis is to use analogies. Analogies can be included by using "generalisations" and other structural features such as proper hierarchy. The generalisations introduced may include general level categories in database such as type of separation and phases present in separation (e.g. gas / liquid), physical properties of components (relative volatility etc.). See the example in section 5. 3.5 Case-Based versus Rule-Based System Rule based systems have been the most commonly used approach to process synthesis. This kind of knowledge can also be included in a case-based reasoning system by defining "general cases", for instance if relative volatility is larger than 1.5 and the decomposition temperature is high for all components, the proposed separation method is distillation. The benefit
1017 of the general cases is that it gives always suggestion even if the specific application area is not well known beforehand or no near cases have been stored in the database (see Figure 1).
Degree 1 of detail
General case
~
Case
Process characteristic
~
9
/ ~ ~ ~ - . ~ Process characteristi~ Figure 1. Structure of the case base (detailed and general cases)
Case
eneral case
Process characteristic "
4. S T R U C T U R E OF C B R A P P L I C A T I O N The hierarchical structure of the database is based on classification of separations presented by Wankat (1990). In the database structure each process document contains general data for separation (feed composition, purification requirements, capacity etc.) and the separations that exist in the separation process train. Every separation process consists of several pieces of process equipment, which have to be selected. Due to this also a lot of equipment specific information needs to be stored as well. Using relational attributes has created link between the process description and equipment specifications. The closer the separations are to one another in the tree hierarchy the more similar they are. The retrieval parameters can be selected and weighted by the user.
4.1 Quality factors One of the most important points to be considered is the definition of the quality factors. These factors describe the value and reliability of the design case. In our application two factors are used: technical maturity and performance (goodness) factor as presented in Table 1. Performance factor describes the proven efficiency of the design. The factors cannot be combined since an existing very proven process can obsolete and therefore a low performer. Table 1. Technical maturity and performance factor Factor values Descriptionof technical maturity 0 1 Process idea or concept exists 2 Process with basic engineering package exists 3 Plant in demonstration scale exists 4 Operating plant exists 5 Process is in wide use
Description of technical performance Failure/unsafe Out of date Modestefficiency Average efficiency Proven good efficiency Best available technology (BAT)
5. E X A M P L E : R E A S O N I N G ON A S O L V E N T R E C O V E R Y S Y S T E M To present the principle of case-based process synthesis the separation of dimethyl formamide (DMF), water, a light, and a heavy boiling component is discussed. The weight compositions and boiling points are given in Table 2.
1018 5.1 Selection for a single separation Synthesis of a separation process can be divided into subtasks of selection of single separations. In the following, the separation of DMF and water is discussed.
Query for specific cases In a CBR system, the user can make different types of queries in the database. The first query is made on DMF and water separation. As a result of this query specific cases on distillation and extraction of these components are found (Table 3).
Table 2. Components in case study Component: Amount b.p. Heavy 3% 165 ~ DMF 9% 153 ~ Water 83 % 100 ~ Light 5% 25 ~
Table 3. Query and results Query Method Separation Component1 Water Component2 DMF Feed kg/h 5000 Feed HK 0.11 Distillate HK 0.0005 Bottoms LK 0.01 Nreal MSA
on case study Found 1 Found 2 Distillation Extraction Water Water DMF DMF 3500 10000 0.2 0.125 0.0005 0.0005 0.001 50 Chloroform
Creative querying of solutions If new or more creative solutions are required, one can apply analogies or more general application cases in the CBR search. For instance, in this case one can query for separations for similar component types: Query for nearly similar boiling point can be done as a b.p. interval (e.g. 150-160 ~ Query for similar polarity can be done based on accentric factor etc. General cases General application cases are general experience based application guidelines, which can be represented as cases in the case base. For instance air stripping can be applied when ~/~p > 35000 kPa in 1000 ppm concentration range. The general application guidelines give a more complete but shallower coverage of the search space than specific cases as seen in Figure 1. Negative cases Negative cases can sometimes cancel the solutions proposed by general application guidelines: Based on general application guidelines it might be possible to apply pervaporation for the separation but a negative case found in the database lists DMF as a component which cannot be separated by pervaporation due to membrane problems. The current situation on these kinds of restrictions should be checked due to development of technologies. Adaptation Since the cases found in the query (Table 3) have different operating conditions (e.g. feed concentration) an adaptation of the case has to be done. For instance, a shortcut way to adapt distillation for different conditions has been given by Douglas et al. (1979). Separation factor S is defined as the ratio of the light (L) to heavy key (H) in the distillate (y) divided by the same ratio in the bottom (x) product. Relation of separation S to average relative volatility a,
1019 number of real stages N, tray efficiency q, distillate rate D, reflux rate L and feed composition z of components lighter than the heavy key is given by Equation 1. S = YL / YH = ct xL / xH x/1 + D / Lz
(1)
Comparison Comparison of the found cases can be done in principle by: 1) Costing (requires dimensioning of equipment). 2) By shortcut comparisons such as the method of Porter and Momoh (1991) that uses column vapour flow to compare cases. This can be combined with the method of Sounders (1964) that gives a generic method of comparison between normal and extractive distillation and extraction. If the latter is applied to this case it seems that extraction is more feasible than distillation since for distillation ce = 3, whereas for extraction ot > 100 in dilute DMF solutions. According to Sounders (1964) ce > 60 is required for extraction to be more feasible than distillation in this case. Therefore, extraction is selected.
5.2 Synthesis of a separation sequence Synthesis of a separation sequence starts from the selection of a single separation as shown above. There are several alternatives to proceed to the selection of a sequence: 1) Finding all possible separation combinations. This is feasible only in small cases. Normally the number of combinations is large. For example for four components and ten separation methods there are 5000 different sequences. The combinatorial explosion takes place quickly when the number of products to be separated is increasing. 2) The use of an optimisation algorithm to find the most feasible separation sequence is a feasible approach as shown by Hurme (1996) by using a genetic optimisation algorithm. Another possible approach is MINLP. 3) Using an upper level CBR to find out the sequence. It is possible to store into the casebase existing cases of feasible separation sequences with the characteristics of the components separated etc. These cases can be retrieved based on the analogies and used for reasoning about the sequence of the current case. In this approach there would be two levels of reasoning by CBR: Upper level which reasons on the sequence and the lower level which concludes the separation method for singe separations in the sequence.
5.3 Combined operations Another point, which should be covered, is the possibility of combined operations. E.g. several products can be separated by a single column by using side streams etc. The approach for this is first to consider single separations and then try to combine them one by one. Another way would be to conclude the possible combination operations from the retrieved cases.
1020 If we use distillation for the water/DMF separation in the case study problem of Table 2, we could conclude for example the separation sequence in Figure 2.
Lights. _( ~ Water ~tllpp~ag F~d_~.~ I Di~ DMF
Lights Dis+llat~n ~ M F ] _illat] ies
Figure 2. Train of single distillation separations
_,[_~HIn eav~ ies Figure 3. The train with a combined operation
Lights Stri~ Water Feed ~ Solvent ~Tj E,~o~iiD , ies Figure 4. The train with extraction separation
Searching for possible combination operations would give us e.g. a case for hydrocarbon separations with uncondensable gases where uncondensables are taken out from the condenser as a third stream. Applying this case would give a combined system shown in Figure 3. If the separation method is not distillation but extraction, there is no combined operation for this but the result would be as in Figure 4. 6. CONCLUSIONS The paper has presented a new process synthesis method; case-based reasoning (CBR). CBR requires a database of existing design solutions, which is at least partly available in companies utilising databases for their engineering information management. The benefit of applying case-based approach is the systematic reuse and storing of the accumulated knowledge. The evaluation of the quality of stored design cases prevents repeating the earlier design mistakes. The approach can also be applied in creative process synthesis. One possible way of including creativity into synthesis is to use analogies by using 'generalisations' as discussed. The approach presented speeds up process design by defining in the early phase the process alternatives to be studied further by rigorous methods. This is more and more important as early design decisions are required in implementing new design paradigms such as process integration, inherent safety and clean technology. In these new methodologies, the major design decisions have to be made as early as possible in the process design. REFERENCES
Barnicki, S. D., Fair, J.R., Ind. Eng. Chem. Res. 29(1990), 421-432. Douglas, J.M., Jafarey, A., McAvoy, T.J., Ind.Eng.Chem. Process Des. Dev. 18(1979) 197-202. Hurme, M., Proceedings of the 2NWGA, Vaasa 1996, 219-224. Hurme, M., Heikkil~i, A.-M., Proceedings of PRES'99, Budapest 1999, 341-346. Koiranen, T. Acta Polytechnica Scandinavica No. 251, Espoo 1998. Porter, K.E., Momoh, S.O., Chem.Engg.J. 46 (1991) 97-108. Sounders, M., Chem.Eng.Prog. 60 (1964) No 2, 75-82. Wankat, P.C., Rate-Controlled Separations, Elsevier, Barking, 1990
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
1021
An analytical process performance model :or batch distillations Silvina D. Zamar, Songlin Xu, and Oscar A. Iribarren Institute for Development and Design INGAR - Conicet Avellaneda 3657 (3000) Santa Fe, Argentina In this paper we propose an approximate analytical prediction of the minimum reflux ratio required by a batch distillation task, arriving to a completely analytical process performance model for batch distillations. The paper presents the derivation of the analytical prediction of Rmin and compares it with the numerical integration predictions, analyzing and quantifying their discrepancy. Also, it puts together the analytical prediction of Rminwith previously derived analytical predictions of Nmin and the Performance Correlation that relates Nmin, Rmin, N and R, to construct the analytical process performance model for batch distillation. 1. I N T R O D U C T I O N The synthesis of separation networks involves the construction of superstructures where the single separation tasks become nodes (Biegler et al, 1997). So, solving separation networks by batch distillation needed the development of short-cut models for the separation task itself. This is so because for multicomponent systems with the recycling of slop Cuts, the use of rigorous models for the single separation task makes the whole problem unaffordable. The move towards simpler models started with the revival of constant volatility models, as in Luyben (1988) and Quintero and Luyben (1990). Then continued with approaches that estimated the instantaneous separation of the column by the Fenske Underwood - Gilliland model. This approach was presented by Diwekar and Madhavan (1991) and further developed for simulation by Sundaram and Evans (1993). At that time, the analytical approaches hold only for binary separations. Bauerle and Sandall (1987) predicted Rim. for binary batch rectification while Chiotti and Iribarren (1991) addressed stripping. Afterwards, in Salomone et al (1997) we proposed an integrated batch approach where the Nmin and Rmin are defined as the N and R required by hypothetical b a t c h columns (operating at total reflux or having an infinite number of stages respectively) to perform the separation task. These quantities are different from the continuous ones for the same separation task. It is shown that N~n continuous is an upper bound for Nminbatch and that Rmin continuous is a lower bound for Rminbatch. The method proposes the use of a Gilliland like correlation that was constructed for batch distillation and relates Nmin and Rmi, with the actual number of stages N and reflux ratio R:
N- N
N+I
II 1"341
= 0.62 1
-
min
-
R+I
(1)
1022 In Salomone et al (1997), we proposed that these Nmin Rmin be computed by numerical integration of the hypothetical columns. Afterwards in Zamar et al (1998), we developed an analytical prediction for Nmin and showed that the distribution of non-key components is satisfactorily approximated by the distribution predicted by the total reflux column. The expression for Nn~n is:
= N min
In I In (1 - / / ' ~ ) 1 In (1 -/'] h~)j In a tk,h~
Where ]]i is the recovery of product i defined by 7~i =d i / f i
(2)
di is the amount of
product i in the distillate after the separation, and fi is the amount of product i in the feed before the separation. Subscripts lk and hk stand for the light and heavy key components. Equation (2) can be rearranged to:
(1--~t~ i)= (1--~t~r)
N min
....
(3)
And be used as a partition function to predict the distribution of the remaining products i, given the recovery of a reference product r where r can be any of the keys. The mass balances at each separation task k can be written: d i,k -- 17i,k fi,k bi, ~ =
(4)
(1 - ~i,k ) f i,k
Where bi is the amount of product i in the bottom after the separation. So the mass balances for the network consist of as many equations (4) as separation tasks are contained in the network. Plus the equations that connect tasks, for example: di,k -- f i,k+l
(5)
f i,k -'bi,k-1 + di,k+l
The first is a simple connection: the distillate collected after performing task k is going to be the feed for task k+l. The second represents a recycle: the distillate of task k+l is going to be added to the bottom of task k-1 to conform the feed of task k. As pointed by Mujtaba and Macchietto (1992), the recoveries of two key components per each separation task, covers the degrees of freedom of the network. We proposed to use them to estimate the Nmin required by each separation through equation (2). Then, we may get the recoveries of the non key components through equation (3). This makes equations (4) and (5) be a linear system of equations. After solving it, we know the feed to each separation task and so we can compute the Rmin required by each. This computation does need the feed composition of both key and non key components to be known. This information allows us to size the columns, and estimate a performance index for optimizing the network. The only procedural step in the method, is the computation of these minimum reflux ratios.
1023 In the present paper we propose an approximate analytical prediction of the Rn~n arriving to a completely analytical process performance model for batch distillation. We first present the derivation of the analytical prediction and then compare it with the numerical integration predictions, analyzing and quantifying their discrepancy. 2. D E R I V A T I O N O F T H E F O R M U L A E F O R Rmin
2.1 Instantaneous separation Consider the instantaneous mass balance of component i between the distillate stream D which has a top composition xi,t and a cross section of the column below stage n, counting n from the top: D x
i,t
--"
V
y
i,n+l
L
--
X i,n
(6)
V and L are the constant molar flowrates of vapor and liquid in the column. We divide eqn (6) by D, resort to the total mass balance V - L + D and use the definition of the reflux ratio R = L / D to rearrange it to the form: Xi, t
( R + l ) Yi,~+ 1
=
-
Rxi, ~
(7)
Now we consider a column with an infinite number of stages (this will generate a pinch i.e. ~, stages with the same vapor and liquid compositions). And introduce our first strong simplifying assumption: all components distribute between the top a n d the bottom o f the column (this will place the pinch at the bottom). Then replacing Yi,n+l = Yi,b and Xi,n = Xi,b in eqn (7) and dividing by Xi,b gives:
ri
=
(R +
1) y''b
_
R
(8)
X i,b
Where ri is the ratio of top to bottom compositions of product i r i =
x i,t / x i,b And here we
introduce our second strong simplifying assumption, we pretend that the equilibrium constant = Y i,b / X i,O in eqn (8) gives:
Ki is a constant, all along the batch process. So replacing: K i
r i
=
(e
+
1)K i
-
e
(9)
Observe that if the assumptions hold, the ratio of top to bottom composition would be a constant during the batch distillation conducted at a constant reflux ratio R.
2.2 Batch distillation The differential mass balance for component i is:
Ot
'
=
-D
x i,t
(10)
1024
Where ni are the moles of component i in the still, and D is the flowrate of distillate. The global mass balance is:
=
/)t
-D
(11)
Where nT is the total moles in the still and as ni = nT Xi,b we may differentiate: On i
n r ~ x i , b -~ xi, b
=
~gnv
(12)
Replacing eqns (11) and (12) into eqn (10) and rearranging gives: OXi'b
=
(r i
- 1) o n r
X i,b
(13)
nr
Integrating eqn (13) from the initial conditions of the feed Xi,b " final condition at the still Xi,b = X i , w , nT = W we get:
Xi,f
,
nT -- F and the
(14)
~ Xi,f Taking the exponential of both sides of eqn (14) and multiplying them by W / F gives: (ri)
Were
Where W i is the moles of component i in the still after the separation and fi the moles of component i in the feed before the separation. Or, in terms of recoveries as:
(1 - 77 i) = 0 77) (ri)
(16)
-
Where 1] is the recovery of total moles
~7 = Z d i / E
f i
2.3 Minimum reflux ratio We obtain the minimum reflux ratio from the expression in equation (9). First we rearrange it to the form: (R+I)K i
=
r i +R
(17)
We apply eqn (17) for the light and heavy key components, and compute the ratio between both:
1025 rl~ + R a
Ik,hk
=
(18)
rhk + R
Rearranging eqn (18) and recalling that this R is the minimum, because of the assumption of ,,~ stages: T lk -- a lk,hk r'hk Rmi n
(alk,hk__l)
-
(19)
After solving the mass balances of the network one computes the total moles recovery of each separation rl, uses equations (16) rearranged to get the composition ratios of the two key components:
:
gn(1
-
77)
=
gnO _ 77)
(20)
Then, one finds the minimum reflux ratio with eqn (19) and finally uses the correlation among N Nmi. R and Rmin in eqn (1) to get the R for an existing column, or to solve the economic tradeoff involved in designing a distillation facility: 3. COMPARISON WITH NUMERICAL PREDICTIONS The Figure in next page shows analytical predictions plotted against numerical predictions (that lie on the straight line). These cases include a ranging from 1.05 to 8 and feed composition of the key components ranging from 0.05 to 0.55 in a 5 components mixture. The assumption that all components distribute between the top and the bottom would in principle produce an over estimation because it places an upper bound for the Rmi, required by the instantaneous separation (King, 1980). However, in the cases when the heavy key component is among the non distributing, which is likely to occur during the initial part of a separation when the light key component is present in large amounts, the assumption predicts distillate compositions for the light key larger than one. This effect is explained for binary systems in Salomone et al (1997) and is responsible for under estimations. 4. CONCLUSIONS Unlike the analytical prediction of Nmin, which did not require further simplifying assumptions, the analytical prediction of Rmin did require two strong assumptions. First that all components distribute, and second that the equilibrium constant Ki remains constant along the batch distillation (an assumption much stronger than constant relative volatility). As a result, the discrepancy between Rmin predicted by the analytical expression and by integrating the instantaneous Underwood column is not that small (analytically predicted Rmin can be as large as twice or as small as half the numerical integration predictions). However they are of the same order of magnitude, and deviations tend to compensate each
1026 80 ..........................................................................................................................................................................................................................., 70
J
J
60 50
@
40 30
@
20
.J
10
i
0
10
i
20
i
30
i
40
i
50
,
60
i
70
i
80
A n a l y t i c a l vs. N u m e r i c a l Predictions of Rmin
other when considering the consecutive tasks to be performed at the column, thus reducing the error impact on the size requirement for the column. We believe that the reduction of the single separation task node to a simple analytical model increases so much the potentiality for exploring appealing more complex separation networks, that it largely compensates for the moderate uncertainty introduced by the approximations needed to arrive to it. 5. ACKNOWLEDGMENTS The authors would like to acknowledge the financial support provided by the National Council for Scientific and Technological Research Conicet of Argentina, the State Commission for Education of the P. R. of China, and Foundation Antorchas of Argentina. REFERENCES Bauerle G. L. and O. C. Sandall AIChE J1 33, 1034 (1987). Biegler L. T., I. E. Grossmann and A. W. Westerberg "Systematic Methods for Chemical Process Design" Prentice- Hall (1997). Chiotti O. J. and O. A. Iribarren Computers chem Engng 15 (1), 1-5 (1991). Diwekar U.M. and K.P. Madhavan Ind. Eng. Chem. Res. 30, pp 713-721 (1991). King, C. J. "Separation Processes" McGraw-Hill New York, 1980. Luyben, W. L. Computers chem Engng 27,642-647 (1988). Mujtaba, I. M. and S. Macchietto Comput chem Engng 16, Suppl. 273-280 (1992). Quintero E. and W. L. Luyben Ind. Eng. Chem. Res. 29, pp 1915-1921 (1990). Salomone H. E., O. J. Chiotti and O. A. Iribarren Ind. Eng. Chem. Res. 36, 130-136 (1997). Sundaram S. and L. B. Evans Ind. Eng. Chem. Res. 32, pp 511-518 (1993). Zamar S. D., H. E. Salomone and O. A. Iribarren Ind. Eng. Chem. Res. 37, 4801-4807 (1998).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1027
Synthesis of Heat Exchanger Networks Considering Stream Splitting and the Rigorous Calculation of the Heat Transfer Coefficient According to the Bell Delaware Method Marcia C. Roque
Liliane M. F. Lona *
Laborat6rio de Amilise, Simulag~o e Sintese de Processos Quimicos - LASSPQ Faculdade de Engenharia Quimica - DPQ - UNICAMP - Cidade Universitfiria Zeferino Vaz C.P. 6066 - Distrito de Bar~o Geraldo - Campinas - S P - Brasil - CEP: 13081-970 e-mail: [email protected] / [email protected]
Abstract. On the context of cost minimization and maximum profit, connected to the interest about the preservation of the environment we live in, the minimization of energy consumption in a chemical industry is placed according to the new market trends. The concept of Pinch Analysis fits in this new scenery as a tool to define the least quantity of hot and cold utilities, as well as the number of heat exchangers to be used in the plant, in order to use the energy of the own process for the many heat exchange operations going on this process. On the present work, a software was developed to work with the heuristics of Pinch Analysis and with the concept of Problem Table, to detect the Pinch temperatures. This software defines a Heat Exchanger Network ( HEN ) for the process and consider the stream splitting taking into account the economic evaluation of this procedure which focus on the maximum energy recovery compared with the results obtained for the case of not splitting the streams. This analysis is based on the calculation of the minimum heat transfer area and the cost of the equipment which are compared with the costs involving hot and cold utilities usage. Another aspect that was taken into account was the calculation of the heat transfer coefficient according to the Bell Delaware method for the shell side and the comparison between the results obtained by using its value. 1. I N T R O D U C T I O N The development of chemical processes consists of several stages, where the main goal is transform raw materials into high value commercial products. These stages begin with the study of the location of the unity, passes through process synthesis and goes to the construction and operation of the plant. One of the techniques employed for reducing utilities costs is Pinch Analysis, which concept is based on the definition of the least quantity of hot and cold utilities, as well as the least number of heat exchangers to be used on a chemical plant, in order to promote the best way of contact between hot and cold streams of the process. In this way, the energy of the own process will be used on the various heat exchange operations the process may require. The Pinch temperature can be defined as being the restraining temperature of the process, which divides the process into two regions. About this topic, it is possible to mention five heuristics that are valid for Pinch Analysis and used on this work:
1028 9 9 9 9 9
Never transfer heat across the Pinch point; Add heat only above the Pinch; Cool only below the Pinch; Always add heat to the lowest possible level of temperature relative to the Pinch point; Always cool the highest possible level of temperature relative to the Pinch point.
2. M E T H O D O L O G Y
2.1. Heat Balances Considering the heuristics presented before, the process is divided into two regions: below and above the Pinch point and heat balances are then made on these two regions. To match two streams, it is necessary to obey two more heuristics, that verify the heat charges in these two regions: 9 Above Pinch, the match is possible only if: FHoTCDtoT < FCOLDCpcoLD 9 Below Pinch, the match is possible only if: FCoLDCpcoLD < FHoTCpHoT 2.2. Stream Splitting A single stream can have enough heat to supply more than one stream and it is necessary to establish some criteria to analyze which streams will be divided and what is the best setting for these streams inside the process. On this work, it was used an adaptation of a methodology presented by Polley, G.T. ( 1995 ), that consists on a series of matrices that show the energetic situation of the process and serve as a guideline to define the best path to stream splitting. These matrices are: 9 8 Matrix: The elements of this matrix show the ratio between the heat capacities of the matched streams and the ratio considered as being ideal. 9 d Matrix: the elements show the difference between the heat capacity difference of the match and the ideal value. 9 D Matrix: the elements show the single arithmetic difference between the matched streams. After each stream splitting, these matrixes are recalculated and show the new situation of the HEN until the achievement of the best energy configuration. 2.3. Cost Analysis To define the cost of each heat exchange unit, it is first necessary to define the minimum area requirement for each unit and this can be done by using the simplified equation presented by Ahmad et alli ( 1990 ). A=
1 Qo ATMLi U o
(1)
where: Qij - heat exchanged between hot stream i and cold stream j; A = minimum heat transfer area; ATMIi = logarithmic mean of the temperature difference. Uij = global heat transfer coefficient of hot stream i and cold stream j defined on equation 2:
1029 1
Uij
=
1
hi
,
1
(2)
hj
Based on these data, it is possible to calculate the cost by equation ( 3 ) presented by S.G. Hall et all. ( 1990 ) for stainless steel shell and tubes heat exchangers:
(3)
Custo ( $ ) = 30.800 + 750 A TM
Since the heat transfer coefficient is very important on the definition of the minimum heat transfer area needed for the process, on this work, in order to get more precise data, this coefficient was calculated by the Bell Delaware method that takes into account the leaks between the baffles and the shell and between the baffle and the tubes, the leaks between the tubes and shell. This method also includes the effect of not only crossed flow and the differences in spacing the extremes baffles. 3. DATA OBTAINED As a case study, it was considered the example presented by Linhoff and Ahmad, ( 1990 ). These data are presented on Table 1 together with the results obtained for the heat balances. Table 1 Process Streams Conditions and heat balances Stream Streams TE Ts FCp Condition ( ~ ) ( ~ ) (KW/~
Hot A 327 40 100 Hot B 220 160 160 Hot C 220 60 60 Hot D 160 45 400 Cold a 100 300 100 Cold b 35 164 70 Cold c 85 138 350 Cold d 60 170 60 Cold e 140 300 200 Reprinted from: Linhoff, B.; Ahamd,S., 1990
Amount of heat above
Amount of heat below
TpINCH
TpINCH
Net heat of the stream ( MW )
( MW ) 16,7 9,6 3,6 0 -15 -0,98 0 -1,2 -30
( MW ) 12 0 6,0 46 -4,0 -8,05 -18,55 -4,8 -2,0
28,7 9,6 9,6 46 -20 -9,03 -18,55 -6,6 -32
In this process, the Pinch temperatures were found by means of the Problem Table presented by Linhoff B. and Ahmad, S. ( 1990 ) and presented next together with the minimum difference of temperature on the exchangers terminals: 9 Hot Streams TpINCH:160 ~ 9 Cold Streams Tp1NCH:150 ~ 9 ATM1N:10~ The possible matches for the streams were verified according to the heuristics presented before.
1030 It was then verified that it was needed to split streams only above Pinch point, where we have 3 hot streams and 4 cold streams. The stream splitting matrix is presented on Table 2 showing the final setting for the HEN. Table 2 Matrix d MCp A B D el e2 e3
MCp 100 70 60 11,33 124 64,67
A1 89,82
A2 10,18
B1 20
B2 16,33
B3 123,67
C 60
2,52 XXX XXX XXX 14,5 XXX
82,16 54,46 21 0,28 104,32 49,53
72,34 44,64 11,48 XXX 94,5 39,71
76,01 48,31 14,85 XXX 98,17 43,38
XXX XXX XXX XXX -9,17 XXX
32,34 4,64 -28,82 XXX 54,5 -0,29
~i 7,66 5,36 28,82 0,82 9,50 4,96
On this matrix, the "X" indicates the matches that were not possible because they did not obey the Pinch Anlysis heuristics. In this way, even considering the stream splitting, just the matches that obeyed these rules were done in order to obtain the maximum energy recovery that was possible on the process. Besides these aspects, it was also considered the stream tick off principle and this principle was assured by using matrix D presented on Table 3: Table 3 Matrix D Q(KW) A B D E1 E2 E3
Q (KW) 15000 980 1200 1700 18600 9700
A1 15000
A2 1700
B1 1200
B2 980
B3 7420
C 3600
CE CE CE CE -11180 -6100
On this matrix, "CE" means that the stream was ticked off. Many settings may be obtained from this technique, and here it is necessary to analyze costs and the possibility of using each of these settings. Here it is necessary the knowledge of the process engineer. Before stream splitting, the heat loads required were: 9 Cold Utility: 93,9 MW 9 Hot Utility: 86,18 MW After stream splitting it was needed to add 17,28 MW and the costs of the exchangers to be placed above Pinch point, as well as their heat exchanger area are presented on Table 4:
1031 Table 4 Heat exchangers position, costs and heat exchange area before and after stream splitting HE Streams Heat Cost ( $ ) Streams Heat Cost( $ ) HE with Exchange with stream without Exchange Without splitting Area without splitting Area with splitting splitting splitting steam (m 2) splitting (m 2 ) TC1 TC2 TC3 TC4 TC5 TC6 Total Area Total Cost
A1 - a A2-el B3 - e 2 B1 - d B2 - b C-e3
3506,72 400 2496 192 147,2 672 5388,16
588.539,17 121.312,64 454.276,56 83.831,66 73.562,84 177.095,30
TC 1 TC2 TC3
A- a B -e C- d
3506,72 3229,60 579,48
32.293,65 552.561,15 160.554,5
7315,8 1.498.618,17
745.409,3
For these calculations, h was considered constant and equal to 0,5 KW / m 2 ~ according to data extracted from Polley, G.T and Pnjeh Shahi, M.H., ( 1991 ). Table 6 shows the detailed design of the exchangers after stream splitting and the changes on heat exchange area using h calculated by Bell Delaware on the shell side .The costs obtained for these values of heat exchange area are presented on Table 5. Based on industrial data for the costs of superheated vapor, it was done an evaluation of the costs with hot utilities load before and after the synthesis of HEN. 9 Costbefore HENS: US$5.745,33 9 Cost after HENS: 1.146,67 Table 5 Heat exchangers position and costs after stream splitting Exchanger Streams Cost ( $ ) Project TC1 A1 - a 460,320.11 TC2 A2 - el 109,445.85 TC3 B3 - e2 984,456.43 TC4 B1- d 109,445.85 TC5 B2 - b 65,916.31 TC6 C - e3 143,743.56 Total Cost
949,467.04
Cost ( $ ) H ( Bell ) 924,502.24 127,058.75 120,011.76 125,429.51 61,068.72 177,095.30 1,535,166.28
Based on industrial data for the costs of superheated vapor, it was done an evaluation of the costs with hot utilities load before and after the synthesis of HEN. 9 Costbefore HENS: US$5.745,33 9 Cost after HENS: 1.146,67
1032 Table 6 Exchanger data for the splitted streams. Exchanger TC1 TC2 Expected area ( h by 6276.15 431.99 Bell ) - m 2 Design a r e a - m 2 2540.05 312.31 Passes on the shell 4 4 Passes on the tubes 8 8 Number of tubes 2787 1078 Internal diameter of 34,8 15.75 tubes - mm External diameter of 38,1 19.1 tubes - mm Tube L e n g t h - m 7.62 4.88 Shell D i a m e t e r - m 2.159 0.9906 Tube P i t c h - mm 47.625 25.4 Number of baffles 10 6 Baffle Cut 0.25 0.35 h ( Bell ) - KW / m 2 0.2 0.48 0C
TC3 364.9
TC4 392.45
TC5 96.08
TC6 672
259.35 4 8 888 15.75
138.84 2 4 488 15.75
115.42 1 2 396 15.75
488.25 2 4 837 34.8
19.1
19.1
19.1
38.1
4.88 0.9398 25.4 6 0.3 0.26
4.88 0.6858 25.4 6 0.45 0.164
4.88 0.59 25.4 6 0.45 1.609
4.88 1.42 47.625 6 0.40 0.5
4. C O N C L U S I O N S Based on the data presented here, it is clear to see that the detailed design affects a lot the heat exchanger area and its cost, providing a better view of the final costs of the project implementation. The Bell Delaware method has proved its efficiency on giving more accurate data for the resistance to heat transfer and the cost analysis gives a tool for the engineer decide the better decision to be made. REFERENCES
1. Ahmad, S., Linhoff, B. and Smith, R., " Cost Optimum Heat Exchanger Networks Part 2", Computers Chem. Engng., Vol. 14, No 7, pp. 751-767, ( 1990 ) 2. Hall, S.G., Ahmad, S., Simth, R. " Capital Costs Targets for Heat Exchanger Networks Comprising Mixed Materials of Construction, Pressure Ratings and Exchanger Types", Computers Chem. Engng., vol. 14, no 3, pp. 319-335, ( 1990 ) 3. Linhoff, B., Ahamd,S. "Cost Optimum Heat Exchanger Networks - Minimum Energy and Capital using simple Models for Capital Cost", Computers Chem. Engng, vol 14, num. 7,pp.729-750, ( 1990 ) 4. Polley, G.T. and Panjeh Shahi, M.H., "Interfacing Heat Exchangers Networks Synthesis and Detailed Heat Exchanger Design", Trans I Chem E, vol.69, pp.445-457, ( 1991 ) 5. Polley, G.T., " Selecting Stream Splits in Heat Exchanger Network Design", Heat Recovery Systems & HCP, vol 15, no 1, pp. 85-94, (1995). *The authors would like to acknowledge FAPESP - Silo Paulo State Foundation for the Research Support, for the support received on the developing of this work.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1033
USING CONCEPTUAL MODELS FOR THE SYNTHESIS AND DESIGN OF BATCH DISTILLATIONS Jos6 Espinosa, Enrique Salomone and Songlin Xu INGAR-CONICET, Avellaneda 3657, 3000 Santa Fe, Argentina e-mail: destila @arcride.edu.ar We present a methodology to estimate the maximum recovery of each component in an azeotropic multicomponent mixture processed at a batch distillation column. Based in the most recent advances in the field of separation feasibility we have adapted an algorithm to predict all the distillation regions for the multicomponent mixture with their corresponding natural and distillation boundaries. The results of the maximum separation problem togheter with those corresponding to conceptual dynamic runs are used to illustrate several important issues related to the synthesis and design of highly non-ideal azeotropic mixtures. 1. I N T R O D U C T I O N One of the typical questions that arise in the conceptual design of distillation systems for non-ideal and azeotropic mixtures is related to what is possible to be obtained from a thermodynamic point of view. This question is trivial in ideal systems where there are no constraints to the "perfect split" separation and every component of the mixture can be obtained as pure as it is desired depending on how much separation power is provided. Azeotropic systems present the particularity that not every component can be isolated and therefore multicomponent cuts will be obtained. Moreover, the composition and sequence of these cuts will typically vary for different initial compositions. Therefore, a very useful piece of information is the estimation of the maximum separation that can be obtained given an initial feed to be processed at a batch distillation column. Hence, the maximum recovery of each species as pure component or as part of one or more azeotropes can be calculated and separating strategies can be proposed to break the azeotropes. After a feasible sequencing alternative is chosen, the next step is to evaluate how much separation power (number of stages and reflux ratio) is needed for a given purity requirement. In this phase of the conceptual design the issue is the trade-off between separation costs and cuts specification. For this task, conceptual models representing the performance of batch columns operating at limiting conditions such as infinite number of stages or total reflux can be used to estimate flows, equipment sizes and utility loads, t~l't21 In this work, we present a methodology to estimate the maximum recovery of each component in a non-ideal multicomponent mixture processed at a batch distillation column. Based in the most recent advances in the field of separation feasibility t31 we have adapted an algorithm to predict all the distillation regions for the multicomponent mixture with their corresponding natural and distillation boundaries. Using piecewise linear approximations for each boundary a mathematical representation is automatically generated. This information is
1034
kept in a hierarchical recursive data structure. In this way, any boundary of the original system can be treated as a system of reduced dimension with its own boundaries that in turn are systems of reduced dimensions. We present the main features of the methodology by describing its application to a couple of examples of highly non-ideal azeotropic mixtures with distillation boundaries. The examples correspond to two quaternary mixtures: Acetone-Chloroform-Benzene-Toluene and Acetone-Chloroform-Methanol-Benzene. Along with the description of the methodology, the examples also illustrate several important issues related to the synthesis and design of distillation separations for this kind of systems by using the results of the dynamic runs representing the infinite stages limiting condition. L21 Moreover, the minimum reflux for a given purity requirement can be also calculated. 2. FEASIBLE CUTS FOR A GIVEN INITIAL FEED
2.1. Algorithm: ComputeMaxRSeparation(F,D,B) A procedure to compute the maximum separation for a batch rectifier for a given feed F, obtaining a top cut D, and a bottom residue B. i. ii. iii. iv.
vi. vii. viii. ix. X.
Set the active distillation space aDS as the one containing F. Set the active unstable node U as the unstable one corresponding to F. Set the top composition D as the composition of the unstable node U. Compute the intersection of the vector F-U with the closest bound limiting the active distillation space. Set the bottom composition B as the composition of the intersection point obtained in the previous step. Set the active bound aBound as the bound intersected in step iv. Mark B as contained in the sub-region defined by the boundary aBound and compute its corresponding unstable and stable nodes within the sub-region. With the compositions of B and D, compute the amount of each cut using an overall material balance. Compute the bubble and dew point for B and D. Return B and D
2.2. Algorithm: GenerateRectifyingCutSequence A procedure to calculate the sequence of feasible cuts to be obtained through a sequence of batch rectifications at maximum separation. Given an initial feed F having nc components, produces the sequence of top cuts {D1,D2 ..... Dnc-1 } and its corresponding bottom residues {B1, B2 ..... Bnc-1} i~ ii. iii.
iv.
Initialize B amount and composition as the initial feed F. Compute the bubble and dew point for B. Compute the unstable node corresponding to B by using a Distillation Line calculation. Compute the stable node corresponding to B by using a Condensation Line calculation.
1035 v. v.1 v.2 v.3 v.4 v.5 vi.
While the boiling and dew temperatures of B are different SetFasB ComputeMaxRSeparation(F, D, B) Seti=i+l Set Bi = B and Di = D Loop Return {D1,D2 ..... Dnc-1 } and {B1, B2 ..... Bnc-1 }
3. EXAMPLES Let us consider the quaternary mixture Acetone-Chloroform-Benzene-Toluene at 1 bar pressure. This system presents two distillation regions. The upper region has one unstable node (Acetone), one stable node (Toluene) and two saddle nodes (the maximum boiling azeotrope between Acetone and Chloroform and Benzene). The lower one has pure Chloroform as unstable, Toluene as stable whilst Benzene and the binary azeotrope act as saddle nodes. Both regions are separated by a distillation boundary containing the binary azeotrope, Benzene and Toluene. Figure 1 shows that the quaternary mixture presents a curved boundary. In this work, the stability of each node is defined in terms of the experiment of open distillation that produces the residue curve maps. However, we calculate it from the stabilities corresponding to the map of distillation lines. Hence, the stability of each one of the pure components and azeotropes can be determined by solving an eigenvalue problem of the Jacobian matrix of the equilibrium in the neighbourhood of each node. [4] Three eigenvalues correspond to each node. Each one of the unstable nodes has three eigenvalues smaller than unity, whilst the three eigenvalues corresponding to Toluene are greater than unity. The saddles have some eigenvalues greater than unity and some smaller than unity. Four ternary subsystems are the natural boundaries for any mixture whose initial composition belongs to the interior of the tetrahedron whilst the boundary formed by the
Axo'
,0L ,0
oo,~U< , ~L _ / / 'z /~/ \ o.~ 086
,.o
.~o_ 9
2ndcut
0.6J \F ~~.~ ACMB
.6
0.21%~
~~,~M
oCt\
_~~
\ \ // !I~
~o_ ~.~ ~.,- o.o '
I \ \~
.8
xe XD3 3
0~t\~ ~c~
o.o
Figure 1. Maximum Recovery Prediction versus Simulation for a quaternary mixture.
06 X,~
] ~
/ /ResidueS.6
o., ~ o . , ~.OBO.O
--
i "~
Figure 2. Quaternary mixture with both stable and unstable boundaries.
1036
Benzene
1.0
_
_..__. _ . L
Acetone cu,
I \
=~
' Te Luene
Chloi )form
0.8
2nd cut r~
~9 0.6
Chlon fforrr, -t
Residu
M
o
0.4
Ace one
0
L
\L
0.2
-'-0.4"~ X~ / / 06x,,\\/ /
0.8 ~ / e ~ O , 2 1.0 B0.0
~-'~6 00.44
~ 0 8 "0 ~176
Figure 3. Another feasible cut sequence for the system ACMB.
0.0
- : : : :':
0.0
: : : :1: : : : ; , :
0.2
: : : :1: : : : .,:
0.4
6
|
0.6
0.8
1.0
Rectification Advance Figure 4. Simulated Maximum Top Compositions for the first example.
binary azeotrope and the two heaviest components appears as a stable one. Each one of the natural and stable boundaries contains information about stability and the mathematical representation of the boundary. The stable boundary has three nodes and hence, the parameters of a plane can be calculated to represent it. Each one of the boundaries has its own sub-boundaries, that in tum are systems of reduced dimensions. Therefore, all the information is kept in a hierarchical recursive data structure. Consider a sequence of batch rectifications at maximum separation with an initial feed F located in the upper distillation field [see XB~ in Figure 1]. The calculation of the distillation line until a node is reached reveals that the corresponding unstable U for this feed is Acetone, therefore a cut of pure Acetone is recovered first at the top of the column. The still composition moves along a straight line through F-U away from the Acetone comer until intersection of this line with the closest bound [XB1 in Figure 1] which is the stable boundary formed by the azeotrope Acetone-Chloroform and the two heaviest components Benzene and Toluene. aBound is in this case a stable boundary [more precisely a linear approximation of the real boundary] that limits the movement of the still path and consequently the top compositions that can be achieved. No calculations of distillation and condensation lines are suggested to update the stable and unstable nodes corresponding to the composition of the residue lying on the stable separatrix. Actually, the calculation of the distillation line starting from XB1 leads to pure Acetone as the unstable node because the azeotrope is a saddle when considering the whole region and the residue is located on a plane that is only an approximation to the boundary. Instead, this information is contained in aBound. For this case the azeotrope is the unstable node, Benzene is a saddle whilst Toluene acts as stable node. The way to determine the stability is done by substracting one eigenvalue greater than unity from each node pertaining the boundary. Only two eigenvalues determine the stability on the boundary because the stable boundary acts like another face of the tetrahedron; i.e. like a ternary system, tS]
1037 During the second step of the distillation and according to the algorithm a cut with azeotropic composition is regained as overhead product. During this step of the process, the state of the residue changes along the linear approximation of the stable boundary until intersection with the binary edge of the heaviest components at xB2. Finally a cut of pure Benzene is obtained and the final residue consists of pure Toluene. Since the dew and bubble temperatures of pure Toluene coincides the outer algorithm is stopped. Figure 2 illustrates the maximum recovery as predicted by the algorithm for a quaternary mixture of Acetone-Chloroform-Methanol-Benzene at 1 bar pressure. The system presents four distillation regions with an unstable boundary formed by AM-MB-CM-ACM-ACMB. The system also has a stable boundary formed by ACMB-MB-B-AC-ACM-M. Whilst the unstable boundary limits the top products achievable in a batch rectifier, the stable one constrains the movement of the still path. Both boundaries are approximated in a piecewise linear manner. This information is also contained in the corresponding subregion and permits checking for intersection for boundaries formed by more points than the necessary to define the mathematical representation. The way we choose to divide the region consists in determining the batch distillation regions for the boundaries. E61 Finally, we choose the difference in the dew and boiling temperatures of a residue in order to stop the outer algorithm. This criterion is more efficient than checking for the stable node of the whole region because any of the pure components or azeotropes could be at the end of the distillation. Figure 3 shows another possible cut sequence for a feed that belongs to the interior of the simplex formed by the nodes AM-ACMB-ACM-AC. Therefore, the first residue composition is located in the plane formed by ACMB-ACM-AC. In this case, the final residue is the azeotropic mixture between Acetone and Chloroform that in turn is a saddle for both the overall system and the subsystem composed by the nodes pertaining the stable boundary. Note that the feed considered belongs to the same distillation region of the feed shown in Figure 2. However, different cut sequences and residues can be achieved. The number of cuts plus one (the residue) equals the number of components in all cases. Returning to the first example, Figures 4 shows the top compositions versus rectification advance at a very high reflux for the system Acetone-Chloroform-Benzene-Toluene at 1 bar pressure. The Figure was generated by using a conceptual dynamic model for an infinite stages column at a very high reflux. E21The corresponding still path is depicted in Figure 1. In Figure 1 it can be observed that pure Acetone can be recovered during more time than that predicted from the maximum recovery algorithm. The difference is due to the shape of the boundary. Figure 4 shows that in the second cut, a stream with variable composition is collected instead of the azeotrope, due to the curvature of the stable boundary. However only the two most volatile components are recovered in this step as predicted by the maximum recovery algorithm. The end of the operation coincides for both the conceptual dynamic simulation and the prediction from the simplified model. From the results of the maximum feasible cuts by considering linearized distillation boundaries, it is clear that the maximum boiling azeotrope between Acetone and Chloroform must be broken to recover all the Acetone and Chloroform in pure form. The conceptual dynamic run on the other hand, predicts that a cut of variable composition must be treated. Under the limiting condition of infinite number of stages and a high reflux ratio, the mean
1038 composition consists of a binary mixture whose composition in Chloroform exceeds the one corresponding to the binary azeotrope. Several ways can be thought to recover both components in pure form. One of them is the use of Benzene as entrainer because Benzene produces a curved stable boundary that can be exploited to recover both the original species and the entrainer. This way of operation requires two vessels and one rectifier. ~71 An estimation of the minimum reflux ratio can be done with the aid of conceptual dynamic runs. As an example, the recovery in the first cut of 83 % of Acetone present in the initial feed will need a reflux ratio about 11. To estimate the minimum reflux ratio and minimum number of stages needed for this separation it is also necessary to define the recovery of the heavy key component Chloroform. In this case we select a recovery about 1%. The minimum number of stages is 8. Details about the methodology for estimating both the minimum energy demand and the minimum number of stages can be found elsewhere. El1'I21 4. C O N C L U S I O N S This work deals with the calculation of the feasible cuts and residue for a given initial mixture charged in the still of a batch distillation column having an infinite number of stages and operating near total reflux. Both stable and unstable boundaries are taken into account. The information about distillation regions is kept in a hierarchical recursive data structure. In this way, any boundary of the original system can be treated as a system of reduced dimension with its own boundaries that in turn are systems of reduced dimensions. Mathematical representations of the boundaries are automatically generated in a piecewise linear manner. Hence, a simple algorithm can be conceived to predict the maximum separation. This information is of great value from the synthesis standpoint because the feasible cuts strongly depend on the initial feed composition. Dynamic runs based on conceptual dynamic models are also considered in this work. They can be used both to refine the predicted distillate cuts because the model can handle nonlinear stable distillation boundaries and to develop separation alternatives to break azeotropes. Also, an estimation of the minimum reflux to achieve a given separation can be performed.
REFERENCES [1] Salomone, H. E., O. J. Chiotti and O. A. Iribarren (1997). Ind. Eng. Chem. Res., 36 (1), 130-136. [2] Espinosa, J. and E. Salomone (1999). Ind. Eng. Chem. Res., 38 (7), 2732-2746. [3] Rooks, R. E., V. Julka, M. F. Doherty and M. F. Malone (1998). AIChE Journal, 44 (6), 1382-1391. [4] Poellmann, P., Bauer, M. H. and E. Blass (1996). Gas. Sep. Purif., 10 (4), 225-241. [5] Safrit, B. T. and A. W. Westerberg (1997). Ind. Eng. Chem. Res., 36 (5), 1841-1854. [6] Bernot, C., M. F. Doherty and M. F. Malone (1991). Chem. Engng Sci., 46 (5), 13111326. [7] Stichlmair, J., and J. Fair (1998).Distillation; Principles and Practice. Wiley-VCH, Chapter 6, 315-321.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1039
Mixed Integer Linear Programming and Constrained Based Search Approaches in a Multipurpose Batch Plant Short Term Scheduling Problem Luis Gimeno a, Maria T. M. Rodrigues b' Luiz C. A. Rodrigues "~and Wesley Alvarenga b aState University of Campinas, School of Electrical and Computer Engineering.CP 6106, 13083-970 Campinas SP,Brasil [email protected] bState University of Campinas, School of Chemical Engineering.CP 6166, 13083-970 Campinas SP,Brasil [email protected] The short term scheduling problem considered consists in batches' allocation inside its processing time windows minimizing earliness in deliveries of final products. Time windows are obtained in a planning phase using final products' due dates and supply of raw materials, leading to a planning frame where Constrained Based Search and Mixed Integer Linear Programming approaches are utilized. CBS techniques are improved using storage constraints and MILP formulations with small degeneracy are developed. 1. INTRODUCTION The problem considered is short term scheduling, where tasks to be scheduled are determined from final products' demand in a short time horizon. Products' demand is characterized by specific quantities to be delivered at specific due dates. The scheduling horizon is constrained by a raw materials availability plan that limits possible release dates for initial tasks in the recipe, and the objective is to minimize final products' earliness. The recipe structure is modeled through its State Task Network (STN) representation. Two approaches are considered: a MILP formulation using uniform discrete time representation and Constrained Based Search (CBS). Application of CBS techniques to batch chemical process scheduling has been reported (Das et al., 1998) since commercial packages such as ILOG Scheduler have been available. The objective is a comparison of these techniques in heavily constrained problems and the main point addressed is how these approaches deal with equipment units sharing and constraints imposed by material balances and storage conditions.
2. CONSTRAINED BASED SEARCH APPROACH. As remarked in (Das et al. 1998) ILOG Scheduler needs a preprocessing material balance phase to determine batches and batch sizes for intermediate products to satisfy final products demand. In fact CBS scheduling techniques applications generally start with: i) a complete definition of the batches to be scheduled (activities); ii) temporal relationships among them linking possible start/end times of batches; and iii) resources required by each batch including equipment units, that is an unique assignment of tasks to units (allowing different tasks assigned to the same unit(s)) is defined previously. Inclusion of assignment choices would be possible but, unless the different choices have large effects on cost function, it would lead to an increase in search effort. All these input data could be obtained from a
1040 previous planning phase once the assignment of tasks to equipment units is defined. A planning level has been developed to determine those inputs Constraint propagation mechanisms need the determination of batches processing time windows since its main function is to update batches earliest beginning times - ebt and latest finishing times - Ift. These time windows can also be obtained from the previous planning phase once a raw materials delivery plan is introduced. In general scheduling decisions in the search tree are ordering decisions, defining an ordering relation between two batches competing for the same resource. An ordering decision of this type can impose changes in both time windows: anticipation on lft of the precedent batch and increase in ebt of the succeeding batch. These changes in processing time windows can trigger further modifications on time windows for unscheduled batches through two propagation mechanisms. Precedence relations among batches propagate these effects through the recipe structure; resources capacity analysis propagates among batches competing for the same resource (disjunctive constraints and edge finding)..Propagation consequences are reduced time windows that can imply in forced orderings between batches competing for the same resource, thus reducing the search tree. To take an ordering decision a pair of batches must be selected. For this purpose generally a bottleneck approach is used focusing in the resource with highest contention and the most constrained subset of batches competing for this resource. Demand for an equipment unit induced by processing time windows of batches assigned to it can be evaluated using concepts like equipment unit aggregated demand (Sadeh, 1991), cruciality function (Keng et al., 1988) or equipment unit slack (ILOG, 1997). The search procedure outlined above can end with a feasible solution or a dead end if time window constraints cannot be satisfied. In the last situation it is necessary to backtrack, in the first the value of the objective function is added as a constraint and the search is reinitialized. 3. M I L P F O R M U L A T I O N A MILP formulation can include in the scheduling problem the planning problem, with the determination of the number of batches of intermediate products and the assignment problem. To put both approaches in the same conditions the MILP formulation with uniform discrete time representation has been modified to work with previously determined unique assignments, number of batches of intermediate products and processing time windows. Binary variables represent batches allocation: w(i,j,p,t)=l means that batch j of task i starts processing at unit p in time slot t. As it is well known uniform discrete time representation generates a great number of binary variables and often in the literature it is discarded for medium and large problems. Nevertheless in the type of problem considered here, namely with time constraints in batch allocation in the form of time windows, it establishes a very useful frame to constrain the scope of equations and reduce the problem dimension. Main characteristics of the MILP formulation are the following: 9 binary variables only exist inside time windows, in fact between ebt and lbt; 9 since assignment is fixed index p is not used in binary variables; batches competing for the same equipment unit are known beforehand; 9 disjunctive constraints, establishing that only one batch can use an equipment unit at each time slot, are only written in slots where competing time windows overlap; 9 mass balance equations are only written for time slots where it is possible to have production or consumption of this state. This information is obtained from time windows;
1041 9 tasks operating in zero wait mode are aggregated in subtrains, and only binary variables for the first task of the subtrain need to be used; 9 equipment units with null or low contention are not modeled; mass balances equations are only written for states whose producing and consuming tasks are assigned to active units. A pegging equation is used to link batches of output and input states not modeled by mass balance equations.
4. PROBLEM DESCRIPTION The plant has seven equipment units used to manufacture three final products. Production recipes' representation is given in Figure 1. Equipment units assigned for each task, intermediate storage conditions and products demand are given in Tables 1 and 2. All raw materials are delivered at time t = 1. Batches processing time windows obtained from the planning phase are represented in Figure 2 together with units aggregated demand.
tasks
Table 1. Assignment of tasks to eqt Lipment unitsand storage conditions equipment states storage tasks equipment states
TA1 TA2 TA3 TA4 TB1 TB2
P1 P2 P7 P2 P3 e P4 P5
A1 A2 A3 B1 B2 C1
produCt ProA ProB ProC
FeedA 0
4hr A1 r-[ TA1 ~ 2o
NIS
NIS UIS FIS (50) ZW FIS (50) ZW
TB3 TC1 TC2 TC3 TC4 TC5
"
P6 P1 P5 P6 P4 P7
Table 2. Products demand mass due date mass 100 32 90 270 32 75 56 lhr ~'[ TA2 ~
A2
2o
3hr
B1
2hr
40
ZW
40
3hr A3 ~[ TA3 ~ ~ ) 5o
B2
lhr
FIS(50)
C2 C3 C4
storage FIS (100) UIS NIS
due date 56
2hr ProA ~[ TA4 ] - ~ 0 2o
ProB
FeedBO-"~TBI~)------~ITB2~---'-~TB3 FeedC O
2hr C1 r-l Tc1 ~ 4o
ZW
FIS(50)
3hr C2 ~.[ TC2 ~ 4o
FIS(IO0)
15 2hr C3 r-[ TC3 ~ is
3hr C4 ~[ TC4 ~ - ~ 0 - ~ 15
NIS
I hr TC5]
Pro C ~-0
is
Figure 1. STN representation
5. CBS and MILP COMPARISON. The situation represented in Figure 2 has been created to analyze CBS and MILP approach when constraints in equipment unit's capacity and intermediate storage are both important. Building a solution using CBS techniques is based on a bottleneck approach. Higher load situations are selected sequentially so that initially unit P1 would be chosen since it has
1042 the smallest slack time: 1 time unit for the set { T A l l 6 - 9, T C 1 / 1 , 2 }. Any ordering at this point must be between a batch of task T A 1 and a batch of T C 1 since batches of the same task are already ordered. Orderings not yet defined by time windows are listed in Table 3 for all the units.
Figure 2. Batches time windows and equipment units aggregated demand. Circles" slots of total reliance; dashed slots represent unavailable time slots. Unit's aggregated demand. Table 3. Orderings to be defined unit P1 P2
P3 P4
orderings to be TC1/1 TC1/2 TA 4/2 TA 4/3 TA 4/6 TA 4/7 none
defined TA1/6,7,8 TA1/7,8,9 TA 2/5 TA 2/6 TA2/9,10 TA 2/9,10
unit P5
orderings to be defined TC2/1 TB2/7
P6
TC3/1 TC3/2
TB3/15 - 18 TB3/17,18
P7
TA3/3 TA3/4
TC5/1 - 4 TC5/3- 5
none
Twelve ordering decisions are possible in unit P 1 . Constraints' propagation after each one of these orderings gives way to the earliness values listed in Table 4. An unavoidable earliness value of 23 units can be obtained from Figure 2 since some latest finishing times have been anticipated due to constraints propagation in the planning phase. It is very likely that whatever the first choice the system will have to backtrack and follow several branches in the search tree as far as several orderings do not increase the value of the cost function.
1043
ordering TC1/1 < TA1/6 TC1/1
Table 4. Earliness values ordering cost TA1/6 < TC1/1 72 23 TA1/7
cost 24 infes infes 23 , , 23 ,,, infes
In fact there is a storage constraint that greatly influences cost function and that does not translate in a capacity problem, so that a bottleneck approach will only deal with it later in the search tree. The problem is a small FIS condition in state B2 given batch sizes for TB2 (40) and TB3 (15). Task TB3 output is a final product so that any anticipation in TB3 batches' latest finishing times will increase earliness value. On theother side task TB3 is assigned to unit P5 together with task TC2, but the load imposed by batches time windows is low as can be seen from Figure 2, thus the selection of unit P5 is not likely to occur at the beginning of the search procedure. Any solution that includes the decision TC2/1 < TB2/7, or that forces this ordering as a consequence of other ordering decisions, will give a high earliness value, the problem being that this can occur lately in the search process. Meanwhile if this fact is detected and ordering TB2/7 < TC2/1 is taken as the first decision the set of remaining ordering decisions is greatly reduced. The example shows that a constrained based search approach can benefit from a view not uniquely centered in bottlenecks. In contrast a MILP approach deals simultaneously with capacity constraints and constraints imposed by mass balances and storage conditions. The problem with a MILP approach is in some way the opposite, it deals with all the constraints, and if constraints are not tight enough this leads to the well-known degeneracy problem. In these situations the Branch & Bound procedure can spent a long time without leading to an integer solution. To avoid this degeneracy characteristic the approach being undertaken has been to work with reduced MILP models where equipment units with null or low competition are not considered. Instead of a local bottleneck approach non-bottleneck units are eliminated. A complete presentation and discussion of MILP model is not possible due to lack of space so that only the pegging equation that links state batches not modeled by mass balance equations is presented (equation 1). Wbit • t) q- gap(b,i,b',i')
(Z t
+ Pi < ( ~ Wb'i't • t)
(1)
t
The term gap(b,i,b' i') gives the time spent in processing the batches of not modeled tasks that link (b, i) and (b' i') in the pegging diagram. This pegging diagram simply represents the mass balance precedence relationships between batches and it is obtained in the planning phase. The gap defined above is a lower bound of the time that must elapse between two batches. If no competition exists in the non-active units, as for example in the case of dedicated units, it is the time that would be spent in a solution of the complete formulation. In other situations, as for example when there is some overlapping between batches in the nonactive units, a complete solution may impose larger times, so that the solution obtained by the reduced MILP can be different, and surely it will be worse. This is an expected result, but as far as competition is low the difference will not be large. MILP formulation considering all the units has been solved with GAMS/OSL. Prob-
1044 lem dimension is: 1006 equations, 189 continuous and 616 binary variables. The solution with earliness of 36 slots was obtained with a gap of 0.01 (optcr) after 15642 iterations and 1430 nodes. Execution time was 97 sec.(Pentium II). A reduced model was solved first considering only units P1,P2,P4,P5 and P7, and in a second step the same global solution was obtained. Gantt chart for the first step is represented in Figure 3. Problem characteristics are: 610 equations, 113 continuous and 403 binary variables. The partial solution was obtained in 2.5 sec. after 692 iterations in 56 nodes minimizing total earliness (optcr = 0.01). F-I 7
-I--q
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............. FigUre 31 Partial solution considering equipment units P1, P2, P4, P5 and P7 ........ 6. C O N C L U S I O N S CBS techniques are primarily oriented to propagation of capacity constraints and could benefit from propagation mechanisms induced by s.torage constraints. MILP discrete time formulation enforces simultaneously all the constraints in the problem and can benefit from a selective utilization where non-bottleneck units are eliminated from the model. REFERENCES 1. Das B.P., Shah N. and Chung P.W.H. (1998). Off-line scheduling a simple chemical batch process production plant using the ILOG scheduler. Computers chem. Engng Vo122, Suppl, pp. $947-$950. 2. Sadeh N.(1991) Look-Ahead Techniques for Micro-Opportunistic Job Shop Scheduling, PhD The-
sis, CMU-CS-91-102, School of Computer Science, Carnegie Mellon University,. 3. Keng N.P., Yun D.Y.Y. Rossi M.(1988) Interaction Sensitive Planning System for Job-Shop Scheduling, in Expert Systems and Intelligent Manufacturing, Ed. M.D.Oliff, Elsevier,. pp. 57-69. 4. ILOG Scheduler 4.0 (1997). User's Manual, ILOG. ACKNOWLEDGMENTS This work was partially supported by Funda~.o de Amparo ~.Pesquisa do Estado de S~.oPaulo (Brasil)
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1045
A Continuous-Time Approach to Short-Term Scheduling of Resource-Constrained Multistage Batch Facilities C. A. M6ndez, G.P. Henning and J. Cerdfi INTEC (UNL - CONICET) Gtiemcs 3450 - 3000 Santa Fc - ARGENTINA E-mail: j [email protected]
ABSTRACT This work presents a new MILP mathematical formulation for the resource-constrained short term scheduling of flowshop batch facilities with a known topology and limited supplies of discrete resources. The proposed MILP approach is based on a continuous time domain representation and accounts for sequence-dependent setups. Assignment and sequencing decisions are independently handled through separate sets of binary variables. In addition, a proper formulation of the sequencing constraints provides a substantial saving in the number of sequencing variables and constraints. By postulating a pair of conditions for the simultaneous execution of processing tasks, rather simple resource constraints requiring a few extra binary variables are also derived. The proposed approach shows a remarkable computational efficiency when applied to industrial applications. 1. INTRODUCTION Different mathematical programming approaches have already been proposed for the short term scheduling of multistage flowshop batch plants. This processing structure consists of multiple processing stages arranged in series, each one containing several units working in parallel. Moreover, the orders to be processed follow the same processing sequence though some of them may skip one or more stages. The problem objective is to develop a production schedule that permits to complete every order at the due date and/or before the end of the time horizon, while satisfying all the plant operating constraints. However, processing tasks usually require renewable resources like utilities and manpower, and a given number of them may be run simultaneously. Since there are limited supplies of resources, it becomes necessary to monitor the total amount utilized by simultaneous tasks since it should never exceed the maximum availability. Reklaitis (1992) made an extensive review on the resource constrained scheduling problem. According to the time domain representation, the scheduling approaches were classified into two types: discrete and continuous time formulations. Discrete representations handle resource constraints in a rather easy way by enforcing them at the discrete time points. However, they require many time intervals and an exceptionally large number of integer variables for industrial applications (Kondili et al., 1993). Similarly, continuous time representations based on the resource-task network (RTN) concept and the partitioning of the scheduling horizon into time intervals of unknown duration usually account for resource constraints (Schilling et al., 1996; Mockus and Reklaitis, 1997). In turn, Pinto and Grossmann (1997) combined LP-based branch and bound and disjunctive programming to deal with the short term scheduling of resource-constrained multistage flowshop batch plants. However, this continuous time approach that extended the formulation of Pinto and Grossmann (1995) for the unconstrained scheduling problem is restricted to
1046 sequence-independent setups. In this work, a new MILP mathematical formulation for the resource-constrained short term scheduling of a flowshop batch plant is presented. It is based on a continuous time domain representation and it can be applied even if sequence-dependent setups and topology constraints are considered. 2. P R O B L E M DEFINITION
Given (a) a multistage flowshop batch plant with a known structure and several units j eJs operating in parallel at each stage seS, (b) a set of orders ieI to be processed at the plant, (c) a set of topology constraints restricting unit interconnections between consecutive processing stages, (d) a set of discrete resources r e R with limited supplies and (e) a specified time horizon, the problem goal is to find a production schedule that allows to manufacture every order before the end of the scheduling horizon, while satisfying the unit allocation constraints and the resource availability limits, at minimum earliness. For each order, it is given the set of required stages, the set of units where it can be processed, the processing time at each stage, the release-time and the promised due-date. To derive the proposed mathematical formulation, it has been assumed that: (1) an unlimited intermediate storage is available between consecutive processing stages; (2) each order comprises a single batch; (3) order split is not allowed and (4) renewable resources like utilities or manpower are discrete, i.e. they are consumed at constant level throughout the processing tasks requiring them. 3. T H E M A T H E M A T I C A L M O D E L 3.1. Notation
(a) Sets I Ij S Si Sii, J Js Jis Jii's Js Jsj,s+l R
set of orders set of orders that can be processed in unitj (Ij c_ I) set of stages set of stages involved in the production of order i set of common stages for orders i and i' (Sii, = Si ~ Si,) set of units set of units that belong to stage s (Js c J) set of units that can process order i in stage s (Jis c J) set of units that can process both orders i and i' in stage s (Jii,s = Jis ~ Ji's) set of units in stage s that are not connected to every unit available in the next stage (s+ 1) but just to some of them (Js c_ Js) set of units belonging to stage (s+l) that are connected to unit j c Js in stage s (Jsj,s+l c Js+l ) set of renewable resources
(b) Parameters di suij ptij "~ii'j g si Oisr
Ar
due date of order i setup time of order i in unitj processing time of order i in unitj sequence-dependent setup time between orders i and i' in unitj last processing stage for order ieI overall requirement of resource r e R by order i~I at stage s maximum availability of resource r
1047
(c) Variables binary variable denoting that order i is processed before order i' (Xii's- 1) or after order i' (Xii,s= 0) in some unitj~Jii,s of stage sESii' Yij binary variable denoting that order i has been allocated to unitj~Ji Wisi's' binary variable denoting that stage s' of order i' has been completed after starting stage s of order i Cis completion time for order i at stage s Xii's
3.2. The Problem Constraints 9 Allocation Constraints Every order i~I should be assigned to a single equipment item at each processing stage s.
~ Yij - 1 j~Jis
V i ~ I , s ~ Si
(1)
9 Sequencing Constraints a) Bound on the starting time of order i at stage sESi due to its release time unit ready time (ruj).
Cis ~ Z Yij (Max [Fuj,roi]+ pto + suo) j~Jis
Vi ~ I,s
~ Si
(roi) and the (2)
b) Bound on the starting time of order i at stage s~Si raised by the orders processed before in the assigned unit.
C i ' s - p t i ' j >_Cis + sui'j + z T i ' j - M ( 1 - X i i ' s ) - M ( 2 -
Y i j - Yi'j)
Vi, i' ~ I , i ' > i,s ~ Sii', j ~ Jii' s
(3)
Ci~ - pto > Ci' s + suo + zv o - M Xii' s - M ( 2 - Yij - Yi' j ) Vi, i'~ I , i ' > i , s ~ S i i ' , j E Jii's
(4)
Constraints (3) and (4) become redundant whenever the orders i and i' are not allocated to the same unit j s J s at stage s sSii, and (Yij + Yi'j) becomes less than 2. Otherwise, both orders will belong to the same processing sequence at stage s and either the order i or the order i' will be processed earlier. Thus, it is no longer necessary to include the equipment index j in the domain of the sequencing variable Xii's and a significant reduction in the number of binary variables is so obtained. Furthermore, whenever Xii's is equal to 1 denoting that order i is processed before order i', the companion variable Xi,is drops to zero, i.e. Xii's + Xi,is = 1. Such a relationship between Xii's and Xi,is has been considered to get an additional saving in both binary variables and constraints. This is achieved by just defining a single variable Xii's for each pair of orders {i, i'}, where by convention ord(i) representing the relative position of i in the ordered list of members of the set I is less than ord(i'). If (Yij + Yi'j) - 2 and Xii,s= 1, order i is processed before order i' in unit jsJii's and (3) will become a binding constraint only if order i is manufactured immediately before i'. Otherwise, Xii's = 0 and, consequently, inequality (3) becomes redundant and (4) will apply on the starting time of order i. Preordering constraints can be easily embedded in the proposed problem formulation to attain a further reduction in the number of sequencing variables and constraints. If the orders are to be sequenced by increasing due dates and order b should be delivered before order a (db
1048 < da), then the variable Xabs can be eliminated from the formulation for any S ESab. Moreover, the constraint (3) for the pair of orders {a,b} can also be removed while the corresponding constraint (4) should be rewritten without the term involving the sequencing variable Xabs.
C) Bound on the completion time of order i at the last stage sit raised by the order due date.
Ci Si g ~---di
'v'i ~ I
(5)
d) Bound on the completion time of order i at stage s~Si raised by its starting time in the next stage (s+l).
Cis ~ Cis + 1-
~ p t o Yij j~Jis+l
Vi ~ I,s
(6)
~ S i - { s i g}
9 Topology C o n s t r a i n t s An order i allocated to unit j~(Jis ~Js) at stage s~Si can be assigned to some equipment itemj'~Ji,s+l at the next stage, only if it is connected to unitj.
Yij"<_
E Yij"
Vi ~ I,s ~ Si, j ~ (Jis ~ ffs)
(7)
j'~(Ji s +lF-hJsj,s+l ) 9
R e n e w a b l e R e s o u r c e Constraints
The maximum availability of a renewable resource r~R can never be exceeded along the production horizon. A pair of orders {i,i'} is simultaneously processed only if : (1) the processing of stage s' of order i' is completed after starting stage s of order i (Wisi's' : 1) and (2) the processing of stage s of order i is completed after starting stage s' of order i' (Wi's'is : 1). In other words, the orders {i,i'} are simultaneously processed only if: Wisi's' + Wi's'is-- 2. Moreover, (Wisi,s,+ Wi's'is ) is never lower than 1. Then,
Wisi's'M>_Ci's'-Cis+ ~ p t i j Yij
Vi, i'~I,i'~=i, s s S i , s'sSi'
(8)
j ~ Jis pisr+
Z ZtOi's'r i'~ I, i':/:i s'~Si'
(misi's'wmi's'is-l)<_Ar
Vi~l,r~R,s~Si
(9)
3.3. Problem Objective Function
For comparison purposes, it was chosen the expression proposed by Pinto and Grossmann (1995) to simultaneously maximize the weighted completion times for all orders in all stages (with higher weights for later stages) and minimize order earliness.
Maximize
[ Z iS~sZ his Cis _ iZ~ I hvi (di _ Ci sie)
where the parameters his and hvi are given by: his = 0.2 [max(di)/di] s, hvi = proposed formulation maximizes (10) subject to constraints (1)-(9).
(10) 10. The
4. RESULTS AND DISCUSSION The example presented in this section is concerned with the scheduling of a multistage multiproduct batch plant first studied by Pinto and Grossmann (1995). The plant comprises 5
1049
i
l
, I 3
,"/12
!
,'
7~
100 110
I
I
I I
I I
4
"
5 I
23 ~
180 190 I
', reaction l fluidization standardizationl
24 25
I I
drying
i
packing
Fig. 1. Multistage batch plant configuration stages, a total of 25 units and an unlimited intermediate storage between successive stages. Some topology constraints between consecutive stages are considered as shown in Figure 1. Five orders are to be processed over a 500 h period. The problem data (processing and setup times for every order at each stage) can be found in the above reference (Pinto and Grossmann, 1995). The example has been slightly modified to incorporate not only sequence dependent changeover times in the reaction and packing stages (see Table 1) but also resource constraints allowing the simultaneous operation of at most two units in the packing stage. A total of 8 workers are available in the packing stage at any one time and 4 are required to operate each packing unit. Figure 2 presents the optimal production schedule for the resourceconstrained example. An analysis of the results reveals that two orders are produced just-intime and the other three exhibit a rather small earliness. The model size, the optimal objective value and the computational requirement using the proposed approach (without preordering constraints) are all provided in Table 2. Similar values corresponding to the solution of the original example (without sequence-dependent changeovers and resource constraints), including those reported by Pinto and Grossmann (1995), are also summarized in Table 2. The examples were solved with ILOG OPL Studio 2.1 (Ilog, 1999), using the embedded CPLEX Optimizer. Table 1. Sequence-dependent changeover times reaction Packing order 1 2 3 4 5 1 2 3 4 1 4.5 2.2 2.8 3.2 1.2 0.9 0.7 2 2.4 2.8 4.1 3.9 1.5 1.1 0.8 3 1.5 1.9 4.1 3.2 1.4 1.0 1.3 4 4.0 1.2 3.5 2.7 0.8 0.4 0.7 5 1.7 2.5 3.7 1.4 0.7 0.4 0.7 1.2 Table 2. Model sizes and computational requirements Example binary vars, objective cont. vars, rows function
5
1.5 0.9 1.4 0.5
CPU time
nodes
92.09
1452
iterations
Unconstrained example Pinto and Grossmann (1995) .............T.h!.s....approa..c.h
161,167, 511
6162.70
.................................................................................................... 1....5......S....,.....2...S....,.....477....................... 6.2.6.8...7.6 ......................... !...-....S...7............................ 20.1 ................................ 7.6...5.. ..................
Resource-constrained example This approach
175, 25,502
5564.26
9.06
1790
7332
1050 Units 4
I
I +
++
+l
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31
5
51
O
11 i
i
i
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I0
.......
32 []
52 []
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14
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Standardization
53
17 19
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12 [] 33 . . . . II - I 13 O~
i23 . . . . . . . . . . . . . . . . . . . . .
34
20
44
21 22 . _ _
24 l
+
I
+
23 24 25 250
r]l 01
Processing Time Setup & Changeover 100 . . . .
Drying
14
54
; OI 3150,
,
,
,
I ~ 400
25 : ,
-
45 55
35- . . . . 101.,,+,~ I 15
01
-
i
Packing
:: I ,
,
I , 450
,
,
i
l , 500
Time
Fig. 2. Optimal schedule for the resource-constrained example 5. CONCLUSIONS
A computationally efficient MILP continuous time approach to the short term scheduling of multistage flowshop batch facilities under limited resource supplies has been presented. It permits to account for sequence-dependent setups without requiring any extra variable or constraint. Moreover, preordering constraints can be easily embedded in the problem formulation. The optimal schedule discovered for a real-world industrial application significantly improves the results reported by previous authors. ACKNOWLEDGMENTS
The authors acknowledge financial support from FONCYT under Grant 14-00356, and from "Universidad Nacional del Litoral" under CAI+Ds 048 and 121. REFERENCES
ILOG OPL Studio 2.1 User's Manual. (1999). Kondili, E., Pantelides, C.C., Sargent, R.W.H. (1993). Comput. Chem. Eng., 17, 211. Mockus, L., Reklaitis, G.V. (1997). Comput. Chem. Eng., 21, 1147. Pinto, J. M., Grossmann I. E. (1995). Ind. Eng. Chem. Res., 34, 3037. Pinto, J.M., Grossmann, I.E. (1997). Comput. Chem. Eng., 21, 801. Reklaitis, G.V. (1992). NATO Advanced Study Institute- Batch Process Systems Engineering. Antalya, Turkey. Schilling, G., Pantelides, C.C., Sargent, R.W.H. (1996). Comput. Chem. Eng., 20, S1221.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1051
A Comparative Study of Bi-linear and Tri-linear Approaches for the Monitoring of an Industrial Batch Process X. Meng, E.B. Martin and A.J. Morris 1 Centre for Process Analytics and Control Technology, Merz Court, University of Newcastle, Newcastle upon Tyne, NE 1 7RU, United Kingdom The empirical monitoring of batch processes has traditionally been performed using the multivariate statistical projection techniques of multi-way principal components analysis and multi-way partial least squares, i.e. bilinear techniques. More recently, tri-linear methodologies such as the extension of factor analysis to three-way problems has been advocated as a possible alternative. A number of papers have been published relating to the advantages and disadvantages of the two methods. This paper contributes to the discussion through a study of an industrial fermentation process where abnormal situations were created in a number of batches. The study emphasizes the necessity for an understanding of the process in order to achieve monitoring models that are fit-for-purpose. 1. I N T R O D U C T I O N The monitoring and control of batch processes for the achievement of reduced product variability, improved product consistency and increased yield is becoming an increasingly challenging issue in multi-product manufacturing. This is complicated by the fact that traditionally there has been a lack of investment in terms of sensors on batch processes. In recent years, the collection and storage of data has become routine and through the application of multivariate statistical techniques, performance monitoring schemes have been developed for the assessment of process behaviour [1-4]. A typical strategy for the development of such a scheme is to build a model from batches relating to acceptable production and through statistical test, monitor new batches for deviations from normal operation. Once non-conforming operation has been identified, remedial action can be taken to ensure that subsequent batches produced are acceptable. Deviations of the process variables from their specified trajectories can arise as a result of errors in the charging of recipes, incorrect addition of raw materials, valve and sensor malfunctions, fouling, impurities and other disturbances. Data collected on a batch process is typically expressed mathematically in terms of a three-dimensional array (batch x variable x time). The traditional tools for the analysis of such data have been Multi-way Principal Component Analysis (MPCA) and Multi-way Partial Least Squares (MPLS) [1-3]. More recently Parallel Factor Analysis (PARAFAC) has been considered as an appropriate analysis tool for the development of monitoring schemes [5]. In contrast to MPCA and MPLS that result in bi-linear models, a tri-linear model is generated. Westerhuis et al (1999) [4] compared the two approaches on a simulation of a The authors acknowledge CPACT for financial support and the ORS Award Scheme.
1052
styrene-butadiene process and an industrial batch polymerisation process. In this paper, multi-way PCA and PARAFAC are applied to an industrial fermentation process and a comparison between the two approaches is made. 2. M E T H O D O L O G Y R E V I E W
2.1. Multi-way Principal Component Analysis (MPCA) Multi-way PCA (MPCA) is an extension to the multivariate statistical technique of PCA. MPCA is able to handle data matrices comprising three or more dimensions [1-3]. The idea of MPCA is to perform a principal components analysis on an unfolded form of the three dimensional data matrix. There are three ways of unfolding the data matrix, that is according to the time trajectory, the process variables or the batch number. The most practical approach is to unfold the three-dimensional matrix according to the time trajectory, that is each time slice (batch• is placed side by side. A large two-dimensional matrix is then obtained. Adopting this approach enables the analysis of the variability between individual batches. A number of metrics can be used for the statistical monitoring of a process. Traditionally the squared prediction error (Q-statistic) and Hotelling's T 2 form the basis of any multivariate statistical monitoring scheme.
2.2 Parallel Factor Analysis (PARAFAC) Parallel Factor Analysis (PARAFAC) was developed based upon the Principle of Proportional Profiles [6]. It can be regarded as one of the simplest three-way generalizations of the traditional multivariate statistical techniques of factor analysis. The PARAFAC model was independently proposed by Carroll and Chang (1970), who named the model CANDECOMP (Canonical Decomposition) [7]. The PARAFAC model of a three-way array can be described as the sum of the outer products from a set of loading vectors. Each set comprises three vectors according to the three modes of the original data. They may also be considered as two loading vectors and one score vector following the concept of PCA, which divides the model into two parts, scores and loadings. F
(1)
Xijk = ~., a i f b j f c k f + e6k f=l
Equation 1 illustrates one approach for expressing the PARAFAC model structure. F is the number of factors included in the model, e(/k is the error between the original data and the data projected down onto the model and ail (i = 1.... I) is the element of the fh factor, for one of the three dimensions.
Likewise, b j f
(j-
1.... J ) and Ckf (k = 1.... K)define the
elements of the factors for the other two dimensions. The objective of the model is to minimise the sum of squared errors. Hence it has the same loss function as for PCA. The argument often given for applying multi-way methods, such as PARAFAC, is that the resultant model is more robust, easily interpretable and simpler [5]. The simplicity of the model refers to the number of parameters that need to be estimated. For PARAFAC, the number of parameters to be calculated is F *(I + J + K ) , where F is the number of the factors and/, J, K are the number of variables for each of the three modes, respectively. In
1053 contrast, for an MPCA model, the number of parameters to be estimated is F * (I + J * K) which is usually much larger than F * (I + J + K). A three-way data set can be considered from each of three dimensions since the data in the matrix can be analysed in each of the three multivariate spaces. However, none of these dimensions can include all the information contained in the other two. This is why MPCA, which unfolds the matrix on one mode, looses some information in terms of the relationship between the variables in that dimension. PARAFAC, as a tri-linear decomposition model, maintains the linear aspects of the relationships for all three modes. Although it models the data using the same loss function as for MPCA, PARAFAC does not perform as well as MPCA in terms of explaining the variance in the data. This can be considered from two perspectives. Firstly MPCA requires the estimation of more parameters than PARAFAC resulting in there being more degrees of freedom to describe the underlying structure and hence a higher proportion of the variability in the data. Secondly, in the calculation of the model structure of PARAFAC, more constraints are imposed than for MPCA. However, an advantage of the PARAFAC approach, which is often cited, is the uniqueness of the solution. Bilinear methods such as multi-way PCA, are affected by rotational freedom, i.e. the direction of the principal components can alter simultaneously without affecting model fitness. Thus external assumptions need to be imposed to determine the direction of the loadings. For PARAFAC, this is not an issue. The 'Principal of Proportional Profiles' underpins the property of uniqueness. The basic idea behind this principle is simple. By comparing the factors extracted separately from two different but related data sets, it should be possible to discover the "real" orientation of the axes in the two solutions. This is achieved by finding that orientation which brings their factor loadings (or factor scores) into parallel, proportional correspondence, across solutions. As long as the two data sets are not equivalent, but possess the same common factors in different relative proportions, there is only one rotational position that will reveal this correspondence [8]. 3. CASE STUDY
3.1 Data Description Data collected on an industrial fermentation process is used to compare the two approaches of multi-way PCA and PARAFAC. Process data was collected on an hourly basis throughout the duration of a batch and was recorded on each of 14 variables for 11 reference batches. By analyzing the final product, the batches were categorized into four groups: normal production, standard production (better than normal production), normal with low production and abnormal production. Batches within the abnormal production category were produced both under normal reaction conditions as well as under modified operating conditions. Since the time duration of batches differ between runs, batch length equalisation techniques were applied. Three methods have been explored previously in the literature [9], the reduction of the duration of the batches to that of the shortest batch, Dynamic Time Warping (DTW) and the introduction of a surrogate variable. DTW is a flexible pattern matching method used previously in the area of speech recognition. The basic idea is to distort the test profile by compressing or stretching it to the same length as a reference profile. In this study, multivariate DTW was applied to the variables.
1054
3.2 Multi-way Principal Components Analysis The next step in the analysis was to apply standard PCA to the unfolded standardised data set. Three models were built. Model lis based on all the process variables (dissolved oxygen, anti-foam oil flow rate, ammonium sulphate flow rate, inlet air flow rate, C-feed, Input/Output 02 flow difference, Input/Output CO2 flow difference, phenyl acetic acid concentartion, water flowrate, pH, head pressure, temperature and agitation speed), while variables 2, 3, 5, 9, 11, 13 are excluded from Model 2 and in Model 3 variables 2, 5, 9 are excluded. For all models, four principal components were retained according to cross validation. For Model 3, approximately 66% of the total variability in the data is explained by four principal components. The variance captured by the four principal components can be summarised in terms of the three dimensions, variable, batch and time (Fig. 1). In Fig. 1, for modes variable and batch, the y-axis is the cumulative sum of squares for the first four principal components. Whilst for the mode time, the time series plots describe the cumulative sum of squares for the first principal component, the first two, the first three and finally the first four. It can be seen that some variables dominate particular components. For example dissolved oxygen (variable 1) dominates PC1 whilst pH (variable 10) dominates PC 2. A similar effect is seen for the mode batch. Batch 7 and to a lesser extent batch 10 dominates PC 1. Time trajectories of the principal component scores and the SPE can also be calculated. 2000
2501 . . . . . . . . . . .
1O(
2001 /
1500
1501
1000
6(
1001 ~ ~ ~ ~ ~
500 C
8(
~
~~
~
4(
500~ O'
1 2 3 4 5 6 7 8 9 10
2( 0 0
1 2 3 4 5 6 7 8 9 10 11
. . 50
. . 100
150
200
Fig. 1. Summation on the modes of variable, batch and time (From Left to Right) The models developed from the 11 normal batches, running under unchanged reaction conditions, formed the reference database. A number of new batches, where controlled changes were made, formed the basis of the test data set. Two of these have been selected as case studies. Batches B 1 and B2 were run at temperatures 2~ above and below the nominal value and were classed by fermentation scientists as abnormal batches even though they both achieved a normal final product. 15 x 10. 4 .
. . .
15
x104,
,
10
10
r " q r"--'a 1
2
3
4
5
6
7
1
2
3
4
5
6
Fig. 2" Sum of Squared Error on the Mode of Variable of Batch B 1 and B2
7
1055 The data from batches B 1 and B2 was then projected onto Model 1. From the sum of squared errors calculated for the mode of variable for both batches, head pressure was identified as the variable indicative of the temperature problems (not shown). Although this particular variable does not directly point to the problem temperature, detailed process understanding leads to the conclusion that a cooling 'problem' had occurred. Model 2 was then used as the reference model to analyze the two new batches. From Fig. 2 (Left) and Fig. 2 (Right), variable 7 (temperature) was identified as the important element in explaining the deviations in both batches. Since the changes in reaction conditions were known for these two batches, it is apparent that the results from projecting the data onto Model 2 are more easily and directly interpreted as a cooling problem. In contrast the results from Model 1, although at first glance may appear misleading, fermentation expertise can help in terms of fault diagnosis and location.
3.3 Application of Parallel Factor Analysis To compare the relative merits of two and three-way methods, a nominal PARAFAC representation was built from the data using the 11 normal batches. The first issue was to decide on the number of factors to include in the model. In two-way methods, such as PCA, latent variables are calculated in a stepwise manner, i.e. the first principal component explains the greatest amount of variability in the data, with the next principal component explaining the next greatest amount of variability and so on. Due to the orthogonality property, including additional principal components will not affect the previously calculated principal components. However, in PARAFAC, selecting a different number of factors can result in a change in the orientation, since the factors are extracted simultaneously. This is a kind of trade-off for uniqueness. A number of techniques have been advocated to address this issue such as residual analysis, cross-validation and split-half. From the plot of residuals versus number of factors, it was found that the residuals decrease smoothly for an increasing number of factors. This makes it difficult to identify the boundary between the signal and the noise. Similarly, the results from cross-validation were not conclusive. Split-half is a type of jack-knife analysis where different subsets of data are analysed independently. Due to the uniqueness of the PARAFAC model, the same result, i.e. same loadings, will be obtained in the non-split modes from models of any suitable subset of data, providing the correct number of factors are chosen. If too many, or too few, factors are chosen, then the model parameters will differ when the model is fitted to different data sets [5]. In this application, factors were not stable on the modes of variable and time when splithalf was carried out on the batch mode. One reason may be that the number of batches included in the reference data set is too small, thus different types of system behavior is captured in the various subsets. Fig. 3 summarizes the loadings of the models using both one and two factors for the reference batches. From the plots of the two-factor model, it appears that the loadings are almost identical for the time and variable modes, whilst for the third mode, the two sets of loadings have similar values but opposite signs. This is an indication of degeneracy [1]. Degeneracy is defined as, the more iterations performed, the closer the correlation between the two components. Adopting alternative procedures, such as a Tucker3 model, degeneracy may be avoided. Other ways to avoid a degenerate solution is to apply orthogonality or nonnegativity constraints to some modes of the model. Inappropriate modelling data and poor data pre-processing can also contribute to degeneracy. A one-factor PARAFAC model was selected and the potential of the representation to identify non-conforming batches was
1056 investigated using the same set of test batches as used in the MPCA study. These batches were projected onto the PARAFAC models using all the process variables of Model 1 and the 7 process variables of Model 2. To investigate possible reasons for non-conforming behavior, the squared errors were summed over the dimension of variable. After comparing the performance with that for the MPCA analysis, it was concluded that the results appear quite similar to the contribution charts resulting from MPCA. 0.2
D
10.I~ 0
D2
0.2
Loadings for Each Dimension,
~
i
Loadings for Each Dimension
0.1 |
i
0
10
50
100
150
200
0
-0.5
-50
-1
'
D 3500~ 0
'
! 5
.
19
,
1i
2
4
6
8
10
12
-200 / 0
I
, 2
, 4
i
, 6
, 8
, 10
12
Fig. 3. Loadings for PARAFAC model with one factor (left) and two factors (right) 4. DISCUSSION The study indicates that MPCA and PARAFAC based performance monitoring approaches show similar fault detection capabilities in this particular application when the appropriate model is selected. In the application discussed, the PARAFAC model is simpler than the MPCA model in terms of the number of parameters to be estimated. However, it cannot explain as much of the variability of the reference data set as MPCA. Selecting the appropriate number of factors is a critical and a problematic issue in PARAFAC modelling. Degeneracy, another distinct phenomenon in PARAFAC, results in a model that is not fit-forpurpose with some factors being highly correlated. Poorly structured models, poor data preprocessing and inefficient computational algorithms can also lead to degeneracy problems and poor overall performance. Overcoming these issues requires other tri-linear modelling approaches to be studied alongside the imposition of constraints such as non-negativity or orthogonality on relevant modes. It is also clear that tasks such as data pre-processing and process-knowledge driven variable selection are essential to the overall success and applicability of the model developed. 5. REFERENCES 1 2 3 4 5 6 7 8 9
K.A. Kosanovich, Ind. Eng. Chem. Res. 35, (1996) 138. P. Nomikos and J. F. MacGregor, AIChE Journal 40, (1994) 1361. P. Nomikos and J. F. MacGregor Technometrics 37, 1995 41. A.J. Westerhuis, T. Kourti and J.F. MacGregor, Journal of Chemometrics, 13, (1999) 397. R. Bro, PhD Thesis, Royal Veterinary & Agricultural University, DK (1998). R.B. Cattell, Psychometrika 9, (1944) 267. J.D. Carroll and J. J. Cheng, Psychometrika, 35(3) (1970) 283. R. Harshman and S. Berenbaum, in D. Eichorn, J. Clausen, N. Haan, M. Honzik, P. Mussen (Eds) "Present and Past in Middle Life", Academic Press, NY, 1981, 435. S.G. Rothwell, E. B. Martin and A. J. Morris, Proceedings 7 th International Conference on Computer Applications in Biotechnology (CAB7), Osaka, Japan (1998).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1057
P l a n n i n g a n d S c h e d u l i n g in a P h a r m a c e u t i c a l R e s e a r c h a n d Development Linas Mockus a, Jonathan Vinson a, and Robert B. Houston b aSearle, a Monsanto Company, 4901 Searle Parkway, Skokie, IL 60077, USA bSolutia, 4400 Lindell Blvd., St. Louis, MO 63108, USA A novel approach to handle all planning and scheduling complexities arising in a pharmaceutical pilot plant is presented. We employ a two-level approach to planning and scheduling: multi-year strategic planning of people resources first, and then daily scheduling of process operations based on this plan. 1. I N T R O D U C T I O N In the face of growing demands and limited resources, planning and scheduling play an important role in the operation of chemical facilities. Specific to the pharmaceutical industry is the need to conduct process design and drug production concurrently. For long-range planning the company needs to make strategic decisions as to which projects are to be pursued. The company looks at marketing and clinical data to help set priority and direction. They must also consider resources available to complete the projects. Uncertainties at this level appear in the timing of projects and when the company plans to register new compounds with the governing bodies. Management may decide to accelerate, delay, or cancel projects depending on clinical trial results, toxicology studies and the pressures of the marketplace. The short-range plan covers production of the active pharmaceutical ingredient (API) and is also subject to complexities. During the early phases of product development, the demand for an API is relatively small and uncertain. As the project matures, the material demands and due dates become more definite. Similarly, the synthetic chemical route for production of the API changes as the process is improved. These improvements usually target the overall yield of the process, but they also affect the utilization of the various pieces of equipment. Because of USA Food and Drug Administration (FDA) regulations for Good Manufacturing Practice (GMP), cleaning of all vessels involved in a process has to be quite thorough. The average batch time is about five days and a piloting campaign is only a few batches, while cleaning and equipment setup for that campaign can take five days as well. In manufacturing plants campaigns run in longer "trains" before cleaning takes place (equipment cleaning takes place at the end of twenty, 36 hour-long batches).
1058 There are several basic job functions in the pharmaceutical R&D environment. These people must work together in various combinations to complete a given project. Chemists develop the synthetic route and define a basic separations framework. Engineers work with the chemists to develop a scalable process. The engineers also conduct lab work, and they prepare hazard reports and other documentation. Analysts develop test methods and specifications for new drugs based on samples prepared by chemists and early piloting studies. These and other tasks occur across most projects, but not all projects have the same set of milestones, nor are they necessarily staffed at the same levels. In addition, many of these activities happen in parallel, only to be definitely complete at the project due date. These human activities are very difficult to mathematically model because of their "fuzzy" nature and the wide array of activities required of each job function. 2. T W O - L E V E L A P P R O A C H
The complexities arising in pharmaceutical research and development can be handled by a two-level approach to planning and scheduling. First, generate a strategic plan for people resources over a multi-year horizon, accounting for intellectual resources such as chemists, analysts, operators, and engineers. Then construct the daily schedule of operations in the pilot plant, accounting for equipment and operators, based on the strategic resource plan. Instead of modeling every single activity, we look at the overall resource requirement of a higher-level milestone, such as completing Phase I clinical studies. This decomposition approach -- breaking down large and complex problem into smaller manageable pieces -- is quite common in operations research. The rolling horizon method decomposes the long-term planning problem into many shortterm scheduling sub-problems by moving the scheduling horizon [1]. The scheduling sub-problems are independent except where scheduling horizons overlap. Subrahmanyam [2] have similarly divided the batch plant design problem into two levels: the Design Super Problem and the Scheduling Sub Stage. The Design Super Problem incorporates the design and aggregate scheduling constraints, while the Scheduling Sub Stage ensures feasibility of schedule. In both approaches, decomposition is based on the structure of the problem, thus requiring integration of various decomposition levels that may violate feasibility. Our proposed approach is based on the decomposition of functional levels, and integration of decomposed sub-problems does not violate feasibility. In addition, this approach mimics our current system for strategic planning and daily scheduling. The strategic plan answers a question like the "For a given process, should we be in phase III of production by the 2 nd quarter of 2000?" In the
1059 detailed plan, this may translate into production of 500 kilograms due J u n e of 2000.
2.1. S t r a t e g i c r e s o u r c e p l a n n i n g The strategic resource plan covers a 5- to 6-year time horizon. It determines when we should be in a certain phase of a project, given the availability of resources and the probability of completing those tasks. The process operation schedule focuses on an 18-month cut of the strategic resource plan and employs the constraints imposed by it. By knowing the phasing of a project, we can calculate the specific production campaigns t h a t need to happen. This is based on either the actual demand for clinical trials (for m a t u r e processes) or historical production needs by phase. The strategic resource plan model focuses on the major milestones of the pharmaceutical life cycle: Phase I, Phase II and Phase III clinical trials and other project related activities, such as validation or demonstration runs and filing support. At this level, we are concerned with the abstract activities of each work function, not pilot plant equipment. Searle has been collecting data that shows how the job functions are devoted to a given phase of development. The planning process starts with allocation of people resources to projects in the development pipeline. Projects that connect be completed by their due date m u s t either be delayed, contracted or dropped. All the uncertainty of pharmaceutical planning and scheduling is lumped into this level and is represented as a probability of success of a given phase. Failure may be caused by poor results from toxicological or stability studies, or from clinical data that show no efficacy. When projects drop or priorities change, m a n a g e m e n t must scramble to reassign resources and potentially resurrect other projects. Our approach is to over-allocate resources based on the probability t h a t some projects will not survive, so t h a t on average the resource constraints are satisfied [3]. We suggest t h a t this method will help the company retain its focus on the important projects. If fewer projects drop t h a n expected, the company has a well-defined need for hiring or buying additional resources (outsourcing). Outsourcing some parts of a project is a common practice due to resource and time limitations. Frequently, because of financial considerations, it is cheaper to contract the production of some intermediates t h a n to hire new employees. In reality, only intermediates t h a t are not critical to a process are outsourced because of the high uncertainty prevailing when the production of chemical is outsourced. By expanding the strategic planning model with financial and uncertainty of outsourcing, we will arrive at more reasonable outsourcing decisions.
1050 2.2. P r o c e s s operation s c h e d u l i n g Given a feasible strategic plan, we are able to set the constraints for a detailed pilot plant schedule without having to worry about the actives of chemists, analysts, and engineers. Since the tightest constraints come from the h u m a n resources, it is not necessary to consider uncertainty at the equipment scheduling level. This greatly simplifies the formulation of the detailed scheduling model of process operations and significantly improves the solution speed. A typical 18-month plan contains over 30,000 variables t h a t represent demands, material balances, equipment and process parameters. The constraints imposed on the 18-month plan are in the area of material demands and timing. At this level, we are dealing only with a detailed scheduling of pilot plant operations, or assignment of operating equipment to each project step. On average, an API requires ten organic transformations (or steps) to reach the final, active form. Production of each step involves unit operations of varied complexity. We are not considering the activities of the formulary, which p u t s the API into tablets or capsules. Complexity arises from the fact t h a t each step can be performed on a different set of equipment. We have developed a novel conceptualization of each step as a sequence of only three major operations: reaction, separation, drying. This helps to reduce the scheduling problem size, but even in this case only dedicated solvers can handle the complexity. In general, only a "good" pseudo-optimal solution is obtained which satisfies our needs. We minimize the m a k e s p a n of a detailed operation schedule based on equipment availability, equipment turnovers, cleaning requirements, t h r o u g h p u t calculations, yield, and inventory. This shows where we can free up resources and increase productivity. Hence, we should see the overall resource profiles decrease or a reduction in the duration of a particular strategic task. 3. A C A S E S T U D Y The purpose of this case study was to locate possible ways of improvement of strategic resource planning and operation scheduling. The impact of project survival probability on the overall plan was determined. Then the effect of different scheduling scenarios was studied on the duration of the project along with asset utilization.
3.1. Strategic resource p l a n n i n g The case study evaluated 17 projects in various phases of development. We note t h a t the resources requirements for each phase may vary by project as each has a different size, be we have kept them the same for simplicity. This can be changed to create a more complex problem.
1061 By planning people resources over the 9-quarter period, we find t h a t one lowest priority project must be outsourced. If we ignore the possibility t h a t projects m a y drop over this period, then four projects cannot be fit into the strategic plan. Clearly, one can adjust demand dates of the projects, or hire new resources, to enable the planner to fit more projects. In this example, the constrained resources were chemists and analysts. However, newly hired people would be used only at certain peak r e q u i r e m e n t times and this is not an acceptable way of using people resources. 3.2. P r o c e s s o p e r a t i o n s c h e d u l i n g Once the strategic plan is set, we can generate the operating schedule for the pilot plant. We only consider those tasks from the previous level t h a t require plant time (in this case projects PR1, PR2, PR3, and PR4). Each step is further broken down into Reaction, Separation and Drying unit operations. While the equipment scheduling is straightforward, there are several modes of scheduling which produce different schedules and resource utilization. These differences relate to how the scheduler sequences batches, and in the t r e a t m e n t of resources for unit operations. Mode I is the standard scheduling mode where all equipment is reserved for the entire batch, and batches of a given step are run sequentially. Mode 2, on the other hand, allows simultaneous batches, and it schedules the resources for each unit operation independently. For each of these scenarios, we determined manpower utilization, and have summarized the results in Table 1. The general trend is t h a t the duration decreases and utilization increases when the degree of concurrency increases from Mode 1 (no concurrency) to Mode 2. This suggests t h a t overlapping steps and unit operations will keep our h u m a n resources active, while also reducing the time to production for a project. Note t h a t the unit operation mode should be applied only to developed processes, as new processes have been r u n less and the times are more uncertain. Table 1. Results Scenario Manpower utilization index ~
PR1 PR2 duration duration (hr.) (hr.) M o d e l 3.93 2664 1092 Mode 2 4.82 2188 972 Total decrease in production time 18% 11% The average n u m b e r of operators required during
PR3 duration (hr.) 2088 1704 18% scheduling
PR4 duration (hr.) 3644 2711 26% period.
4. C O N C L U S I O N S AND F U T U R E W O R K We have demonstrated how to incorporate the complex n a t u r e of h u m a n activities into the planning decisions without detailing the individual tasks and
1062 how those tasks impose constraints on the planning model. To this end we have demonstrated an approach for combining a strategic resource planning with short-term operation scheduling. The real secret of success in this process is having data available to make the necessary decisions and tightly integrating the model with this data. Strategic resource planning and process operation scheduling models are tightly connected. While it is possible to consider people resources, such as chemists and engineers, at the scheduling level, it unnecessary complicates and overburdens the model. The "fuzzy" nature of people resources disappears when aggregated over a long period and it is becomes possible to distinguish patterns. Our approach exploits the structure of the problem to reduce the complexity. This work represents a significant improvement in planning and scheduling of activities in our research and development group. Improvement of this framework will continue in several areas. Resource planning and operations scheduling should be more tightly integrated. Sensitivity analysis of both the strategic resource plan and the operation schedule would provide solutions that are more robust. To determine the robustness of the resource plan and the operation schedule, one may simulate them by perturbing project survival probabilities and processing times. Another option for improving production time at the operation level is to allow different equipment choices for each unit. As the algorithms we employ improve, we may begin adding documentation preparation and other project related activities to the detailed operation scheduling level. These tools are meant to help the company determine both the strategic plan for the coming years and the operation schedule on the short term. At the strategic level, one can look for human resource bottlenecks and examine the effect of priorities and survival probabilities. At the operations level, management can study equipment and manpower limitations more easily. REFERENCES
1. Basset M. H, J. F. Pekny, and G. V. Reklaitis, Decomposition Techniques for the Solution of Large-Scale Scheduling Problems, AIChE Journal, 42 (1996) 3373-3387. 2. Subrahmanyam S., J. F. Pekny, and G. V. Reklaitis, Ind. Eng. Chem. Res., 33 (1994) 2688-2701. 3. Samikoglu, O., S. J. Honkomp, J. F. Pekny and G. V. Reklaitis, Sensitivity Analysis for Project Planning and Scheduling under Uncertain Completions, Computers Chem. Engng., 22 (1998) $871-$874.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1063
Mixed Integer programming techniques for the scheduling of fuel oil and asphalt production J.M. Pintot and M. Joly Department of Chemical Engineering, University of S~o Paulo, S~o Paulo - SP, Brazil 05508-900 This paper addresses the development and solution of mixed-integer (MIP) optimization models to a real-world fuel oil and asphalt production scheduling problem at the PETROBRAS REVAP refinery, which is responsible for approximately 80% of all fuel oil consumed in Brazil. 1. INTRODUCTION Systems for refinery scheduling emerge as necessary tools for keeping competitivity at a worldwide level. On the other hand, the use of commercial packages do not show satisfactory results due to the fact they many times do not offer the scheduling quality [ 1] and/or are unable to represent plant particularities [2]. Hence, refineries have been directed towards developing their own scheduling systems. The approach taken in the development of an integrated scheduling system for the short-term scheduling at the PETROBRAS REVAP refinery is to concentrate on separate sections of the plant, to work the problems independently and further integrate them [3]. This paper addresses the development and solution of MIP optimization models for the fuel oil and asphalt production area, which involves the optimal management of continuous operations such as mixing, storage and distribution. The special importance of this problem resides on: a) the plant produces approximately 80% of all fuel oil consumed in Brazil; b) the plant has relevant storage limitations and c) the Brazilian oil-sector is under a gradual end of a 43-year-old constitutional monopoly. The problem is first modeled as a nonconvex MINLP [4], which has the drawback that no global solution is guaranteed by conventional MINLP algorithms, although this difficulty was partially circumvented in [5]. A rigorous linearization of the previous model is then proposed to provide a rigorous lower bound for the objective.
2. PROBLEMDEFINITION Figure 1 illustrates the system configuration which includes 1 deasphalting unit (UDASF), 1 cracking unit (UFCC), 2 storage tanks for diluents, 15 storage tanks for final products, 4 charging terminals and 2 oil pipelines as well as all their interconnections. During the scheduling horizon, asphalt residue (RASF) is produced in the UDASF, as bottom product, and further diluted on-line with at least one of the following diluents: decanted oil (OCC) and light cycle oil (LCO), with the purpose of producing four grades of fuel oil (FO 1, FO2, FO3 and FO4), or with another diluent, heavy gasoil (HG), aiming to produce two asphalt specifications (CAP 07 and CAP 20). Moreover, the plant produces two grades of ultra-viscous oil (UVO1 and UVO2) which must have only pure LCO from UFCC as RASF diluent. The UDASF production must also satisfy a Author to whom correspondence should be addressed. E-mail: jompinto(a~,usp.br. The authors ackalowledge support received from FAPESP under grant 96/02444-6 and from Petrobras. _
_
_
1064 minimal demand of pure RASF to the refinery oil-header (roh). The major specification of all final products is the viscosity range, which has to be adjusted by proper dilution with the available diluents. The OCC (from UFCC) and the HG supply streams are totally consumed by the plant; the HG supply stream is directed to storage in TK-42221 and the OCC stream from UFCC is either directly utilized for RASF dilution or directed to storage in TK-42208 (LCO and OCC mixed) since these two operations can not occur simultaneously. Unlike the above described, the LCO stream from UFCC must be directed to the plant only when necessary, i.e., when it is required to charge TK-42208 or when UVO 1 (or UVO2) must be produced. In this case, to assert that pure LCO is being utilized to dilute RASF, the TK-42208 level must increase at a proper rate while pure LCO flows in the dilution line (see Fig. 1). A strategy of allocating the fuel oil production temporarily in a tank is usual but undesirable since it implies additional process steps, as viscosity adjustment and homogenization steps. Storage tanks cannot be charged and discharged simultaneously; exception is the HG storage tank, which is continually charged. The distribution of a given product, by oil-pipeline or trucks, requires that two tanks that contain it are connected (we say "aligned') to the same line; exceptions are TK-44108 and TK-43307, which operate individually. Hence, there is the option of replacing the supplier tank in case of urgent necessity of receiving material. The distribution of UVO/asphalt is only performed by trucks, from 6:00 a.m. to 6:00 p.m. The fuel oils are only dispatched by oil-pipelines. Demands are defined by the refinery planning and should be exactly met. ~
I~
HG
....
RAW MATERIAL
MIXED ~.~-KK ~-~'~
.... .
FROMVACUUMUNIT OC L O
t. . . . I~ii ~tI~K l ~ J ~"--L~" i~ J ~TK? .", ~ --T~ ,.,=~02 OUV/ASPHALTAREA _~_~ _'~-L ~
'
"
DILUTI.ONLINE .
UDASF
--
RASF TO THE REFINERY OIL H E A D E R
_
M=xR-.~_
-
-
'
u cc
OCC'TORASFDILUTION L~O
~~ ~~I - T~4_-3fT3~K ~K~.o~ TK "fTK
r
~---_ TK __0~ 33
FUEL-OIL AREA
LOCALOIL-PIPELINE
LCO
LCO TO'DIESEL AREA
PRODUCT STORAGETANK(S) HG TK-42221 OCC/LECO ....... . . . . . . . OD .TKTK-44111/ . . .42208 . . . . .12. . UVOI
CAP20 TK-44110/15/16 FOI TK-43501/02 F04 TK-45507 OIL-PIPELINE TO SAO PAULO
Fig. 1. Schematic representation of the plant. 3. SCHEDULING OPTIMIZATION MODEL It is assumed that: (A1) the system is isothermal at ambient temperature; (A2) the mix between RASF and diluents is ideal. All products are incompressible fluids; (A3) there is pre-assignment between tanks and products, as shown in Fig. 1; temporary storage is not allowed; (A4) the viscosity of the oil mix is based on the weighted volumetric flow of input streams in the mixer (see Fig. 1) and their original viscosity values; (A5) perfect mixing in the mixer; (A6) the demands are known a priori. There are no deadlines. (A7) changeover times are neglected. Unlike [6], where task processing times may vary considerably, this problem can be modeled under discrete time, as in [7]. Due to the dispatching timetable for UVO/asphalt, two modeling rules are imposed: (R1) the scheduling horizon (SchH) must start either at 6:00 a.m. or 6:00 p.m.; (R2) SchH must be a natural multiple of 12 hours. Table 1 provides the model nomenclature. The objective and the major constraints of the MILP model are reported below.
1065
Table 1 N o m e n c l a t u r e o f the M I P optimization models. Indices and Sets b=l,2,..., DT.(T/12) auxiliary index; d = 1,2,...,D diluent storage tanks; i = 1,2,...,I fuel oil storage tanks; o = 1,2, .... O oil-pipelines; p = 1,2, .... Pfuel oil grades; q = 1,2,...,Q UVO/asphalt storage tanks; s = 1,2, .... S diluents: v = 1,2,...,VUVO/asphalt grades; t = L2,...,T fame interval (T is given by T = SchI-I / DT); P/set ofp that can be stored in I; Sd set of s that can be stored in d; Vq set of v that can be stored in q. Parameters (costs, demands and rates are given in volumetric basis, unless where explicitly mentioned) CBi pumping costs, per unit flowrate, between tank i and any oil pipeline: CD~ unit cost of diluent s; CIDa inventory cost coefficient of storage in d per volume and time units; CII~ inventory cost coefficient of storage in i per volume and time units: CIQq inventory cost coefficient of storage in q per volume and time units; CR RASF unit cost: DMFOo,p demand ofp in market fed by o during the scheduling horizon: D M R A minimal demand of (pure) RASF in roh during the scheduling horizon; DMUVv demand of v during the scheduling horizon; D T time span. Must be a natural divisor of 12, i.e., DT ~ { 1,2,3,4,6,12}; F n~ maximum flowrate produced in the mixer; FDDd .... maximum unloading flowrate for d; F H G HG feed flowrate to TK-42221; FID "~ maximum unloading flowrate for i, Vi; FLCO/FOCC LCO and OCC nominal production rates by UFCC, respectively; F o j n / F O f l ~ flowrate lower and upper bounds in o, respectively; FQD "~ maximum unloading flowrate for q, Vq; FRA "~" minimal RASF flowrate from UDASF to feed the mixer in t, Vt; FRAM nominal RASF production rate by UDASF: FSEC minimal LCO flowrate directed to TK-42208 while an UVO is produced: HTb auxiliary 0-1 parameter to model the timetable of UVO/asphalt unloading; MD~ viscosity of s; M/~ viscosity specification for i; MQq viscosity specification for q; MRA RASF viscosity; VDZd initial volume in d; V/Z~ initial vol. in i; VQZq initial vol. in q; VDff"/VDam~ lower and upper volumetric capacity bounds of d, respectively; Vli"~"/Vlff~ lower and upper vol. capacity bounds of i, respectively; VQq"~"/VQqm~ lower and upper vol. capacity bounds of q, respectively; Binary Variables XDCt denotes if TK-42208 is charged at ~ XlCi, t denotes if i is charged at t: XlDi, o,t denotes if i unloads to o at t: XQCq, t denotes if q is charged at ~ XQDq, t denotes if q unloads at t; XDRAt denotes if the dilution line (see Fig. 1) transports HG at t; XLCOtdenotes if the RASF dilution is done with pure LCO (from UFCC) at t; XZt denotes ff the OCC stream (from UFCC) charges TK-42208 at t; XWt denotes if CAP-20 is sent to its charging terminal at t. Continuous Variables (the flowrates are given in a volumetric basis) FRAAt RASF flowrate from UDASF to roh at t; FRAUt RASF flowrate from UDASF to mixer at t: FDRAd,t flowrate from d to mixer at ~, FIRA u flowrate from mixer (RASF+OCC+LCO) to i at t; FQRAq, t flowrate from mixer (RASF+HG or RASF+LCO) to q at t; FDCs, t flowrate of diluent s to storage (in the dedicated tank) at t: FIDi, o,t flowrate from i to o at t; FQDq, t flowrate from q to respective charging terminal at t; FOo,p,t flowrate ofp in o at r, FOCCRt OCC flowrate from UFCC to mixer at t; FPLCOt LCO flowrate from UFCC effectively used by file plant at t: FRLCOt LCO flowrate from UFCC to mixer at t; FIRAKi, t/FIDKi, o,t/FQRAKq, t/FQDKq, t a u x . variables for viscosity calculations: VD~t diluent level in d at r, Vli,t product level in i at t; VQq,t product level in q at t; VIKi,t auxiliary variable for viscosity calculation of i at t; VQKq,t auxiliary variable for viscosity calculation of q at t.
Minimige (A) Operating Cost = (raw-material cost) + (inventory cost) + (pumping cost) T
S
O C : Z ( ( Z (TD, .FDC~,t + CD 2 9FOCCI~ +CD s 9F R L C q + CR. FRAM) + t=l s=i i Q D I 0 + ( Z (TIIi "VIi,t + Z (TIQq .VQq,t + Z CIDd Und, t ) + ( Z Z CBi "FIDi,o,t )) 7=1 q=l d=l i=l o=1
(1)
~$ubiect to:
(B) Material Balance Constraints. Constraint (2a) illustrates v o l u m e t r i c b a l a n c e for fuel oil storage t a n k s Similar balances hold for U V O / a s p h a l t and diluent t a n k s t'
O
l/In 'in < Vii, c = VIZi + ~.[YlRAi, t - Z(YlDi, o,t)] < VI m'~ t=l o=I
Vi,t'
(2a)
1066 (C) Foreseen Demand Supply of Plant Products. The demands for fuel oil, UVO/asphalt and to the r o h are as follows: T
T
DA/1FOo, p = Z ( F O o , p,t)
T
V v ; D M R A <_~ ( F R A A t )
Vo, p ; D M U V v = Z
Z ( F Q D q , t) t=l q EVq
t=l
(3a; 3b; 3c)
t=l
(D) Operating Rules. Storage tanks can not load and unload at the same time, with exception of TK-42221. O
XIC,.; + ~ (XID,,o,t) <_l
Vi, t " XQCq, t +XQDe, t < I
(4a; 4b)
Vq, t
o=1
Constraints (4c-4d) impose the a l i g n m e n t for fuel oil, UVO and asphalt unloading. ATDi.o, t - XIDi+l,o, t = 0
i = 1,3,5;
Vo,t"
q = 1,3; Vt
XQDq, t - XQDq+I, t = 0
(4C; 4d)
Due to the a l i g n m e n t , if CAP-20 is unloaded, then XWt=I (0 otherwise), as in (4e). 8
Z (XQDq, t) - 2. X W t = 0
(4e)
Vr
q=6
The dispatching of UVO/ asphalt includes (4t). .YQDq, t <_HTb
b = ] ..... [T/(12/DT)];
(12/DT).(b-1)+1<_t<_(12/DT).b;
Vq
(40
The asphalt and UVO should contain proper RASF diluents, as in (4g; 4h). 8
4
Z ('u
t ) -- XDRAt : 0
Vt
q=5
9 Z (XQCq, t) - X L C O t = 0 q=l
(4g; 4h)
Vt
(E) Material Flow Constraints. The RASF from UDASF can be sent to the r o h or to the mixer, as in (5a). Also, (5b) imposes a lower bound to the RASF flowrate to the mixer: F R A M = FRAA t + FRA U t
Vt ; where FRA
gt ~
ff/~c/min
(5a; 5b)
Vt
The material balance in the mixer is stated in (5c). Pump limitations are imposed by (5d-5e). I
Q
D
Z [glRfl~,t -b Z F'Q)RAq,t =- F R A Lrt + Z FDRAa,t + FOCCRt + FRLCOt l=l q=l d=l 0 <_F i l l , t <_~]()i,t" F m a x
Vi, t ; 0 <__FQRAq, t <_XQCq, t 9F max
Vt
where
(5c) (5d; 5e)
Vq, t
Constraints (5f-5g) state the unloading conditions for TK-42221 and TK-42208, respectively V t " 0 < FDRA2t <_min{1- XDCt,1 - X D R A t } F D D max Vt (5t"; 5g) The OCC production is either stored in TK-42208 or directed to dilute RASF, as in (5h-5i).
0 N FDPt~I, t < X D R A t 9F D D ; max
V t ; F O C C R t = ( 1 - X Z t ). F O C C Vt (5h; 5i) The LCO charges TK-42208 and/or to produce an UVO, as in (Sj). In the latter case, while the UVO is produced, the TK-42208 must be fed with a minimal LCO rate ( F S E C ) , as stated by (Sk). F[)(72, t = X Z t 9F O C C
F P L C O t = FD(73, t + F R L C O t
V t ; FDC3, t >_FSE, C . X L C O t
(Sj; 5k)
Vt
The flowrate in oil-pipelines has lower and upper bounds, as in (51). i'_6
i>6
FOomm( EO. 5"XID,,o, t + E X I D i , o,t)
Vo, p,t"
(51)
(F) Viscosity Constraints. The linearized form preserves the number of 0-1 variables of the MINLP model and relies on disaggregation of the stream variables into two sets. One of them performs material balances and the other considers the viscosity characteristics of the streams. The resulting linear constraints are:
1067 t'
0
I:IKi, t, = VIZ i .MI~ + Y'~(FIRAKi, t - ~;'~FIDKi,o,t ) t=l
V i,t" VI,, t .MI, =VIK~, t
(6a; 6b)
Vi,t
o=I
FIRAK,, t < ( F m ~ M I i ) . X I C , , t
Vi, t" FIDe,o, t .MI, =FIDK,,o, t
Vi, o,t
(6c; 6d)
t'
VQKq,t, = VQZq .MQq + Z(FQRAKq,t -FQDI~,t)
Vq,r " VQq,t .MQq =VQKq, t
Vq, t
(6e; 6 0
t=l
FQRAI%t <(FmaX.MQq).XQCe, t D
FRA Ut .MILd+ Z
Yq, t" FQDq, t.MQq=FQDKq, t
s<_:2
V q, t I
(6g; 6h) Q
Z FDRAa,t "AIDs + FOCCRt "MD2 + FRLCOt "MDs = ZFIRAKi,t + ~ YORAhq,t
d=l seSct
i=l
Vt
(6i)
q=l
4. COMPUTATIONAL RESULTS The modeling system GAMS [8] was used in order to implement the MIP optimization models and their solution methods. The outer-approximation code DICOPT++ [5] solved the nonconvex MINLP model while the solution of the MILP model was obtained with an LP based branch and bound (BB) search performed by OSL [9]. The latter was also used to solve the master problems of DICOPT++ while the GRG method solved its NLP subproblems. Table 2 Main plant data for the real world example (SchH = 3 days; D T - 2 hours) Fuel-oil storaee tanks i=1.2.3.4.5.6 i=7 I UVO/Asvhalt storage tanks o=1.2.3.4.5.6.7.8 ..... Capacity limitation (lO-3.m3) 2-30 2-65 I Capacity limitation (lO-3.m3) 0.5-4 Max. unload, flowrate (m3/h) 167 167 Max. unload, flowrate (m3/h) 83 Diluent storaee tanks d=l d=2 I DiL and R A S F svecific, s=l s=2 s=3 RASF Capacity limitation (lO-3.m3) 2-50 2-50 [ Viscosity range 20 4 35 Max. unload, flowrate (m3/h) 208 208 i Plant production (m3/h) 25 67 67 150 Product sveciFu:ation v=l v=2 v=3 v=4 v=l v=2 v=3 v=4 Viscosity range 14 16 18 20 18 24 30 32 Demand (10-3.m3)* 0.7/3 2/0.8 3.5/0 0/3 1.2 2.2 0.9 2.9 "Relative to the local oil-pipeline (o = 1) and oil-pipeline to S~o Paulo (o=2), respectively.
I
A real world example based on maximum plant capacity (200,000 m3/month) is presented. The main plant data are reported in Table 2. Model dimensions are shown in Fig. 2. Figures 3 and 4 illustrate main results. Four cases of unavailability of the oil pipeline to Sao Paulo is also considered: unavailability between t=O and t=9 (case A), 9St_<18 (case B), 18_q_<27 (case C) and 27_
Table 3 Computational performance (Pentium266 Mhz). Case A B C D
model MILP M1NLP MILP MINLP MILP MINLP MILP MINLP
nodes 937 1296 764 1197 -
iter. 15674 13815 16626 15508 13086 23792 23080 12845
CPU time,(s) 570.46 335.36 711.01 391.45 490.86 531.98 851.78 299,30
Objective 969.61 966.99 965.72 961.14 954.99 956.99 950.65 959,49
9 0-1 variables [] constraints [] continuous variables 4514 2 62 9
6890 ~4465 ~ i i i l - q,
! !
MINLP model
MILP model
Fig. 2. Dimension of the MIPs
]
1068
Fig. 4. Production schedule and storage information. 5. C O N C L U S I O N S The nonconvexities of the original MINLP model can be avoided through rigorous linearization. Although this increases the model size, it has the advantage of providing a rigorous lower bound to the objective. A real-world refinery example was presented and solved. The solution of the nonconvex MINLP can be accomplished with the augmented penalty outer-approximation code DICOPT++ while a LP based BB performed by OSL efficiently solves the MILP model. The computational expense of the MIP models is similar and, in general, does not allow reaching global optimality. Nevertheless, a valuable decision tool to the scheduler is provided. REFERENCES [1] Steinschom, D.; Hofferl, F. (1997) Refinery Scheduling Using Mixed Integer LP and Dynamic Recursion. In: NPRA Computer Conference, New Orleans, LA. [2] Moro, L.F.L.; Zanin, A.C.; Pinto, J.M. Comput. & Chem. Engng., 22 (1998) S1039-S1042.
[31 Magalh~es, M.V.O.; Moro, L.F.L.; Smania, P; Hassimoto, M.K.; Pinto, J.M.; Abadia, G.J. (1998) SIPP - A Solution for Refinery Scheduling. In: NPRA Computer Conference, San Antonio, TX. [41 Joly, M.: Pinto, J.M. (1999) Scheduling Optimization of Fuel Oil and Asphalt Production. In: E('('E 2, Montpellier, France. [5] Viswanathan, J.; Grossmann, I.E. Comp. & Chem. Engng., 14 (1990) 769-782. [6] Moro, L.F.L.; Pinto, J.M (1998) A Mixed-IntegerModel for Short-Term Crude Oil Scheduling. In: AIChEAnnual National Meeting, Miami Beach, FL. [7] Lee H.; Pinto, J.M.; Grossmann, I.E.; Park, S. Ind & Eng. Chem. Res., 35 (1996) 1630-1641. [8] Brooke, A.; Kendrick, D.; Meeraus, A. (1992) G A M S - A User's Guide. The ScientificPress, Redwood City, CA. [9] IBM (1991) OSL - Guide and Reference - release 2. IBM, Kingston, NY. [10] Pinto, J.M.; Joly, M. (1999) MILP Model for Scheduling of Continuous Multiproduct Plant Distribution. In: 15t~ Int. Conference on CAD/CAMRobotics & Factories of the Future, Jkguas de Lind6ia (Brazil).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1069
A Time-Windows Approach for Enhancing the Capabilities of Batch Scheduling Systems: An Application to Simulated Annealing Search Luiz C.Rodrigues a, Mois6s Graells b, Jordi Canton b, Luis Gimeno a, Maria T.Rodrigues c, Antonio Espufiab and Luis Puigjaner b "~School of Electrical and Computer Engineering, Universidade Estadual de CampinasUNICAMP,PO Box 6106,13083-970 Campinas SP, Brasil. [email protected] bChemical Engineering Department, Universitat Polit~cnica de Catalunya, U.P.C. E.T.S.E.I.B., Diagonal 647, E-08028, Barcelona, Spain (graells,canton,lpc,aec)@eq.upc.es CSchool of Chemical Engineering, Universidade Estadual de Campinas-UNICAMP PO Box 6166, 13083-970 Campinas SP,Brasil (rodrigue,maite)@desq.feq.unicamp.br Simulated Annealing (SA) is utilized for a short term scheduling problem where batches have to be scheduled inside processing time windows obtained in a planning phase. It is well known that SA, when used for solving a short-term scheduling problem, can generate a large quantity of infeasible candidates in heavily constrained situations, since it is difficult to filter out infeasible candidates based only in their structure, basically an ordering relation between the tasks. It is shown that tools from Constrained Based Search scheduling techniques can be utilized to implement a filtering procedure for the problem, determining which interchanges lead to feasible situations. 1. INTRODUCTION Simulated Annealing (SA) has shown to be a good technique for solving scheduling problems that can be treated as job sequencing problems. In SA procedure, a job is randomly selected and its position in the sequence is also randomly changed. The new schedule is evaluated, being accepted or not according to Metropolis criterion (Metropolis et al., 1953; Kirkpatrick et al., 1983). SA, as other evolutionary techniques like Genetic Algorithms or Tabu Search, can generate a great number of infeasible candidates if the problem is highly constrained. When sequencing of tasks in equipment units has to fulfill other constraints besides equipment occupation and simple precedence constraints, SA scheduling applications benefit from introducing some type of filtering procedures to check candidate feasibility prior to schedule evaluation. For example in (Graells, 1996) candidates feasibility is tested in front of Finite Intermediate Storage constraints. In this paper the problem considered is short term scheduling, where tasks to be scheduled are determined from final products demand in a short time horizon. Products demand is characterized by specific quantities to be delivered at specific due dates and the scheduling horizon is constrained by a raw materials supply plan. The objective is to schedule tasks inside their time windows minimizing final products' earliness. A planning system determines the quantity of batches of each intermediate product to fulfill final products demand and batches latest finishing times - lft. Batches earliest beginning
1070 times - ebt are determined from raw materials availability plan. In this way the scheduling problem is constrained by mass balance relations defined by products recipe, intermediate products storage conditions and processing time windows (eft- Ift) given by the planning phase. This paper shows how time windows analysis at the planning phase permits the identification of precedence relations between batches, and hence the reduction in the set of possible batches permutations in the SA procedure. This reduction results in a pruning of the scheduling opportunities since only pairs of batches without precedence relationship will be allowed to switch positions. Time windows' analysis is being implemented as a new filtering procedure for the simulation module of a general package (MOPP from Polytechnic University of Catalonia; Graells et al., 1998). This time windows module is projected as a flexible multipurpose procedure to be used either manually or automatically. The module is proposed for two different purposes: 9 as an information provider system allowing the user to analyze different production scenarios, hence permitting easier and deeper insight into the feasible scheduling alternatives 9 as an automatic tool for pruning the number of possible sequences analyzed by the optimization procedures, in this case S A. 2. PROBLEM DESCRIPTION The case study proposed next is from Papageorgiu and Pantelides (1993) with some additional changes. The plant has seven units used to manufacture three final products. Production recipes representation is given in Figure 1. Equipment units assigned for each task, intermediate storage conditions and products demand are given in Tables 1 and 2. All raw materials are delivered at time t = 1. A1 TA1 , ~ 0 - - - ~
4hr FeedA
~
2o
FeedBO
NIS
ZW
2o
2o
so
4o
lhr
FIS(50) ProB
FIS(50) 15
C1 3hr C2 2hr C3 TC1 ] ~ 0 - ~ TC2 ~ TC3 ~ 40 ZW 4o FIS(IO0) 15
2hr
~
2hr ProA TA4 ~ - ~ 0
B1 B2 3hr 2hr ~ITBI~:~--~TB2~)---~TB3 4o
FeedC
lhr A2 3hr A3 TA2 " , ~ : ~ - - ~ TA3 ~
3hr C4 1hr Pro C TC4 ~ - - ~ 0 - - ~ TC5 ~ - ~ 0 is NIS 15
Figure 1. STN representation for the proposed case study 3. PLANNING PHASE A planning phase has been developed to determine the specific time window for each batch. It considers mass balance restrictions and equipment units constraints using the tools developed in the field of Constrained Based Search (Le Pape, 1994, Baptiste and Le Pape, 1995, ILOG, 1997). To take into account specific storage policies often found in process in-
1071 dustries (NIS, ZW, FIS) some additional constraint propagation mechanisms have been introduced. The resulting time windows are represented in Figure 2. Figure 3 shows a measure of equipment units load given through units aggregated demand (Sadeh, 1991). Table 1. Assignment of tasks to equipment units and storage conditions. tasks
equipment
states
storage
tasks
equipment
states
storage
TA1 TA2 TA3 TA 4 TB1 TB2
P1 P2 P7 P2 P3 e P4 P5
A1 A2 A3 B1 B2 C1
NIS UIS FIS (50) ZW FIS (50) ZW
TB3 TC1 TC2 TC3 TC4 TC5
P6 P1 P5 P6 P4 P7
C2 C3 C4
FIS (100) UIS NIS
Table 2. Products demand. produCt ProA ProB ProC
mass 100 270 75
due date 32 32 56
mass 90
Figure 2. Batches processing time windows. Circles denote slots of total reliance (Sadeh, 1991) while dashes denote unavailable time slots
due date 56
Figure 3. Unit's aggregated demand shows tasks competing for the unit
Unit P 1 is responsible for the reduction in time windows of batches involved in P r o C since the first task in its recipe utilizes also unit P 1. Aggregated demand show time intervals of high contention in units P2 and P7 as a result of constraint propagation.
1072
Reduction in processing time windows can give way to forced orderings between batches that must be obeyed by any candidate in the SA procedure in order to become a feasible scheduling solution.
4. FILTERING P R O C E D U R E ON SA CANDIDATES. Orderings obtained from the planning phase can be represented in a graph where all the batches are represented as nodes. Arcs between nodes are directed arcs if the planning phase has detected a specific ordering, and disjunctive arcs when their order is not defined. Directed arcs come from: 1. orderings detected in equipment units by constraint propagation mechanisms, 2. pegging orderings from mass balance considerations, and 3. precedence orderings resulting from imposing time ordering between batches of the same task. Let us call these arcs forced arcs, Figure 4 represents the situation in equipment units P2 and P7.
Figure 4. Forced arcs (dashed) and disjunctive arcs (in gray) in units P2 and P7 A feasible candidate is a graph solution where all the nodes in each equipment unit are linked through directed arcs and no cycle exists. To generate a new candidate two batches are randomly selected and their positions are interchanged. Feasibility of this candidate is analyzed through the following steps: l- For the first batch, all directed arcs linked to it and corresponding to batches in the same equipment unit are eliminated if they do not belong to the set of forced arcs. Eliminated arcs can or cannot become disjunctive arcs. They constitute the set of candidate arcs. 2- All the candidate arcs linked to the selected node are tested for feasibility. If imposing one direction in the candidate gives way to a cycle, it is discarded. The set of remaining (disjunctive) arcs connected to this node show the possible (feasible) interchanges in the equipment unit. If the set is empty go to step 1. 3- A second node is randomly selected from the set of feasible interchanges. 4- Arcs housekeeping: arcs initially directed to (from) the first node are directed to (from) the second node and reciprocally. The procedure is straightforward as far as it suffices to use a cycle detection function working on the entire graph. Suppose for this procedure (Figure 5) that last candidate orderings in unit P7 are those shown in Figure 5a and that batch TA3/3 has been randomly selected. Orderings TC5/1 < TA3/3 and TA3/3 < TC5/2 are eliminated since they are not forced
1073 arcs (Figure 2). Possible disjunctive arcs are represented in gray in Figure 5b. Arc TA3/3 <>TC5/4 is discarded (Figure 5c) as it originates a cycle in case TC5/4 < TA3/3 (TA3/3, TA3/4, TC5/4). If batch T C 5 / 4 is selected, only disjunctive arc T C 5 / 4 <> T A 3 / 4 is retained (Figure 6) since ordering T C 5 / 4 < T A 3 / 3 for the other possible disjunctive arc would lead to a cycle (TA3/3, TC5/2, TC5/3, TC5/4)
Figure 6. Disjunctive arcs with selection T C 5 / 4 . To illustrate filtering performance the whole situation represented in Figure 2 is considered now. The first node (task,batch) randomly selected is restricted to the set of nodes with disjunctive arcs connecting to other nodes in the same equipment unit. Orderings detected in the planning phase restrict this first choice to 29 nodes out of 85 total nodes. Table 3 contains its distribution in equipment units. When a node of Table 3 is randomly selected in the initial solution given by Table 3, the algorithm described in Section 4 gives the feasible interchanges presented in Table 4 together with the total amount of possible interchanges if no filtering were used. In fact this last figure would be larger since first node selection would not be restricted to 29 nodes. Table 3. Remaining disjunctive arcs after planning phase and initial solution unit
nodeswith disjunctive arcs
quantity
unit
ordering in equipment unit
P1
TA1/6 - 9; TC1/1,2
6
P1
TA1/1-6; TC1/1, 2; TA1/7- 10
P2
TA2/5,6,9,10; TA4/2,3,6,7
8
P2
P3 P4 P5 P6 P7
-
0 0 1 6 7
P3 P4 P5 P6 P7
T A 2 / 1 - 4 ; TA4/1; TA2/5; TA4/2; TA2/6; TA4/3-5; T A 2 / 7 - 9 ; TA4/6; TA2/IO; TA4/7- 10 TBI/1 - 7 TB1/1 - 7; TC4/1 -5 T B 2 / 1 - 7; TC2/I,2 TB3/1 - 18; TC3/I - 5 TA3/1,2; TC5/I; TA3/3; TC5/2,3; TA3/4; TC5/4,5
TB2/7; TC2/1 T B 3 / 1 5 - 1 8 ; TC3/1,2 TA3/3,4; TC5/I-5
1074
5. C O N C L U S I O N S . A filtering procedure has been developed to reduce the amount of candidates generated in Simulated Annealing in heavily constrained situations in terms of equipment units' capacity. A planning phase is used to determine forced orderings between batches and a graph procedure is utilized to eliminate infeasible candidates in the interchange phase. This work shows that a convenient handling of the information available at a planning phase may be of great help to the user and that the use of time-windows may lead to a significant reduction in the complexity of the problem. Table 4. Feasible interchanges. first node TAll6 TA I /7 TA1/8 TAI/9 TC1/1 TC1/2 TA2/5 TA2/6 TA2/9 TA2/10 TA 4/2 TA 4/3 TA 4/6 TA 4/ 7 TB2/7
feasible interchanges TCI/1 TC1/1,2 -
TA1/6 TA1/7 - 9 TA 4/2 TA 4/3 TA 4/6 TA 4/6, 7 TA2/5 TA 2/6 TA2/9,10 TA 2/I O TC2/1
N of feasible N ofpossible first node interchanges interchanges 2 TC2/I TB3/15 2 2 2 TB 3/16 2 TB3/17 TB3/18 1 10 TC3/1 3 10 TC3/2 1 10 TA3/3 1 10 TA3/4 1 10 TC5/1 2 10 TC5/2 1 10 TC5/3 1 10 TC5/4 2 10 TC5/5 1 10 1 2
feasible interchanges TB2/7
N of feasible N ofpossible interchanges interchanges 1 7
2
TC3/1,2 TB3/15-18
4
TC5/I - 3 Tc5/3- 5 TA 3/3 TA 3/3 TA3/4 TA 3/4
3 3 1 1 1 1
5 5 5 5 18 18 5 5 4 4 4 4 4
REFERENCES. Baptiste P., Le Pape C. (1995). A Theoretical and Experimental Comparison of Constraint Propagation Techniques for Disjunctive Scheduling, Proceedings 14 'h International Joint Conference on Artificial Intelligence. Graells M., Espufia A. anti Puigjaner L. (1996). Evolutionary Identification of Best Schedules for Optimum Production Planning. Second International Conference on Computer Integrated Manufacturing in the Process Industries, Eindhoven, Holland.
Graells,M.,J. Cant6n,B. Pesehaud and L.Puigjaner (1998). General approach and tool for the Scheduling of complex production systems. Computer chem. Engng. Vol.22, Suppl.,pp $395-$402 ILOG Scheduler 4.0 (1997) User's Manual, ILOG. Kirkpatrick, C.D. Gelatt, P.M. Vecchi (1983). Optimization by Simulated Annealing. Science. 220:671-180. Le Pape C. (1994). Implementation of Resource Constraints in ILOG SCHEDULE: A Library for the Development of Constrained Based Scheduling Systems, Intelligent Systems Engineering, Vol3 No. 2, pp. 55-66. Metropolis, W., A. Roenbluth, M. Rosenbluth, A. Teller, E. Teller (1953) Equation of the state calculations by fast computing machines, Journal of Chemical Physics, 21:1087-1092. Papageorgiu L.G. and Pantelides C.C. (1993) A Hierarchical Approach for Campaign Planning of Multipurpose Batch Plants, ESCAPE 2, Suppl. to Computers chem. Engng, Vol 17, pp. 27-32. Sadeh N.(1991) Look-Ahead Techniques for Micro-Opportunistic Job Shop Scheduling, PhD Thesis, CMU-CS91-102, School of Computer Science, Carnegie Mellon University.
ACKNOWLEDGMENTS. This work was partially supported by Funda~o de Amparo ~ Pesquisa do Estado de S~o Paulo (Brasil)
European Symposiumon ComputerAidedProcess Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
1075
A Hierarchical Approach to Real-Time Scheduling of a Multiproduct Batch Plant with Uncertainties Guido Sand a, Sebastian Engell a, Christian Schulz b, Andreas M~irkert c and Rtidiger Schultz c aprocess Control Laboratory, Department of Chemical Engineering, University of Dortmund, 44221 Dortmund, Germany, {g.sand I s.engell} @ct.uni-dortmund.de bprocess Systems Enterprise Ltd., 107a Hammersmith Bridge Road, London, W6 9DA, United Kingdom, c. schulz @ psenterprise.com CDepartment of Mathematics, Gerhard-Mercator-University of Duisburg, 47048 Duisburg, Germany, schultz @math.uni-duisburg.de In this contribution, a two level hierarchical approach to real-time scheduling of a real world polymer production plant is presented. On both levels optimisation problems are formulated as mathematical programs and solved by non-standard algorithms. The focus of this paper is on the upper level planning problem where uncertainties are modelled explicitly. The key features of the model and of the solution algorithm are explained and numerical results are presented. 1.
INTRODUCTION In the processing industries, multiproduct batch plants, which usually consist of a number of units which are grouped in stages, are used for the production of products with similar recipe structures, e.g. modifications of one type of polymer. Scheduling deals with assigning resources, in particular processing units or material, to tasks over time to fulfd certain production goals. The flexibility of these plants offers the possibility to react quickly to changes of the market with reasonable costs. For this reason computer based scheduling tools are necessary to fully exploit the flexibility of the plant despite its complexity. Although mathematical programming techniques provide methods to solve scheduling problems systematically, in a large part of the corresponding work the aspect of uncertainty is neglected. If it is addressed, this is usually either done by modification of nominal schedules, which were computed based on deterministic data depending on the actual situation, or the generation of robust schedules which are insensitive to a priori supposed uncertainties [ 1-3]. In our contribution, a real time scheduling algorithm for a real world problem is presented, which focuses on the determination of the next decisions within a rolling horizon and reflects
This research was funded by the Deutsche Forschungsgemeinschaft under grants EN 152/7 and SCHU 1029/3 which is gratefully acknowledged.
1076 uncertainties in a stochastic fashion. A two level hierarchical model is formulated, which combines a stochastic long term planning model with a deterministic short term scheduling model; both mathematical programs are solved by non standard algorithms. After a description of the benchmark process, the basic solution concept is introduced. In the sequel, the focus is on modelling and solving the upper level planning problem, which contains a stochastic description of the uncertainties. Finally, numerical results and the intended future work are discussed. 2.
BENCHMARK PROCESS In this work the production of expandable polystyrene (EPS) is used as a benchmark process. In addition to common properties of recipe-driven multiproduct batch processes (e.g. limited capacity of equipment items, shared and non-shared intermediates, different types of storage policies) there are some features which lead to special difficulties. In the plant shown schematically in fig. 1 two EPS-types (A and B) with five grain size fractions each are produced from several raw materials (E). The recipe-driven plant consists of the main stages "preparation", "polymerisation" and "finishing". The preparation stage provides three types of intermediates for the EPS-producing polymerisation stage, which is, as wll as the preparation stage operated in batch mode. After a polymerisation is terminated the batch is transferred to one of two mixing tanks, which are exclusively assigned to one EPS-type each. The mixing tanks serve as buffers and provide the separation units with continuous feeds. As long as a finishing line is running, both the mixer content and the feed must remain within upper and lower bounds (the mixer is "on duty"). If the mixer content is falling below the lower bound, an expensive shut down procedure has to be executed for the connected separation unit (mixer "off duty"). The separation unit is essential for the process, since for each polymerisation batch of the desired EPS-type one main grain size fraction can be chosen by selecting one out of five recipes, but a significant byproduction of the other four fractions is inevitable. The production goals are to fulfil all customer orders (grain size fractions) with minimum delay, to produce minimal amounts of undesired gram 9 0 A2 size fractions, and to start up ~A5 and shut down the finishing A4 lines rarely. A5 The scheduling problem is influenced by considerable uncertainties: Market requirei ~>B1 B2 ments are partially unknown . B3 (demands vary), the chemical B4 reactions are not completely ~ri 1 B5 reproducible, the processing Preparation I" times vary and breakdowns of Fig. 1. Polystyrene Production single units are possible.
E -I%
0000
mo.
1077 3.
SOLUTION CONCEPT The difficulty of solving real world scheduling problems arises mostly from the combinatorial complexity resulting from the discrete decisions. An additional difficulty of the process discussed here is the following non-convex non-linear equation which describes the mixing process:
(1)
Cf/C - Ff / F with C = Z Cf and F = ~, Ff f where
Cf
f
denotes the mass of grain size fraction f
inside the mixer and
Ff
the feed of
fraction f . Due to the strong coupling of the scheduling decisions and the uncertainties, a high scheduling performance requires the consideration of a horizon of several weeks. On the other hand, the complex constraints necessitate a resolution of at most one hour to guarantee the feasibility of the schedule. Therefore, the scheduling task is a large, stochastic, mixed-integer and non-linear real-time optimisation problem. Real world scheduling tasks belong to the class of NP-hard problems, and there are no known solution algorithms of polynomial complexity in the problems size [3]. Consequently there is a need for approximative strategies, e.g. the formulation and solution of simplified substitute problems. In the proposed approach, the overall problem is structured hierarchically into a long-term, stochastic, linear planning problem and a short-term, deterministic, non-linear scheduling problem. The planner and the scheduler are elements of a feedback structure where information about changes of the demand profile and stochastic events in the process is available online. In the planning algorithm, a simplified process model for a horizon of two to four weeks is used (see following paragraphs). The algorithm generates scheduling guidelines for a horizon of four to eight days. Each optimisation run is based on deterministic data of the actual process state and the demand profile as well as probability distributions of the uncertain parameters. It is started each evening and should be finished after ten hours computing time (over night). Consequently, in a moving horizon a set of guidelines is valid for the next 24 hours only. The scheduler is based on a detailed deterministic process model, which comprises all scheduling decisions and the above non-linear model of the mixing process. If new information about the demands or the process becomes available it re-schedules the process with respect to the guidelines. In the moving horizon approach only those scheduled decisions are actually implemented on the plant that can be realised before the next stochastic event occurs. For short-term scheduling a MINLP-model with a tailored solution algorithm is used, which was developed in previous work [4]. The model is based on a continuous time representation and contains the non-linear mixer equation (1). It is solved by a depth first search algorithm, which uses the commercial solvers CONOPT 2 [5] and CPLEX 6.0 [6] to solve NLP- and MILPsubproblems, respectively. For a scheduling horizon of e.g. 6 days, near optimal feasible solutions can be found in less than one minute CPU-time, which satisfies the real time requirements.
1078
4.
SINGLE SCENARIO PLANNING PROBLEM The planning problem is formulated as a two-stage stochastic program which includes discrete scenarios of the evolution of the uncertain parameters. The model is of the MILP-type and derived from a formulation for one scenario, which can be regarded as a quasideterministic base model. In order to be able to consider a planning horizon of some weeks and to meet the computing time limitations at the same time, the degree of accuracy of the model has to be reduced relative to the scheduling model. 4.1. Modelling Approach To limit the size of the model a problem specific approach is used, which exploits specific process properties. It can be characterised by the following three key features: 1. The model highlights decisions with long-term effects. These are the discrete mixer states and the choices of the recipes for the polymerisations. 2. The model is based on an aggregated representation with i max aggregation intervals t agg of equal durations. Consequently, similar decisions in one interval are grouped and modelled by a single variable. The state of mixer p in interval i is described by Zi,p ~ {0;1}, the number of recipes r started in i is represented by N i , p , r E I N .
3. The non-linear mixing effects are approximated linearly by constant delays for each input batch. This approximation causes only small errors and does not affect the feasibility of the schedules.
4.2. Process Constraints The constraints can be formulated in an aggregated fashion, which guarantees the generation of feasible plans without losing degrees of freedom. Key elements of the model are the capacity restriction of the polymerisation stage and the restrictions of the mixer contents. The number of polymerisation starts in one aggregation interval is constrained not only by the number of available reactors but also by the number of polymerisations that are still running at the beginning of the interval. It should be noted, that the sequence of intervals with active upper bounds is not known a priori. The capacity restrictions are modelled by constraining the number of polymerisation starts for all intervals i to k by the a priori computed parameters Nimkax" k
}-" }-" N j, p,r < Ni,mkax j=ip,r
Vi,k lk > i
(2)
The properties of the process ensure that the restrictions of the content of a mixer p can be satisfied during an interval i, if they are satisfied at the boundaries of the interval. The mixer contents at the beginning of the planning horizon are regarded as known, and the feasible contents at the interval boundaries then follow from the mass balances around the mixers. The
1079 mixer input is modelled by the variables
Ni,p,r, and the feeds F are restricted by formulating
the mass balances as inequalities with lower and upper bounds Ci,p at the end of interval i is constrained by
F rain and F max . The content
Ci-l,p + Z Ni,p,r - Zi,p 9Fmin tagg > Ci,p >- Ci-l,p + Z Ni,p ,r - Zi,p 9Fmax tagg F r " ~i=1
-
" ~i=1
Vi,p
(3)
4.3. Production Goals The production goals "meet the due dates" and "avoid unnecessary by-production", are often reflected by penalising the deviations between the production and the demand profdes only at certain times, typically at the due dates. In addition to this objective function, the planning model contains alternative formulations which reflect the production goals more realistically and more intuitively, as the following example shows. If the due date of a demand b for fraction f of EPS-type p cannot be met, it is not the
primary goal to minimise the product shortage at the given due date but to produce the full demanded amount Bb, p, f with minimal delay. The core of the formulation of this scheduling goal is the following mass balance around the product storage:
M i,p, f = Mi_l,p,f + Z (tgf,r "Ni,p,r)- Z (Ub,i "Bb,p,f) r b
Vi, p, f
(4)
The variables M ~ IR + denote the storage contents; the storage inputs are calculated as the product of the number of performed recipes of class Ni,p, r and the relative amount Pf,r of fraction f variables
in a polymerisation batch, which is produced according to recipe r. The binary
Ub,i indicate the interval i in which the demand b can be covered. The delays are
minimised by the minimisation of a weighted sum of all u's. 5.
TWO STAGE STOCHASTIC PROGRAM For real-time scheduling under uncertainty a two-stage stochastic program is a very promising modelling approach [7]. In the first stage, the "here and now"-decisions have to be made based upon the knowledge of the probability distribution of the future stochastic events. The decisions of the second stage, in which the uncertainties are regarded as realised, are a recourse for the effects of the first stage decisions as a function of the realisations. The probability distributions of the uncertain parameters are modelled by discrete scenarios co with fixed probabilities PoJ- For linear models and cost functions a two-stage stochastic program
can be formulated as a large MILP:
min{ cx+~-'~(P~176176176176176176
X, y o ) ~ r v c o }
(5)
1080 where c, q, x and y are vectors of coefficients and unknowns, A, T , W, b and h are given matrices or vectors, and X and Y denote polyhedral sets comprising integrality requirements. Up to now only uncertain demands are considered, so the stochastic 14 2 325 86 331 program emerges from the base model mainly by 14 3,5 171 48 189 defining the variables as co-dependent and co28 2 735 170 816 independent ones. Typical sizes of single Tab. 1. Size of single scenario problems scenario problems are shown in table 1; by the stochastic extension the problem size grows approximately proportionally to the number of scenarios. The two-stage stochastic program is solved by a dual decomposition algorithm based on Lagrangian relaxation [8]. The emerging subproblems and Lagrangian dual problems are solved by CPLEX [6] or NOA [9], respectively. The numerical performance of the stochastic programs mainly depends on of the formulation of the objective function and the choice of the first-stage variables. Good results are achieved if the production goals are modelled by minimising a weighted sum of product shortages, the number of polymerisations and the number of mixer state changes, and if the first stage is restricted to the mixer states. Stochastic planning models with e.g. a horizon of two weeks, aggregation intervals of two days, 20 scenarios and a first stage of three intervals were solved optimally within ten minutes CPUtime on a SUN Ultra 2. The numerical performance becomes poorer for formulations including the explicit computation of delays or if the number of recipes of each class are defined as firststage variables. 6.
FUTURE W O R K The current work deals with solving the more critical instances efficiently. Furthermore, the model will be extended by uncertainties of the reaction times and yields and the plant capacity. REFERENCES 1. D.J. Mignon, S.J. Honkomp and G.V. Reklaitis, Comp. Chem. Engg., S19 (1995) 615. 2. K.B. Kanakamedala, G.V. Reklaitis and V. Venkatasubramanian, Ind. Engg. Chem. Res., 33 (1994) 77. 3. N. Shah, Proc. FOCAPO98, Snowbird, USA (1998) 75. 4. C., Schulz, R. Rudolf and S. Engell, Proc. FOCAPO98, Snowbird, USA (1998) 224. 5. A. Drud, ORSA J. Computing 6 (1994) 206. 6. ILOG CPLEX Division, Using the CPLEX Callable Library (1999). 7. C.C. CarCe, M.P. Nowak, W. R6misch and R. Schultz, Proc. 13th Power Systems Computation Conference, Trondheim, Norway (1999) 1086. 8. C.C. CarCe and R. Schultz, Oper. Res. Lett., 24 (1999) 37. 9. K.C. Kiwiel, User's Guide for NOA 3.0, Warsaw, Poland, 1994.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1081
Closed-Loop Implementation of Optimal Operating Policies in Batch Distillation Massimiliano Barolo* and Paolo Dal Cengio Istituto di Impianti Chimici, Universit/t di Padova, via Marzolo 9 I-3 5131 Padova PD (Italy) A strategy for closed-loop optimization of batch distillation columns is proposed. The optimal reflux rate is calculated off-line for several feed compositions, and is correlated to the composition profile that is found in the column at the end of the startup phase. Since this composition profile can be estimated on-line by a state observer, it is possible to determine on-line the closed-loop optimal reflux ratio to be used with any feed of unknown composition. Results for binary and ternary systems indicate that, by using this procedure, the column performance can be improvedby as much as 30% with respect to a conventional open-loop optimal strategy. 1. INTRODUCTION The use of batch distillation as a separation process has become increasingly popular in the fine chemistry, pharmaceutical, biochemical, and food industries. While the capital investment (i.e., fixed costs) needed for building a batch column is lower than that requested for a train of continuous columns, the operating costs of batch distillation are higher, since this unsteady operation usually consumes a large amount of energy. Optimizing the column operating policy is the way to maximize profits. This amounts to determining the value of the reflux ratio (or sequence of reflux ratios) that, for the nominal feed composition, results in maximization of a prescribed profit function. In the present work, we refer to the simplest (and yet most frequently used) policy, which is based on the use of a single, optimal value of the reflux ratio during both the main-cut production phase and the off-cut removal phase (Muhrer and Luyben, 1992). Solving the dynamic optimization problem on-line (Bosley and Edgar, 1993) is computationally expensive, even with the current availability of computing hardware. On the other hand, when the optimal operating policy is determined off-line to reduce the demand for on-line computing facilities, an issue arises on how to implement this "open-loop optimal" policy in a closed-loop fashion (Edgar, 1996). In fact, the actual composition of the feed charge during one batch is usually different from the nominal value, and might not even be known, because it results from the mixing of fresh feed with off-cuts of unknown composition. Thus, implementation of an open-loop optimal operating policy may result in a significant loss of performance, since the sensitivity of batch distillation to perturbations in the feed composition is large (Barolo and Botteon, 1998). The purpose of this paper is to present a simple, yet effective, method for implementing an open-loop optimal operating policy in a closed-loop fashion. The main objective is to have the time-consuming step (i.e. the dynamic optimization of the column performance index) performed off-line once and for all, while letting only "simple" calculations be performed online. Thus, the need for dedicated on-line computing hardware can be significantly reduced. Author to whom all correspondence should be addressed. Email: max@po i ochi. cheg. unipd, i t
1082 2. COLUMN MODEL AND OPERATING PROCEDURE Following Quintero-Marmol et al. (1991), a fairly simple model of a batch rectifier is employed; details about the model equations and parameters are reported by Dal Cengio (1999). The operating procedure is the same described by Luyben (1988). The column is started up at total reflux until the liquid in the reflux drum reaches the specification for the lightest fraction. Then, the removal of the distillate product is started at a constant reflux rate (whose value is to be determined by optimization), and the withdrawal of main cuts and slop cuts from the top of the column proceeds sequentially using a single value of the distillate rate. The operation is stopped when the composition of the liquid in the bottom, plus all of the liquid draining from the column trays, meets the specification for the heaviest product. If a slop cut is not being removed from the top when the heavy product specification is met, a check is made on the composition of the liquid contained in the reflux drum: if mixing of this liquid with the cut being collected from the top of the column (]-th cut) results in an onspecification cut, then the reflux drum content is actually added to the j-th cut. The performance index to be maximized is the column capacity factor (CAP; Luyben, 1988), defined as CAP = ~.,~ P~/(td~st + t~tch), where N f i s the total number of fractions recovered on specification, P~ is the amount of the i-th fraction recovered on specification, t aist is the distillation time, and t~tch = 0.5 h is the switching time needed for charging and discharging the column. 3. BINARY MIXTURES The separation of a non-ideal ethanol/water mixture is considered. The proposed procedure for closed-loop optimization comprises the following three steps. As a first step, the optimal reflux rate is determined off-line for several different compositions of the feed. Then (second step), a correlation between the optimal reflux rates and the evolution of the composition profile in the column during the startup phase is derived from off-line analysis of the dynamic process data. Finally (third step), detection of the composition profile in the column during the startup phase is performed on-line, thus allowing implementation of the closed-loop optimal reflux ratio. The results presented in this section refer to a nominal feed composition xF = 0.4 (ethanol mole fraction), while the specifications for the light and the heavy products (P~ and P2, respectively) are Xp] = 0.84 and Xp~ = 0.99, where Xp~ is the requested mole fraction of the i-th (i.e., dominant) component in the i-th product. 3.1. Step 1: open-loop optimization For several compositions of the feed, the value R ~ of the reflux rate that maximizes CAP can be found by open-loop optimization under the hypothesis that the product and tray compositions are known exactly at any time. The optimization results are illustrated in Figure 1. It can be seen that the optimal reflux rate decreases roughly linearly with the feed composition for xp < 0.37 and for xp > 0.4 However, /?open changes dramatically for feed compositions within the range [0.37; 0.40]. The "jump" of the optimal reflux rate is related to a shift in the location of the CAP maximum, as is illustrated in Figure 2. For relatively diluted feeds (x R < 0.37), it is convenient to operate the column at low reflux rates, so that the distillation time is quite short, and a small amount of distillate product (P~) is obtained. However, for lighter feeds, it is better to increase the reflux rate, in such a way as to obtain a larger amount of product, with less slop cut, at the expense of a larger distillation time. ~" " o p t
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3.2. Step 2: correlation development In this step, a way is sought to correlate the optimal reflux rate to a parameter that can be measured (or estimated) on-line while starting up the column. Figure 3 shows that both the and reboiler composition and the reboiler temperature at the end of the startup phase (x_ena B T ~ "d, respectively) are biunivocally related to the feed composition, and therefore to the optimal reflux rate. Therefore, since a relationship between x F and ~"R"opt ~ is available (Figure 1), --opt R ~ . can . be easily . . . correlated either to x B d or to Tend -8 9For this binary system, a simple piecewise-linear correlation between -Te"d and --opt R ~ B was derived. 3.3. Step 3: closed-loop optimization The separation of several feeds of unknown composition is considered in this step. For each feed charge, the closed-loop optimization proceeds as follows. The batch column is started up as described previously, and at the end of the startup phase the reboiler temperature is detected. By using the correlation developed in the previous subsection, the optimal reflux rate is determined on-line, and the removal of the distillate product is started at the relevant distillate rate; then, the operation proceeds and terminates as usual. The results obtained with this strategy are compared to those coming from a conventional open-loop optimal operation, where the reflux rate determined off-line for the nominal feed composition (xp = 0.4 ) is always employed, whatever the actual feed composition. In order to compare the performance of the two procedures for a certain value of x F , the following performance index was used A = 100 • (CAP - CAPm~) / CAPm~ , where CAPm~ is the maximum achievable value of CAP, that is the value of CAP that would be obtained if the "truly" optimal reflux ratio for the current value of the feed composition were employed with perfect composition estimations. It should be noted that, whatever the strategy employed, it is required to know on-line the distillate and bottom product compositions, since these compositions are necessary to detect the end of the startup phase, as well as to stop the accumulation of product P~ and the whole batch itself. In an actual plant, composition measurements are usually provided by gas chromatographs. However, these devices are known to suffer for high investment and maintenance costs. Moreover, they provide delayed responses, which can have a detrimental effect on the performance of the control system. Therefore, in this work it was assumed that
1084 the product compositions can not be made available from direct measurements, but they need to be estimated on-line from available "secondary" measurements. To this purpose, an extended Luenberger observer that uses two tray temperature measurements from the "plant" was employed to reconstruct on-line the unavailable composition measurements (see Barolo and Berto (1998) for details about the observer). The observer was initialized with the nominal feed composition. ,
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Figure 4 shows that the improvement obtained with the proposed procedure is indeed remarkable when diluted feeds (x r < 0.4 ) are to be separated. Using a single, suboptimal value of the reflux rate for any composition of the feed may result in as much as 26% reduction in CAP. Instead, virtually no loss of performance is exhibited for diluted feeds when the optimal reflux rate is determined on-line according to the proposed procedure. For xe> 0.4, it was found that getting very good estimates of the product compositions becomes harder. This leads to evaluate incorrectly the time at which the withdrawal of product P~ should be stopped, therefore diminishing the value of CAP with respect to the maximum achievable. This is why the proposed procedure settles on A ___--4% for xR> 0.4, with no remarkable improvements over the open-loop optimal operation. Improved performance could be obtained if a more reliable composition estimation technique were employed. 4. TERNARY MIXTURES In ternary separations, at most three products and two slop cuts are obtained. The separation of an ideal mixture with relative volatilities c~i = 4 / 2 / 1 is considered here. The nominal feed composition is zR,i = 0.3333/0.3333/0.3334, while the product specifications are x~ = 0.95 for each product. Three tray temperature measurements were fed to the observer for estimating on-line the product compositions; as in the binary case, the observer was initialized with the nominal feed composition on all trays. Due to lack of space, only a short outline of the procedure devised for ternary mixtures will be illustrated. 4.1. Step 1: o p e n - l o o p optimization
For ternary separations, two "product regions" can be located in a (CAP vs. reflux rate) plot for any given feed composition. In fact, depending on the value of the reflux rate, either two
1085 (P~ and P3 ) or three products can be obtained at the end of the batch. A maximum of CAP can be achieved within each product region. For the nominal feed composition, the absolute maximum of CAP lies inside the three-product region, but for feeds leaner in the intermediate component, the absolute maximum may well be located into the two-product region. It can also be shown that the values of R ~ in the two-product region are significantly different (lower) than those in the three-product region. ~" " o p t
4.2. Step 2: correlation development From open-loop analysis, it was verified that the mole fraction xB, i -e,d of component i in the reboiler at the end of the startup phase linearly correlates to the mole fraction xp,i of the same component in the feed. This allows mapping each feed composition point (F-point) in a triangular composition diagram into one (and only one) bottom composition point (B-point, evaluated at the end of startup) in the same diagram, and to correlate each B-point to the relevant value of R ~ At the end of the startup phase, the reboiler composition can be estimated through the observer; we shall indicate this composition with xB, i'e"d and the corresponding point in the triangular composition diagram with B. The "distance" (1-norm) between B and each of the B-points can be easily calculated on-line, and the inverse of this distance can be used as a weight for the calculation of the closed-loop optimal reflux ratio RCtOSed Note that it is important to ensure that only the distance of t3 from those B-points that opt lie in the correct product region (i.e., either two-product or three-product) is included in the calculation. In order to preliminarily estimate whether a certain feed should be separated into two or three products, a a correlation was developed off-line to determine the limiting value x B,2 ~n of the mole fraction of component 2 in the reboiler at the end of the startup phase for which only two products should be recovered. ~, L o p t
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4.3. Step 3: closed-loop optimization At the end of the startup phase, it is checked whether two or three products should be recovered from the current batch. All the B-points lying on the "wrong" product region are discarded, and the remaining B-points are used to calculate R ct~ Several feed mixtures of unknown composition were used to test the proposed procedure. Four regions in the feed composition space were considered, which correspond to feeds with composition close to the nominal one, and feeds rich in each of the three components. Results for feeds with an actual composition close to the nominal one (runs 1 through 7) are reported in Figure 5a, while Figure 5b shows the results for feeds rich in the heavy component (runs 8+11), intermediate component (runs 12+15), and light component (runs 16+19). The conventional open-loop strategy brings about a reduction of CAP ranging between --4 and --8% for most feeds, but reaches 30% for feeds rich in the light component. Conversely, no significant loss of performance is obtained with the proposed closed-loop strategy. For some runs, the proposed procedure seems to lead to a value of CAP even larger than the maximum. This is due to the fact that, due to small initial errors in the composition estimation, the end of the startup phase is slightly anticipated with respect to the relevant reference case. This leads to a reduction of the total distillation time, and eventually to an improvement of CAP. However, this is a minor effect that can be neglected for the purpose of this investigation. ~" L o p t
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(a) (b) Fig. 5. Performance comparison between open-loop and closed-loop optimization for several feeds of unknown composition (ternary system) 5. CONCLUSIONS A three-step strategy for closed-loop optimization of batch distillation columns operated at constant reflux ratio has been proposed. The first and second steps are performed off-line once and for all, while the third step is performed on-line during each batch. Results for binary and ternary systems have indicated that the improvements with respect to conventional open-loop optimal strategies may be as high as 30%. The proposed procedure does not require knowing in advance the actual composition of the feed that is going to be processed. Therefore, a self-optimizing control of the operation can be obtained, provided that the compositions in the column can be estimated during the batch; in the examples presented, these composition estimations have been obtained through an extended Luenberger observer. One further advantage of the proposed procedure is that, since it performs off-line the timeconsuming step (i.e. the dynamic optimization of the column performance), the need for computing hardware and software to be dedicated to the control of the operation can be significantly reduced. ACKNOWLEDGEMENTS This work was carried out in the framework of the MURST Project "Ottimizzazione Dinamica e Controllo dei Processi Chimici" (Cofinanziamento 1998). REFERENCES
Barolo, M. and F. Berto (1998), Ind. Eng. Chem. Res., 37, 4689-4698. Barolo, M. and F. Botteon (1998), Chem. Eng. Sci., 53, 1819-1834. Bosley, J. R. and T. F. Edgar (1993). In: DYCORD+ '92, (J. C. Balchen et. al, Eds.) Pergamon Press, Amsterdam (The Netherlands), pp. 303-308. Dal Cengio, P. (1999), Ma.Sc. Thesis, Istituto di Impianti Chimici, Univ. of Padova (Italy). Edgar, T. F. (1996), J. Proc. Contr., 6, 99-110. Luyben, W. L. (1988), Ind. Eng. Chem. Res., 27, 642-647. Muhrer, C. A. and W. L. Luyben (1992). In: Practical Distillation Control (W. L. Luyben, Ed.), Van Nostrand Reinhold, New York (U.S.A.), pp.508-528. Quintero-Marmol, E., W. L. Luyben and C. Georgakis (1991), Ind. Eng. Chem. Res., 30, 1870-1880.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScience B.V. All rights reserved.
1087
Development of an efficient system for scheduling large-scale industrial process and consumer goods factories with multipurpose and flexible storage Christoforos Charalambous a, Turaj Tahmassebi b and Khalil S Hindi a a Department of Systems Engineering, Brunel University, Uxbridge, UB8 3PH, UK. e-mail : [email protected] b Unilever Research, Port Sunlight Lab, Quarry Road east, Bebington Wirral, Mersey Side L63 3JW, UK. e-mail: [email protected] 1 INTRODUCTION This paper is concerned with the short-term scheduling of multistage, multipurpose manufacturing systems found in the process industry. Such systems produce a large number of low-volume, high-value products such as pharmaceuticals, detergents and food products. Demand for such products is likely to be irregular and sensitive to market changes. Therefore, the ability to develop efficient short-term schedules for these systems is of paramount economic importance. Process industry plants comprise several stages that consist of parallel (possibly identical) processing units. Each stage is dedicated to a specific type of operation (e.g. producing, postdosing, packing) and each unit in a stage can perform that operation on given materials according to a utilisation profile (mode of operation, production rate, etc.). For a product to be produced, it must pass through at least some (usually all) stages. In addition, storage stages may be incorporated in the system for the intermediate buffering of intermediate and finished materials, enhancing plant flexibility. Different connections are defined between adjacent stages for material transfer, thus forming a system network. A set of structural and operational constraints may also be present. Given a portfolio of product demands, the aim is to generate an optimal (or near optimal), factory-wide schedule with respect to an objective criterion such as makespan or cost. Due to inherent complexity, there has been relatively limited research on the general version of the problem [7]. The majority of previous work concentrates on the development of Mathematical Programming (MP) formulations for specific problem instances. Kondili et al made a significant contribution to the MP approach with the definition of the State-Task Network (STN) [4] for the process representation. Representing materials with state nodes and processing operations with task nodes, STN achieves an unambiguous framework for defining product recipes in the process industry context. Based on the STN framework, several formulation improvements have been proposed by various authors using discretised [8, 9] or continuous [6, 3] time representations. However, to maintain a tractable formulation size, most work tends to focus on specialised problem instances with relaxed system * The authors would like to thank Unilever Research for the provision of case studies, as well as for partially funding this research.
1088 constraints. On the other hand, when the problem is considered under a sufficiently general setting (for example, see [10]), practical applicability is limited and only theoretical conclusions can be drawn. Surprisingly, attempts to tackle the problem using heuristic based approaches have been scarce, even though several researchers advocate that this line of attack is more promising for hard scheduling problems [2, 5]. A notable example has been presented by Deal et al [2], who start from an initial schedule, then improve it using an exchange heuristic. However, the study again considers a simple, specialised problem instance. In general, schedulers for process industry systems have been either too specialised or prohibitively slow. In fact, as far as the authors are aware, theirs is the only work that has so far succeeded in tackling the general process industry short term scheduling problem, leading to a tool that can be readily used in an industrial setting. The approach adopted here is two-level. At the first level, a means of developing system schedules is provided based on a generic, object-oriented modelling paradigm and an efficient schedule generation function. This function is used by the second level of the proposed approach, which is responsible for the examination of the solution space and the identification of high-quality schedules. 2 SYSTEM M O D E L L I N G The purpose of the developed model is to enable the encapsulation of the majority of manufacturing systems found in the process industry in a concise and unambiguous manner. The overwhelming complexity of the systems considered prohibits the use of standardised models (such as jobshop), making the development of a holistic scheduling approach extremely hard. Instead, the proposed approach identifies and models the constituent parts of a manufacturing system, postulating that this collection of models implicitly provides the complete system model. Four basic constituent parts are identified; namely, materials, transformers, buffers and connectors. Any material that may be present in the system, whether a final product, a raw material or an intermediate product, is considered as a material object that is defined in terms of its physical characteristics (density, demand...), volatility (the minimum and maximum period of storage before further processing), its production requirements and the set of operations on different transformers where it can be developed. The production requirements are defined in terms of a set of constituent materials that are needed for the development of the material object described. For each constituent material, the mass required for the development of a unit mass of the defined material object is also provided. The production requirements can be viewed as an 'immediate' material recipe, defining the requirements local to that material state. Transformer objects represent the physical system units (such as packing lines, mixers, etc.) that are capable of producing different materials. A transformer can be viewed as a collection of the operations that it can carry out. Each operation is defined in terms of the material that it produces, its mode (continuous vs. batch, fixed rate/time vs. variable rate/time) and its characteristics (production rate if continuous, size and time if batch). Each transformer also has an operational state that describes its behaviour within the scheduling timespan. The state definition consists of a series of time intervals that cover the timespan. Within each time interval, the transformer may be unavailable (cleaning, etc.), idle, or performing a certain operation at a given rate.
1089 Buffer objects are system units that can temporarily store materials for future usage, such as tanks or silos. These are defined in terms of their storing capacity, the set of materials that they can store and their storage policy (whether more than one material can be stored concurrently). A state representation is also required, indicating the storage utilisation of the buffer in the scheduling timespan. This is again achieved by the use of variable-size time intervals. Within each interval, the fluctuation of material stored is recorded. Since only constant operation rates are permitted for the units feeding or being fed by the buffer, this fluctuation can be represented by a series of (time, quantity) points. Connectors enable the material transfer between different system units. Each connector is defined in terms of the system units that it connects, the materials that it can transfer and their corresponding transfer rates. An operational state is also required that is defined in a manner similar to that of a transformer. Crucial to the success of any modelling paradigm is the means it provides for constraint representation. The proposed approach provides a flexible framework for the definition and monitoring of constraints. A differentiation is first made between dedicated and system constraints. The former can be defined with respect to a single system object, whereas the definition of the latter requires more than one system object. Dedicated constraints, such as changeover times between processing of two different materials or maintenance periods, are defined as object properties; thereby extending the object definitions described earlier. A wide variety of system constraints can be present in the manufacturing systems considered including connectivity constraints between system units, parallelism and concurrency constraints, flow constraints and common system utilities (manpower, electricity, etc.). To represent system constraints, a new 'resource' object class is introduced, with the aim of providing a unified framework for constraint definition. The resource object is associated with a capacity and a set of rules that define how it is consumed by operations taking place at other system objects. An operational state is also defined in a format similar to that of a transformer object, indicating in each time interval the level of resource consumption with respect to the resource capacity. 3 SCHEDULE GENERATION The purpose of the scheduling function is to construct a complete and feasible system schedule, using a problem definition that is based on the modelling paradigm described earlier. This task is exceedingly complex due to the large gap between the abstract nature of the model (needed to allow the encapsulation of the variety of systems considered) and the detailed definition of the system schedule. The development of an efficient scheduling algorithm constitutes the core of the proposed approach and its detailed description is beyond the scope of this paper. Instead, an overview of the underlying algorithm is provided. The scheduling algorithm first divides the portfolio of product demands into a set of sublots. Each sublot is treated as an indivisible unit, in the sense that all operations needed for its production must be carried out without pre-emption. Division of demand lots into sublots is necessary to enhance flexibility and avoid clogging of system units for inordinately large utilisation periods. The resulting sublot set is then arranged into a sequence and the complete schedule is developed by considering each sublot in turn and identifying the best feasible set of factory-wide operation allocations that would lead to the sublot production. The scheduling of each sublot increments the partial schedule obtained by the sublots already scheduled.
1090 Sublot scheduling is carried out in a back-propagation manner, backtracking when deadends are encountered. The algorithm starts by identifying all system unit intervals that could generate the material represented in the sublot (named candidate intervals). These intervals are then sorted on the basis of a semi-deterministic heuristic and the most preferable is chosen. The algorithm then identifies the materials defined by the production requirements of the material considered. For each of these materials, the set of candidate intervals are identified and sorted. A test is then conducted to establish whether the new interval chosen, together with the ones already chosen, can provide a feasible set of operations for all materials considered to this point. Feasibility is achieved if the operational characteristics of the system units involved are respected and none of the system constraints are violated. If this is the case, the operation described is repeated for the materials defined in the production requirements of the materials last considered. If a feasible set of operations cannot be found then the algorithm backtracks and chooses the next preferable interval until feasibility is re-established. The process is repeated until feasible operations are identified for the raw materials associated with the considered sublot. 4 EXAMINATION OF SOLUTION SPACE The schedule corresponding to a random sublot sequence is unlikely to be of high quality. This is because scheduling decisions are conducted in a greedy and localised manner, prohibiting high-level optimisation. Experimentation has indicated that the quality of the resulting schedule is highly dependent on the sublot sequence used. Therefore, the remaining task is to identify the sublot sequence that corresponds to an optimal or near-optimal schedule. The solution space considered corresponds to all permutations of the sublot set. As a typical sublot set has more than 20 members, any exact method is impractical. Instead, the search is conducted using a metaheuristic; namely, Simulated Annealing (SA). A key feature of the developed scheduling approach is that it requires a 'fast' search, in the sense that a high-quality schedule must be obtained by examining only a very limited number of solutions from the search space. This is due to the inordinate complexity of the solution evaluation function (i.e. the sublot-sequence schedule generator) that leads to large execution times. To compensate for this, modifications were made on the standard SA paradigm to enable a more efficient investigation of the search space. The proposed SA includes a modified Boltzmann distribution that allows easier movement at early stages but is more restrictive later in the search, as well as variable neighbourhood structures. A detailed description of the proposed modifications and their rationale, as well as experiments validating the improved performance yielded by the modified scheme can be found in [ 1]. 5 E X P E R I M E N T A L RESULTS A large number of case studies has been carried out on real-life commercial consumer goods factories. For each factory, the proposed approach was tested for different scheduling scenarios and results were compared with existing schedules developed by experienced plant managers. In all cases, optimal, or near-optimal solutions were obtained. A representative case study that addresses a powder detergent plant is provided below. The plant comprises five stages as shown in figure 1. The first stage consists of a single making unit that is capable of producing 4 different base materials. The making unit operates at a continuous rate that is fixed for each material production. It is directly linked to 6 silos of
1091
different capacities that can temporarily store the base materials generated. The silos are grouped in pairs and the making unit is not allowed to feed both group members simultaneously. Similarly, silos of the same group cannot simultaneously feed subsequent system units. All silos are linked to four identical post-dosing units that can process a base powder to create a variant. A total of six variants exist. There is no limit on the number of post-dosing units that can be fed simultaneously from a given silo but the total outflow for each silo cannot exceed a certain value. Post-dosing units also operate in a continuous manner but are allowed to assume variable rates. ................
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r - - - - - - - " - ~
Storage Area
Figure 1: Plant layout for examined case study. As shown in the graphical representation, each post-dosing unit is directly linked to a number of packing lines that produce the final products. There are twenty-nine different products, most dedicated to a specific packing line. Packing lines also operate continuously. To enhance the factory flexibility, a limited capacity storage area is introduced between the post-dosing and the packing line stages. The storage area is capable of holding several variants simultaneously and can be fed by three of the four post-dosing units. The feeding is performed manually and the material transfer rates are variable, depending on the actual unit and its operational status. If the post-dosing directly feeds a packing line then the transfer rate is halved. If more than one line is fed simultaneously, no transfer to the storage area can take place. A similar arrangement exists between the storage area and the packing lines, where most of the packing lines can be fed from the storage area. The latter is also connected to a dedicated Bulk line. Due to limitations imposed by the manual handling of the storage area, a total maximum operational flow is set both for the connections leading into the storage area and those going out of it. Product demands are generated on the basis of a weekly forecast and the scheduling task is to develop a corresponding schedule that respects all system constraints while minimising the makespan. A makespan of less than 120 hours is considered acceptable and the best solution developed by plant managers is 116 hours. The plant was modelled according to the modelling paradigm described in section 2. The resulting model consists of 39 material objects (4 bases, 6 variants and 29 final products), 15 transformers (1 making unit, 4 post-dosing units and 10 packing lines), 7 buffers (6 silos and 1 storage area) and connectors for all the connections present in the system. In addition, several resource objects are defined to capture the variety of constraints that govern the system. These include 8 resource objects to represent the flow constraints (one for each silo output plus one for incoming connections to the storage area and one for outgoing), 6 resources to represent the silo parallelism constraints and 3 resources to capture the rate
1092
constraints from the post-dosing units to the storage area. For each resource object, its capacity and the rules defining its consumption are defined. Test runs were made using a Pentium II 450MHz processor. Examining 10,000 sublot sequences, an average makespan of 115.374 hours was achieved. The average computational time was 12 minutes and 33 seconds. Tests were also conducted for the same case study employing different scenarios such as different product demands, incorporation of labour constraints and reduction of the storage area capacity. In all scenarios, high quality schedules were generated. 6 CONCLUSIONS A novel approach has been presented for the short-term scheduling of multistage, multipurpose scheduling systems found in the process industry, which is based on an abstract modelling paradigm for capturing problem definitions and a heuristic scheduler for the development of feasible schedules. The developed tool is generic, in that is capable of dealing with a wide variety of process industry plants with different operational characteristics and constraints. In all the examined case studies, the tool reached high-quality solutions in relatively low execution times, varying according to problem complexity and size. Three stage systems (upstream, downsteam and intermediate storage stages) could be solved within seconds and, even for more complex problems, execution times were below one hour. Since industrial schedulers are expected to require computational times of up to eight hours, clearly, the presented approach can be considered as both effective and efficient. In addition, the tool can be treated as a 'black-box', since the algorithms and heuristics used are independent of the problem instance investigated. This guards the user from any scheduling considerations and extends the applicability of the tool to the vast majority of manufacturing systems found in the process industry.
REFERENCES [ 1] C. Charalambous, T. Tahmassebi, and K.S. Hindi. Modelling multi-stage manufacturing systems for efficient scheduling. European Journal of Operational Research, accepted April 1999. [2] D.E. Deal, T Yang and S. Hallquist. Job scheduling in pertochemical production: Two-stage processing with finite intermediate storage. Computers and Chemical Engineering, 18(4): 333 - 344, 1994. [3] M.G. Ierapetritou and C. Floudas. Effective continuous-time formulations for short-term scheduling II: Continuous and semi-continuous processes. Industrial and Engineering Chemistry Research, 37:4360 4374, 1998. [4] E. Kondili, C.C. Pandelides, and R.W.H. Sargent. A general algorithm for short-term scheduling of batch operations - I. MILP formulation. Computers and Chemical Engineering, 17(2): 221 - 227, 1993. [5] S.D. Mokashi and A. Kokossis. The maximum order tree method: A new approach for optimal scheduling of product distribution line. Computers and Chemical Engineering, 21 :$679 - $684, 1997. [6] J. Pinto and I.E. Grossmann. A continuous time mixed integer linear programming model for short-term scheduling of multistage batch plants. Industrial and Engineering Chemistry Research, 15(11): 741 - 748, 1991. [7] G.V. Reklaitis. Review of scheduling process operations. AIChE Sumposium Series, 78:119 - 223, 1982. [8] N.V. Sahinidis and I.E. Grossmann. Reformulation ofmultiperiod MILP models for planning and scheduling chemical processes. Computers and Chemical Engineering, 15(4): 225 - 272, 1991. [9] N. Shah, C.C. Pandelides, and R.W.H. Sargent. A general algorithm for short-term scheduling of batch operations- II. Computational issues. Computers and Chemical Engineering, 17(2): 229- 244, 1993. [10] M.G. Zenter and G.V. Reklaitis. An interval based mathematical model for scheduling resource-constrained batch chemical processes. Proceedings of NA TO ASI on batch processing systems engineering, pages 779 807, 1992, Anatalya Turkey.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1093
Optimal Cyclic Operation of Biomass Production Ben H.L. Betlem a, Pieter Mulder b, Brian Roffel a aDepartment of Chemical Engineering, University of Twente, The Netherlands bDepartment of Applied Physics, University of Twente, The Netherlands; 1 The rate of biomass production is optimised for a predefined feed exhaustion using the residue ratio as a degree of freedom. By means of the Production Curve, the transitions between continuous, repeated batch, and repeated fed-batch operation are determined. The optimal modes of operation are indicated by analytical expressions for the bioreaction kinetic parameters. The key measures "degree of difficulty of conversion" and "degree of exhaustion" are introduced to define the optimal modes in more general terms. The "degree of difficulty" describes the effect of the kinetic parameters and the feed substrate concentration on the conversion; the "degree of exhaustion" describes the desired final condition. In fedbatch operation, the proposed constant feed policy approximates the optimal feed policy closely. 1. INTRODUCTION For the production of biomass, three basic modes of operation are possible: continuous, batch or fed-batch operation. Continuous operation is not always the best option. Extreme exhaustion of the enriched feed can only be realised when the feed is halted. In the batch and fed-batch mode, a part of the bioculture mass must be left in the reactor for the next batch. This residue, defined as the fraction of the maximal reactor volume, is a degree of freedom for optimisation. In the literature, some articles restrict themselves to a single fed-batch run and pay special attention to the use of Pontryagin's maximum principle or Green's theorem [1-2]. Several studies have been published concerning cyclic batch or cyclic fed-batch biomass production. In [3], the fed-batch cyclic operation is optimised for a prescribed final biomass and feed concentration. In [4], an automated fermentation system has been developed to implement repeated batch and repeated fed-batch operations for a methanol-utilising bacteria. In [5], the optimal cyclic operation has been studied as a two-objective programming problem. The noninferior solutions are identified for multiple objectives: the biomass production rate and the substrate conversion. In [6], cyclic operation is used with a fixed residue ratio to study the kinetic parameters of fermentation processes. The fixed ratio will rarely result in an optimal production rate. In this article, the following new items will be discussed. - The optimal control mode is determined for the total range of the kinetic parameters, feed concentrations, and required final concentrations. - The transitions between the optimal modes of operation are analytically defined. Currently with the department of Chemistry and Process Engineering, New Castle University, UK.1
1094 -
Key measures for the performance and exhaustion are introduced to describe the areas of optimal operation in a more general way. This method has been applied earlier in a similar fashion for other cyclic processes such as batch distillation or cyclic gas separation [7-8]. A sub-optimal fed-batch control policy is introduced, since this operation is easier to implement and is compared with the optimal feed policy.
-
2.
PROCESS
BEHAVIOUR
Substrate is converted by biomass into additional biomass. It is assumed that the reactor is ideally mixed. The biomass and substrate are represented by their concentrations in the culture, called X and S respectively. The balances for the bioreactor are: d S = it {a }. s - F i n . S dt V '
da : - it {a } . x -~-(a F - a ). 5-----n-n dt Y V '
(1)
d--~-V = F~. - F
dt
ou,
Where Y is the specific biomass yield and It{S} is the specific growth rate, which depends on S. Y can also be a function of S, but in this work, Y is assumed constant. In addition, it is assumed that both biomass decay and maintenance requirements are negligible. The growth rate It{S} relates the change in biomass concentration to the substrate concentration. Two types of relationships for IriS} are commonly used in the literature: the substrate saturation model (Monod equation) and the substrate inhibition model (Andrews inhibition equation). Both can be given by:
s
(2)
It{S}: Itmax Ks + S + S 2 / K !
Where Ks is the saturation or Monod constant, K1 is the inhibition constant and Itmax is maximum specific growth rate. The value of Ks expresses the affinity of the biomass for the substrate. The Monod growth kinetics is a special case of the substrate inhibition kinetics. Its equation is derived from Eq. (2) when the inhibition term in the denominator is neglected, thus when KI ~ ~. Substrate inhibited growth rate has a maximum for S = Sopt "-- ~/KI Ks 9 3. O P T I M A L C Y C L I C O P E R A T I O N The goal in this work is to maximise the biomass production rate. In [9], it is shown that for a stationary cyclic operation the relationship between X and S develops to a final state where the trajectories of refreshing and conversion have equal slopes: X = Y(SF-S). Therefore, during cyclic operation, a relatively simple relationship exists between the substrate and biomass concentration. Then, for continuous and batch operation the specific production rates can be defined by: Pgcont : JL~g f }. Y(gF - g f )
'
PRbatch --
Y(S F - S T).(1-r/) T
with
r/=
V~ max
(3)
The cycle time T depends on the control strategy and is derived in [5]. The constraints for the production optimisation are the substrate feed concentration, SF and the remaining substrate concentration in the product, Sf.
1095 0.16
3.5 E
repeated fed-
3
'T
E 0.14
"9 m
r
2
fed-batc~~~
"~ ~. 1 ~ o.5 0
continuous
0.2
0.4
A.
n [-1
0.6/" / ,
0.8
0.1
~ 0.08 ~ ~ ; : d b a t c ~ 0.06 / /repeated "~ L. 0.04 J batch r9 0.02
~1.5
._ oo
~o
repeated
' e - 0.12
1~2.5
1
~ o
1
continuoous
0
0.2
o.4
'n [-]
o.6
o.8
B.
Fig. 1. Specific production for CONT, RB, RFB-OPT, and RFB-CF. (]2max=l [h-l], Y=0.45, SF--10 [kg.m-3], Sy=0.01 [kg.m-3]). A. weak inhibition (Ks=0.03 [kg.m3], K/=2 [kg.m-3]), B. strong inhibition (Ks=3 [kg.m3], K/=0.02 [kg.m-3]). Three modes of operation have been studied: continuous, repeated batch, and repeated fedbatch operation. Table 1 gives an indication of the application areas. For fed-batch, two different feed policies have been applied: dynamic optimal feed rate and sub-optimal constant feed. Fig. 1 shows the specific production rates for these four modes in case of weak and of strong inhibition. The first case is similar to an example from [3]. The results show that the relative differences between the production rates increase when the production gets more difficult. Table 1 Application areas of modes of operation. Monod growth
Inhibited growth
continuous / /
high Sf/SF
high Sf/SF
repeated batdh
low Sf/SF
low Sf/SF,weak inhibition
not applicable
low Sf/SF,strong inhibition
/
repeated fed-batch / /
4. AREAS OF OPTIMAL OPERATION MODE In this section, the optimal mode of operation is determined by means of the Production Curve. The curve, shown in Fig. 2, represents the specific production as a function of the cycle time. To determine the optimal control mode for a particular final state, the required final state has been taken here as a starting point. This representation is possible, because for each set of reaction kinetic parameters, only one unique Production Curve exists. This has two reasons. In a cyclic operation, there is a fixed relationship between substrate and biomass concentration. Further, the specific production rate at every point is independent of the final state to be reached. This not only applies to repeated batch, but also to repeated fed-batch as the feed rate follows the singular arc. A batch cycle starts at V= Vmax and S> Sopt (at the right hand side in Fig. 2). During exhaustion, first, the conversion rate increases and after passing the inflection point at SIp, it decreases and is exhausted until Sf is reached (at the left hand side). According to Eq. (1), the biomass growth is maximal, when the mathematical product ,u{S}.X is maximal. Consequently, the maximum production rate does not coincide with the maximum growth
1096 [
,~ _ ,',
] / :
d
~'~\(b?"~/S > Sopt' v = vmax
,.~~ .......i
I /
IContinuous
fed-batch
~~/""/repetacthe
/
/
repeated
....
;~- ~)~ V < V . ~
"
......
v
~ S o p t at V = Vmax
lJ ~ l ~ SCONT,RB I Fed" b a t c h ~ ~ , ~ . , ' \ , . , :sIP(maximal production rate) .. S < Sopt, V = Vmax ,'~ : SRB,RFB
Ieatch
, ~ ...... .........
!/
cycle time T Fig. 2. General Production Curves for batch and fed-batch operation
rate: SIp < Sopt. Since continuous operation is the limit case of batch operation for T--+ 0, the slope of the tangent at the Production Curve equals the continuous production rate. A fedbatch cycle starts at V< Vmo~ and S-Sopt. Following the singular arc, the growth rate remains the same until V= Vmo~. From this point, the fed-batch develops equal to the batch process. The optimal operation for a certain Sf can be found by taking the maximal tangent to the Production Curve in the point SU [10]. When the tangent to the Production Curve has a maximum in SUitself, then continuous operation in SUis optimal with a specific production rate of dP/dt. When the tangent contacts the curve at another point, then cyclic operation is maximal. In Fig. 2 the points Sb and ST indicate the start and final substrate concentrations of the optimal cycle. S, is the final state requirement, and Sb is realised by the proper residue ratio. Then, the specific production rate equals (P{ tf}-P{ tb})/(tf-tb). From the curve it follows that three areas of modes of operation can be distinguished. For this classification, final state values higher than Sopt are not considered, since they are not realistic. When Sf> Sop, the production rate becomes lower whereas the exhaustion becomes less. C o n t i n u o u s operation: SCONT,RB~ Sf< Sopt When SU is larger than the inflection point Sin, then continuous operation is optimal. Therefore, the transition point between continuous and batch operation, SCONr,RS, corresponds with SIp that can be found by taking d2X/dt2=O. Then from Eq. (1) it follows that this is satisfied if d(#{S}X)/dt=O. Using the relation for cyclic operation X= Y(SF-S), X can be eliminated and the following condition is obtained: 9
~)X ~Su
() (I.t{SI }" Y(SF - S• ))= O , thus S I = ScoNr,=8 = ()Sf
+Xs +S;/K,)-K= I+SF/K I
9 Repeated batch operation: SRB,RFB~ Sf< SCONT,RB Repeated batch operation is optimal if Sf is located between the point where the tangent contacts the Production Curve at Sopt and the point SIp. At Sopt the Production Curves of fedbatch and repeated fed-batch diverge. Thus, SRB,RF8 can be determined from the condition that the tangent at Sop, dP/dt, equals the production rate for the cycle Sopt to Sf, (P{ Su}-P{ Sopt})/T. Consequently, for repeated batch operation Sb is between Sf and Sopt. 9 Repeated fed-batch operation: 0 < Sf< SRS,RF8 In this case repeated fed-batch is the optimal mode of operation.
1097
Fig. 3. Optimal mode of operations related to the degree of difficulty (6) and the degree of exhaustion (e) for five degrees of inhibition ((Ks [kg.m-3], KI [kg.m-3])= (0.0003,200), (0.003,20), (0.03,2), (0.3,0.2), (3,0.02). The following scaling factors for Sf and SF are proposed to combine the optimal operation areas for all different kinetic parameters into one figure (Fig. 3):
Sopt - S f
e = ~ ,
Sop,
S
2
1--ScoNr,RB//(KsKI )
~=___L_F.
Sop,
(5)
2 + S CONr.RB/ Ks
Where e can be interpreted as a measure for exhaustion of the feed. It is scaled between minimal (e = 0) and total exhaustion (e = 1). Values higher than Sopt are not realistic. The parameter fi has been chosen such that e{ ScoNr,ir }+~= 1 and can interpreted as a degree of difficulty of the conversion. The denominator of the right hand term is a measure for the affinity of the biomass to the substrate concentration. The nominator reflects the influence of the inhibition and if KI--->,,,,, it becomes one. Fig. 3 shows the optimal modes as a function of exhaustion and conversion difficulty for five cases with the same Sopt. If KI becomes larger and the exhaustion is high, the transition between batch and fed-batch shifts to the right. For Monod kinetics fed-batch operation is no option and Eq. (4) with KI ~ ,~, is sufficient to determine the optimal mode of operation. 5. S U B - O P T I M A L O P E R A T I O N W I T H R E C Y C L E S Repeated fed-batch operation with optimal feed control can be approached by an operation with constant feed control, which is easier to implement. The procedure applied for the cyclic operation is the following. A finished batch is refreshed with SF such that the growth rate at start-up equals #eo of Pmax. Next, the feed rate is set at Fin= lu~.#{Sopt} rIVmax. During the conversion phase the substrate concentration will only decrease. First, the growth rate increases from p% to Pma~. Next, it decreases and the feeding is stopped when the growth rate has returned to peo. The residue ratio, r/, is used to ensure the final volume becomes Vmax. When p~ is taken as 100%, the reactor will be filled up such that at start-up: V= Vma~ and S = Sopt. This agrees with the point where repeated fed-batch becomes repeated batch operation.
1098 Fig. 1 shows the specific production rate for RFB-CF, RFB-OPT, and RB mode. The constant feed policy performs nearly as well as the optimal feed policy. For weak inhibition, the maximum is reached at approximately #%=99.5% and for strong inhibition applies #~o = 98%. It appears that the maximum of the RFB-CF policy does not differ much from the maximum of the RFB-OPT policy. For weak inhibition, both are identical and for strong inhibition, the optimal cycle period of RFB-CF-becomes shorter than for RFB-OPT. 6. CONCLUSIONS A method to describe the optimal operation has been developed for bioreactions with substrate inhibition. Based on the Production Curve, the transitions from continuous to repeated batch and from repeated batch to repeated fed-batch with dynamic optimal feed rate have been determined. It has been shown that the optimal control mode can be described by the combination of a term that indicates the performance level (degree of difficulty) and one term for the degree of exhaustion. Simulation studies show that fed-batch with constant feed is not much inferior to fed-batch with optimal feed control, because the residue ratio brings the substrate concentration in the area of optimal growth. SYMBOLS CONT F
continuous reactor feed [m3.h-1] KI inhibition constant [kg.m3] Ks saturation (Monod) constant [kg.m3] P specific production[kg.m -3] PR P rate [kg.m3.hl] RB repeated batch RFB repeated fed-batch RFB-OPTRFB with optimal feed control RFB-CF RFB with constant feed control S substrate concentration [kg.m3] T cycle time [h] V volume reactor [m3] X biomass [kg.m3] Y specific biomass yield [-]
E 7/ /2
Indices 0 cont b f F IP opt
degree of conversion difficulty (def: (5)) degree of exhaustion (def.: (5)) ratio of residue and max. volume [-] growth rate [hl] percentage of maximum la [h1] after emptying continuous after refreshing at end of cycle feed at inflection point at maximal growth rate
REFERENCES 1. K.Y. San and G. Stephanopoulos, Biotechnol.Bioengng, 26 (1984) 1261-1264. 2. L. Cazzador, Biotechnol. Bioengng, 31 (1988) 670-674. 3. W.A. Weigand, Biotechnol. Bioengng, 23, (1981) 249-266. 4. P.J. Henningan, Ph.D. Thesis, Purdue Univ. (1983). 5. M. Matsubaru, S.Hasegawa, and K. Shimizu, Biotechnol. Bioengng, 27 (1985) 1214-1222. 6. B.M. Wincure, D.G. Cooper, and A. Rey, Biotechnol. Bioengng, 46 (1995) 180-183. 7. B.H.L. Betlem and B.Roffel, ICSC-WMC 97, Auckland, NZ (1997) 428-434. 8. B.H.L. Betlem, H.C. Krijnsen, and H. Huijnen, Chem. Eng. J., 71 (1998) 111-126. 9. P.K. Shukla and S. Pushpavanam, Chem. Engng Sci., 53 (1998) 341-352. 10. D.W.T. Rippin, Computers and Chem. Engng, 7 (1983) 137-156.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1099
Short-term Scheduling and Recipe Optimization of Blending Processes Klaus Glismann and Gtinter Gruhn a a
Department of Process and Plant Engineering, Technical University Hamburg-Harburg Schwarzenbergstrasse 95, D-21071 Hamburg Phone: +49-(0)40/42878-3241, Fax: +49-(0)40/42878-2992 e-mail: glismann @tuhh.de
The main objective of this paper is to present an integrated approach to coordinate short-term scheduling of multi-product blending facilities with nonlinear recipe optimization. The proposed strategy is based on a hierarchical concept consisting of three business levels: Longrange planning, short-term scheduling and process control. Long-range planning is accomplished by solving a large-scale nonlinear recipe optimization problem (multi-blend problem). Resulting blending recipes and production volumes are provided as goals for the scheduling level. The scheduling problem is formulated as a mixed-integer linear program (MILP) derived from a Resource-Task Network representation. The scheduling model permits recipe changeovers in order to utilize an additional degree of freedom for optimization. By interpreting the solution of the scheduling problem new constraints can be imposed on the previous multi-blend problem. Thus bottlenecks arising during scheduling are considered already on the topmost long-range planning level. Based on the outlined approach a commercial software system for in-line blending and batch blending processes has been designed.
1 Introduction This paper presents a strategy to coordinate short-term scheduling of industrial blending processes with nonlinear recipe optimization. The focus is on blending processes, but the strategy is also applicable to other multi-product processes having typical criterions of blending processes. Blending processes themselves can be characterized by the following key features: - Blending stocks of widely different properties are supplied continuously or in batches. They are sent to intermediate tanks. Properties and flowrates of the components usually vary over time. - The different components are blended according to recipes in an in-line blender or in a batch tank. - The blends are stored in tanks and/or are delivered directly. - The recipes must guarantee an on-spec product with minimum give-away. Product property limits are often restricted by law. - Similar products can be blended by applying different recipes.
1100 A common field for blending processes is the production of gasoline and gas oil in refineries. Nevertheless, blending applications can also be found in several variations throughout all branches of process industry. Today's widespread approach for the scheduling of such processes is to use intuitive graphical user interfaces combined with discrete-event simulators (Bodington, 1995). Heuristics related to operating policies can be incorporated to speed exploration of alternate policies. However, each scenario still has to be constructed and rated manually. Mathematical programming techniques for short-term scheduling of multi-product plants have been extensively studied in the past years, but not much is reported about the application of these techniques to the short-term scheduling of blending processes. Even though the process has a simple structure and therefore should be well suited for creating an appropriate optimization model. The scheduling of crude oil can be named as a related application that is mentioned in literature (Shah, 1996), (Lee et al., 1996). An approach based on a mathematical model offers a user-friendly treatment of the underlying scheduling problem: User-defined constraints and objectives can be included in a straightforward way. Thus in this paper a strategy based on a combined nonlinear programming (NLP) and mixed-integer linear programming (MILP) formulation is developed. Planning the operation of blending processes covers proper coordination of feedstock and products with market requirements and economics. However, short-term scheduling of blending processes is more complicated than scheduling of most other processes because of the option to blend a product in many different ways: Consideration of recipe optimization and short-term scheduling within an integrated approach becomes necessary. In order to avoid arising nonlinearities in scheduling, an iterative scheduling strategy is developed so that the problem can still be modeled favorably as a mixed-integer linear program based on a Resource-Task Network (RTN) representation (Pantelides, 1994). Nonlinear recipe optimization is carried out separately within long-range planning but can be integrated into the overall strategy. 2
Basic planning and scheduling approach
Within each company of process industry three hierarchical business areas can be identified: planning, scheduling and controlling. Planning and operating of blending processes can be understood according to these levels. This hierarchical model can be described by the following features: Detailing and reliability of information increases from top to bottom. - The longest planning horizon can be found at the top. The horizon shortens rapidly when moving down towards the process control level. - Complex planning and scheduling tasks are broken into simpler ones that are solved within each level. - Results of each level are forwarded to the attached levels (in both directions). -
The developed strategy is built up according to this hierarchical view. A long-range plan for blending processes usually covers a horizon of about one month. Therein roughly scheduled customer demands are balanced with the available blending component volumes. State of the art models are multi-period models that consider multiple blends simultaneously (multi-blend optimization) (Rigby et al., 1995). They embody a
1101 nonlinear recipe optimization problem. During the optimization run the running times of each operation mode for all upstream facilities (e.g. the reformer) can also be determined. At this level a large-scale NLP has to be set up and solved. The usual objective is maximum profit given by the sum of sold products minus the value of spent feedstock. The free variables are -
the component volumes related to each product and each period, the running times of each possible operation mode for the upstream facilities.
Constraints arise from -
-
-
the blending process structure (flowsheet, tank and blender data, etc.), the forecast on the component production defined by each operation mode for the upstream facilities, the product delivery-dates, the nonlinear and linear blending models, the planning periods, given by product demands and specific planning priorities.
The obtained solution is transferred to the short-term scheduling level: The calculated product quantities are the goal quantities that have to be met applying the previously optimized recipes. At this short-term level specific attention is paid to the delivery dates and the allocation of the blenders. The planning horizon is shortened to one week. The main scheduling priorities are (in the given order): a) to obtain a feasible schedule satisfying all product demands, b) to meet the goals set by the long-range planning, c) to optimize the operation of all blending facilities itself (e.g. to minimize product and recipe changeovers). An appropriate MILP formulation derived from a RTN process representation can be developed in order to fulfill the named goals. The most important feature of this model is that alternate recipes for each product and period can be provided. A recipe changeover becomes a free variable for optimization. So, which one of the alternative recipes is preferred in a particular situation results from the optimization run. The mathematical model will be described in detail within the next chapter. After processing the scheduling problem, deviations from the goals can occur because of the following reasons: - The more precisely considered delivery-dates in scheduling require an earlier production, because within long-range planning product demands were defined for periods and not for precise delivery dates. The necessary number of changeover operations can not be determined within long-range planning. No equipment item can be assigned to different operations at the same time anymore. Simultaneous allocation of equipment can not be excluded within long-range planning. In long-range planning, material is balanced according to the defined periods. A violation of given tank limits inside a period can not be determined until scheduling is done. -
-
-
1102 When one of the given goals can not be met within scheduling, actions according to three different strategies are available: a) The resulting feasible schedule is accepted in spite of the deviations. The closest approximation to all given goals can be guaranteed mathematically. b) Within scheduling a modified problem is constructed in order to shift deviations between goals. This can be accomplished by applying different weights to each single goal. c) Finally, a strategy coordinating short-term scheduling with long-range planning can be applied. The scheduling level can be left and a modified multi-blend problem can be solved utilizing knowledge of the bottleneck in scheduling. The new goals for scheduling are more likely to be met. This strategy leads to an integrated optimization of planning and scheduling. Selection between the given strategies depends mainly on the current situation and the given scheduling priorities. Figure 1 illustrates all alternatives explained above.
II ~ il
Long-term Planning Goals: Product Quantities and Recipes Short-term Scheduling
I~.~ ]i
Goals can be reached or
Deviations can be accepted Goals cannot be reached
Modified Formulation of Scheduling Problem
i[
End of Short-term Scheduling
I
]]
Goals: Schedule, Recipes, Throughputs, Tank levels Blend Control
]]
Figure 1" Short-term scheduling strategies After passing the scheduling level a schedule which can be visualized graphically as a Ganttchart is recovered. Set-points for the process control level can be derived from it. The blending process itself is carried out within this operative level. The received operating instructions are transformed into control strategies for the process control system. Advanced control of all blenders, respectively blend tanks, adjusts their operation to the given set-points taking into account the current situation that can differ from the assumed situation in planning.
1103
3 Scheduling model The scheduling model is based on a Resource-Task Network representation. Figure 2 shows an example of a simple in-line blending process with 1 blender, 2 products and 3 components.
Component 1
~
Blender Product 1
Recipe P1 I Product 2 ~ m ~ ' l
RecipeP2 I
-O 9
Figure 2: Resource-Task Network of a typical blending process The mathematical model can by characterized by the following key features: - Time is modeled according to a uniform discretization. - Tasks can be given a temporary validity in order to adjust scheduling to the periods of long-range planning. - The maximum count of a resource is 1. Resources with similar characteristics are treated as different resources. This assumption simplifies resource balances. To formulate a mathematical model for optimization requires a deliberate consideration of how time can be modeled. For semi-continuously operated blending processes the duration of each single blending operation is not known a priori. The same is true for batch blending processes with variable batch sizes. However, a continuous-time problem formulation is not appropriate since a nonlinear mixed-integer program would result from modeling the continuously refilled blending component tanks. The application of a model based on uniform discretization of time is less crucial because the most significant simplification can be lessened: Previous fixing of the duration for an individual blending operation is necessary, but by subdividing a blending order into several smaller tasks corresponding to the discretization of time, blending becomes more flexible. In order to avoid unreasonable short running times of a recipe and too many recipe changeovers constraints can be added to enforce a minimum running time for particular recipes. Forced recipe changeovers due to the periods defined within long-range planning are not restrained by these constraints. Additionally, the technique of goal programming is applied in order to minimize changes in throughput of all blenders between intervals. By making use of a RTN representation it is possible to define a different product recipe for each period defined in long-range planning. This is achieved by providing different tasks at different points of time. This temporary validity contributes to the operating strategy of running the blending process optimally as initially planned. Even better fulfillment of the targets of a long-range plan can be accomplished by providing alternative recipes for each product and period. Calculation of these alternative recipes is carried out in the multi-blend optimization on the long-range planning level. But it is not possible to calculate additional recipes in advance since the planning model is incapable of taking scheduling tasks into
1104 account. So these recipes are added to the second scheduling problem after deviations occurred in the first problem following the proposed coordination strategy. 4
Summary
and
Discussion
This paper has presented a strategy to coordinate short-term scheduling of blending processes with nonlinear recipe optimization. The recipe optimization problem is treated as a NLP and its results, the recipes and tank goals, are forwarded to the scheduling problem. The scheduling problem is formulated as a MILP based on a Resource-Task Network representation. The scheduling model - is capable of switching between alternative recipes during optimization, - can take recipes into account that are defined according to the long-range planning periods, - uses a combined strategy consisting of additional constraints and a special objective that avoids unreasonable short running times of recipes and that minimizes recipe changeovers. Deviations in the given goals of long-range planning can be transferred back to the recipe optimization problem according to the presented strategy: The NLP is modified based on an analysis of the solution resulting from the scheduling problem. So, bottlenecks that can not be foreseen in long-range planning can be included. The altered NLP is solved and new goals are obtained that are more likely to be met within scheduling. The proposed planning and scheduling approach has been integrated within a commercial system for the overall optimization of blending processes. A high efficiency and sufficient ease of use could be proved by solving several problems of industrial magnitude. The software system has just been licensed to a refinery, which uses the system to optimize the blending of gasoline. References
Bodington, C. E. (ed.), "Planning, Scheduling and Control Integration in the Process Industries", chapter 6, Mc-Graw-Hill (1996). Dash Associates Limited, XPRESS-MP Release 11, Warwickshire, UK (1999). Lee, H., Pinto, J. M., Grossmann, I. E. and Park, S., "Mixed-Integer Linear Programming Model for Refinery Short-Term Scheduling of Crude Oil Unloading with Inventory Management", Ind. Eng. Chem. Res., Vol. 35, pp. 1630-1641 (1996). Pantelides, C. C .... Unified Frameworks for Optimal Process Planning and Scheduling", Proceedings of the Second International Conference on Foundations of Computer, CACHE Publications (1994), pp. 253-274. Polke, M. (Hrsg.), ,,ProzeBleittechnik", 2. Auflage, Mtinchen, Wien (1994). Reklaitis, G. V .... Scheduling Approaches for the Batch Process Industries", Working Paper No. CIPAC 95-9, School of Chemical Engineering, Purdue University, West Lafayette (1995). Rigby, B., Lasdon, L. S. and Waren, A. D., "The Evolution of Texaco's Blending Systems: From OMEGA to StarBlend", Interfaces, Vol. 25, No. 5, pp. 64-83 (1995). Shah, N., "Mathematical Programming Techniques for Crude Oil Scheduling", Computers chem. Engng, Vol. 20, Suppl., pp. S1227-S1232 (1996).
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1105
Planning and Maintenance Optimization for Multipurpose Plants C.G. Vassiliadis a, J. Arvela a, E.N. Pistikopoulos ~* and L.G. Papageorgiou b :'Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College, London SW7 2BY, U.K. bDepartment of Chemical Engineering, University College London, London WC1E 7JE, U.K. The interactions between maintenance and production planning are studied in this paper. A preventive maintenance model is proposed coupled with a multiperiod planning model, which provides the basis for the simultaneous optimization of maintenance and production planning for multipurpose batch plants. The overall problem is formulated as a mixed integer linear programming (MILP). The applicability of the proposed framework is demonstrated by a numerical exalnple. 1. I N T R O D U C T I O N Tl~e key characteristic of multipurpose process plants is that different products, or even different batches of the same product, can follow different production routes using different units in the plant. To organize the tirnely production of the required amount of products at minimum cost, a number of planning and scheduling frameworks can be introduced to handle the allocation of utilities, resources and production tasks. The degree of utilization of assets and resources, however, is critically associated with the level of availability of equipment components, which is determined by the initial reliability characteristics and the implemented maintenance policy. In such a multipurpose operating environment, maintainability decisions such as the timing of maintenance must be made having accounted for maintenance opportunities arising from the fact that equipment idle item can be often incurred due to the production pattern. In addition, flexibility in selecting different production can significantly mitigate the adverse effect of equipment failure on the production process. In this respect, the determinations of optimal maintenance and production policies are problems, which clearly depend each another. If a production plan is fixed and used as an input to the optimization formulation for the determination of the optimal maintenance policy, it is likely that a different production plan may facilitate a better maintenance policy. On the other hand, if a maintenance schedule is fixed and used as input for the determination of the optimal production plan, it is likely that a different maintenance policy may facilitate a better production plan. To overcome these concerns and quantify the interactions between production and maintenance planning models, proper linking mechanisms between the two models must be established so that a simultaneous strategy is developed. The importance of considering reliability and lnaintenance criteria in process manufacturing, design and operation has been recognized over the last fifteen years (see for Tc) whonl correspondence should be addressed
1106 example, [1-9]). Most of the previous work focuses on continuous processes, with only a few works specifically concerned with multipurpose operation (for example, [10-13]). A common lheme that is emerging from previous work is the need for introducing consistent and rigorous system effectiveness criteria to characterize the performance of a process system from both the availability and the productivity point of view. In this work, we propose an integrated system effectiveness optimization framework for the simultaneous optimization of maintenance and production planning for multipurpose batch plants. The key elements of our approach are: (i) an aggregate production planning model, describing the process related characteristics within a long time horizon of operation, (ii) the maintenance model, describing the reliability characteristics of the system and the effect of maintenance policies on the availability of the equipment components, and (iii) the linking variables, that provide the mechanism for the quantification of the interactions between production and maintenance planning by associating the utilization of process assets and resources with the availability of equipment as determined by the maintenance model. This paper is organized as follows. First, a summary of the aggregate multiperiod production model adopted here is given. Then, a novel maintenance planning model is proposed for modelling equipment deterioration and preventive maintenance activities in a ~nuitil-~urpose process operating environment. Finally, the maintenance planning model is linked with a multiperiod production planning model into a single optimization formulation; a numerical example is also presented to illustrate the key features of the simultaneous afq-~roach. 2.
MULTIPERIOD
PRODUCTION
PLANNING
MODEL
An aggregate multiperiod production planning model based on State-task Network (STN) framework [14], is adopted in this work as a process model representation for multipurpose plants. The time horizon of interest is discretized into a number of time periods of equal duration,/-/. The key mathematical constraints of the multiperiod model are given below. Rd,VOllr(:d Utilization Constraints:
E Fi N~jt <_ U jt
(1)
'~,j, t
i~ l,i
R!nin _ nmax ~:e Nijt <- Biit < uut Nijt
Ccq)acity Constraintsv
,
(2)
Vi,.j e K i t Vs, t
i6 T. j~ K i
Drain < D ~'t < D smax st t
DeiJ~altd Colzstraints:
-
-
,
-
(4)
VS, t
-
Pi -]
uti/#,, o,,,,,,,-,i,,t,.,
y_, y_,
(p,,ij oNij ,
(3)
iE T,. J~ K i
W,,,
(5)
i .je K; (o=0
where N,i,, B o, are the number of batches and the amount of material being processed, respectively, of task i in unitj over time period t; S.,., is the amount of material in state s at end
1107 o1 period t; D.,v is the amount of material of state s delivered to external customers over period t: and U i, is the expected uptime of unitj during period t. The resource utilization constraints (1) ensure that the total processing time on a unit cannot exceed the expected uptime of the unit while the capacity constraints (2) suggest that batchsizes are allowed to vary between minimum and maximum values. The material balances (3) state that the amount of material in a certain state at the end of a time period equal to the amount in storage from the previous period, adding the amount produced and subtracting the amount consumed and delivered. Finally, the demand constraints (4) state that demand fluctuates between lower and upper bounds, while the utility constraints (5) ensure that the utilization level of utilities such as steam and cooling water does not exceed the corresponding availability levels.
3. P R E V E N T I V E M A I N T E N A N C E M O D E L In this section, an analytical preventive maintenance model is described, assuming that all multipurpose equipment units are in the wear-out phase, i.e. their failure rate is increasing with time. The underlying assumptions of the proposed maintenance policy are the following: . Equipment failure rate is monotonically increasing with time. . Each preventive maintenance activity restores the component to an As-Good-As-New (AGAN) status. 9 In the event of failure, minimal repair is performed to restore the failed unit to an AsGood-As-Old (AGAO) status. 9 According to equipment specifications, preventive maintenance should be performed to each unit.j at least every r.i time periods, i.e. each unit cannot operate more than "c.i time periods without maintenance. Based on the above maintenance policy assumptions, it is possible to construct analytical expression describing unit failure rate as a function of initial equipment reliability characteristics and the maintenance optimization variables. In particular, we introduce two sets of binary variables: X iI : I if preventive maintenance is performed on unit.j during time period t; 0 otherwise Z i~o : 1 if unitj has been maintained for last time 0 periods ago; 0 otherwise.
Therefore, the equipment failure during time period t,,a,jt, can be described by the fo Ilowing constraints: "/'j
Vj, t
~jt -- E ~YjoZ,jtO
(6)
0=1
Z jt 0 < X .j,t_0
Vj, t, 0 = l..-c,j
(7)
T.i
y__, z./,o = l
v i, t
(8)
0=1
Expected equipment uptime is defined as the expected period of time during which an item is able to perform its intended function. Naturally, uptime depends on the expected number of failures and on the duration of maintenance, which are determined by the failure and
1108
~naintenance characteristics of equipment. In the case of minimal repair and AGAN preventive maintenance policy, the expected number of failures, N i T , of unit.j operating for T time units is given [15] by: T NiT = I)].(s)ds
(9)
0
Assuming that equipment can fail both during minimal repair and preventive maintenance, the expected equipment uptime for unitj during period t is given [ 10] by:
U.i; = H ( 1 - Ac.Zjt) -./
A/?jX jt
V,i , t
(10)
where AI} and Ac'; are the preventive and corrective maintenance duration
for unit .j,
respectively. 4. S I M U L T A N E O U S O P T I M I Z A T I O N OF P R O D U C T I O N AND M A I N T E N A N C E PLANNING
By combining all the appropriate terms and expressions (1)-(10), derived in previous sections, the problem of simultaneously identifying the optimal production and maintenance 191:lla corresponds to the following optimization problem (P1). pi-1
lYlaX (I) - Z/TsID,.t- Z Cut Z Z st
ut
- Z c;'x ;, - Z
.it s.l. constraints (1)-(10)
Z (fll,ij,oNij ' + a,,ijroBij t ) i .je K i go=0
)/at)
- v ; , - A{; x .
.i;
(11)
The first term of the objective function (11), see also [10], represents the profit generated by 1he delivered products, the second term denotes the cost of utilities while the third and fourth terms correspond to the direct costs of preventive and corrective maintenance, respectively. Note that (11) is closely related to a system effectiveness or performability measure [9, 16, 17]. Problem (P1) corresponds to a mixed integer linear programming (MILP) formulation that can be solved using standard branch-and-bound techniques. The applicability ot: the proposed framework is illustrated next with a numerical example. 5.
NUMERICAL EXAMPLE
(7onsider the process described by the STN [10], shown in Figure 1. The operating horizon is two years, which is discretized in 24 one-month time periods. The demand for products B a~(t U is between 5000 and 20000 units for each month period with unit price per period equal Io G; = ().5 for both B and C. The failure rate and maintenance characteristics are the same for all three units, as follows: r.i - 9
A';.- 6h, C": - 50 and C } ' = 1000.
7.jl - 0 , 0 0 2 h - 1
,
7,jo = 7.i,o-1 +0.001
~
2 < 0 < r.j --
_
~
k c. - 24h .1
'
1109 2 hr Make_B ] . @
3 hr
@ ;I
I 2.5hr Make_C I
~-@
Fig. 1 State-task Network (STN) for Example The capacity of units and storage facilities as well as the types of tasks performed by each unit are given in Table 1. Table 1" Resource Details
Unit
Capacity
Suitability
Unit 1 Unit 2 Unit 3 FTank BTank CTank
200 50 40 oo oo ~
Make A Make_B, Make_C Make_B, Make_C Feed B C
The above example was modelled in the GAMS modelling system [18] using the CPLEX optimizer for the solution of the MILP model. The optimal preventive maintenance policy, as depicted by the solution of problem (P1), is shown in Figure 21 Unit/Time
I
2
3
4
5
6
9 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24
7 8 i T!!!i!!ii!i !@ii
;::;;;:~:;;~7;;;;;;;;;;;;~ :iii[iiiiiiiili!;i;iiiiiiii)ilill ::::.=:~.~:::~ ................. ............................:ie!:!e!zeii!i!!i!?~:!?ii!!is!eii!
;iiiiiiiiiiiiiii{)ii!iiiil;i!]]! ~as~sz~a~s~; ............................
';;;~'~'~'~';~';~';~'~'~' i}ilililiiii}}iiil))iTiiiiii!i)i! ii?771iii;i77i;iiTiiiii;iiii ;)r162162
i'i i i i i i i i'i i'i i i i i'i i~)i~iii~i~i~i~ii;{ii~i~ili~!~ii 7!i!i!iiiiii!711;I!II;II~III iiiiii!i!iiiiiiiiiii!ii{{!{?iii;{ iai~i~ia?~iai~i~!~?~iai~!~iais!~!a
;;~';;~';~';~'~'~'~: !){iii!iglgiiiiiiii[!{iigil)!iiii !iii!iiii}Tiiiiiil;ii]iiiiiiii?ii i{?ir
:................................. !~sii!!il;;;;~i;~i~:~:;~i ii771=!iliTigiiTiiiili?iiiii7 .................................
i~i~i~i~i~i~i~i~iii~iii~i~i~iii~ iiTil!ii@~!iTi!iiii77~gi iii72i!iii!i71i!?ii!:!i!iii!iiii 77!i!!{{ii77ii{{{i{~7g2i{ .................
Fi,,~. 2: Optimal Preventive Maintenance Schedule Note that Unit 1 has a large capacity and can, therefore, easily produce the required volumes of A during each period. Therefore, a very high expected uptime for Unit l is not essential and less preventive maintenance actions are scheduled. On the other hand, Units 2 and 3 have smaller capacities thus very high uptimes are required. Consequently, a large number of maintenance actions are required to produce the required volumes of B and C. Unit 2, in particular is more important to the timely production of B and C as it has larger capacity than (h~it ,7. Therefore, Unit 2 is maintained more often than the other two units. It is worth mentioning that the solution of the resulting MILP problem was obtained after 2.3 seconds (SUN SPARC10 workstation) by solving only the corresponding relaxed linear i Thc i~rcvcntive maintenance activities are denoted by shaded boxes.
1110 programming (LP) problem. This property has been observed for other numerical examples studied; however, its theoretical proof (for model unimodularity) is still under investigation. Extensions of this work to include other types of maintenance policy such as Markov models, and other type of failure mechanism have also been considered [19, 20]. 6.
CONCLUDING REMARKS
~i'l~is work described a system effectiveness optimization framework for the simultaneous determination of the optimal production and maintenance plans in multipurpose plants. The expected process profitability, defined as the balance between maintenance benefits and costs, is i~troduced as a system effectiveness criterion. A novel maintenance planning model has also been described, which has been integrated with an aggregate multiperiod production planning model. The resulting optimization problem corresponds to a mixed integer linear l-)rograi-nming (MILP) model. Current research focuses on (i) the extension of the proposed MILP model for the simultaneous identification of the optimal design together with the production and n~aintenance planning policies for multipurpose plants, and (ii) the effect of uncertainty on the interact ions between production and maintenance policies. REFERENCES Van RUn, C.F.H., FOCAPO'87 Proc., (1987) 221. Valdez-Flores, C. and R.M. Feldman, Naval Res. Logistics, 36 (1989)419. Straub. D.A. and I.E. Grossmann, Comp. Chem. Eng., 29 (1990) 967. Pistikopoulos, E.N. and T.A. Mazzuchi, Comp. Chem. Eng., 14 (1990) 991. Limnios, N., IEEE Trans. on Reliabili~., 41 (1992) 219. Grievink, J.K., R. Smit, R. Dekker and C.F.H. Van Rijn FOCAPO'93 Proc., (1993) 133. Thomaidis, T.V. and E.N. Pistikopoulos, IEEE Trans. o17 Reliability, 44 (1995) 243. Tan..l.S. and M.A. Kramer, Coral). Chem. Eng., 21 (1997) 1451. Vassiliadis, C.G. and E.N. Pistikopoulos, Proc. Annual ReliabiliO~ and Mainminabili~ Syllq)osium, (1999) 78. (). Dedopoulos, I.T. and N. Shah, Chem. Eng. Res. Design, 74(1996), 307. 1. Rotstein, G.E., R. Lavie and D.R. Lewin, Comp. Chem. Eng., 20 (1996) 201. 2. Sanmarti, E., A. Espuna and L. Puigjaner, Comp. Chem. Eng., S19 (1995) $565. 3. Sanmarti, E., A. Espuna and L. Puigjaner, Comp. Chem. Eng., 21 (1997) 1157. 4. Kondili, E., C.C. Pantelides and R.W.H. Sargent, Comp. Chem. Eng., 17 (1993) 211. 5. Balrlow, R. and L. Hunter, Oper. Research, 8 (1990) 90. 6. A yen, T., Reliability Engineering and System SaJe~, 41 (1993) 259. 7. Suhner, R., K.S. Trivedi and A. Puliafito, Performance and Reliability Analysis of Computer Systems, Kluwer Academic Publishers, 1996. 18,. Brooke, A., Kendrick, D., Meeraus, A. and R. Raman, GAMS: A User's Guide, GAMS Development Corporation, 1998. 19. Vassiladis, C.G., M.C. Vassiliadou, L.G. Papageorgiou and E.N. Pistikopoulos, presented at 2000 Annual Reliability and Maintainability Symposium, Los Angeles, USA, 2000. 2(). Vassiliadis, C.G., J. Arvela, E.N. Pistikopoulos and L.G. Papageorgiou, paper submitted lo hJd. E~I~,. Chem. Res., 2000.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier ScienceB.V. All rights reserved.
1111
A Mathematical Programming Approach for the Optimal Scheduling of Heat-Integrated Multipurpose Plants under Fouling Conditions Michael C. Georgiadis a and Lazaros G. Papageorgiou b* ~'Chemical Process Engineering Thessaloniki, Greece.
Research
Institute,
P.O.
Box
361,
Thermi
57001,
I'Department of Chemical Engineering, University College London, Torrington Place, London WC 1E 7JE, U.K. This work presents a systematic mathematical framework for scheduling the operation of multipurpose plants involving heat-integrated unit operations under fouling considerations. Based on a uniform time discretization, the overall problem is formulated as a mixed integer non-linear programming (MINLP) model. An iterative procedure has been developed for the solution of the resulting non-convex MINLP model, involving the solution of a series of mixed integer linear programming (MILP) and non-linear programming (NLP) subproblems. An exainple problem is presented to illustrate the applicability of the proposed approach. 1. I N T R O D U C T I O N Fouling is a major unresolved problem of significant interest in the field of heat transfer design and operation. Fouling affects nearly every plant relying on operation of heat exchangers and introduces costs, which are ultimately related to the conservation of energy, operation and capital investment. The common practice to mitigate fouling is to implement Cleaning-In-Place (CIP) operations. Several analytical methods for determining the optimal cleaning schedules for single heat exchangers have been proposed in the literature [1-3]. The main di-awback of these approaches is that they are restricted to a single equipment item. However, in process plants complex Heat Exchanger Networks (HENs) exist with many interactive processing units. In that case, analytical methods for the determination of CIP policies cannot be applied. Recently, a cyclic cleaning scheduling approach applied to heat exchanger networks under rapid fouling conditions has been presented [4]. Despite its clear importance, the minimisation of cleaning and external utilities cost in multipurpose plants, exploring heat integration opportunities has not been extensively considered in the literature. In most of the approaches presented so far, the performance of the equipment remained constant with time and the effect of fouling on the production schedule has been neglected [5-7]. This work proposes a mathematical framework for the introduction of fouling considerations during the heat integration of batch plant operation. In particular, it demonstrates how fouling aspect can be incorporated within a general mathematical
Author to whom correspondence should be addressed
1112
l:ormulation [7] for the scheduling and heat integration of multipurpose plant operation thus ensuring that all relevant scheduling aspects are taken into account. 2.
M O D E L L I N G OF F O U L I N G C O N S I D E R A T I O N S
The mathematical formulation is based on the discretisation of the time horizon into a number of time intervals of equal duration. System events (task starting and finishing times, changes in resources availability levels, HTM temperature and hold-up profiles etc.) are allowed to occur only at the boundaries of time intervals. The key variables of the formulation characterise the utilisation of resources over time horizon and the objective function includes the value of the products and the cost of external utilities. The mathematical formulation relies on a number of constraints, including (a) the allocation, mass balance, capacity and utility utilisation constraints [8], (b) rigorous mass and energy balanced on each HTM storage tank, (c) energy balances on heat-integrated operations, and (d) capacity and operability constraints. In the context of heat integration of multipurpose plants, fouling is taken into account by considering the overall heat transfer coefficient decaying (with time) profiles between HTM and processing material. For example, the energy balance for a heat-integrated operation (see, [7]) is given as follows: Pi-1
qot - U ii kij
X
E T- i j o Wij,t_o -
(~ ~"jt + "-" ijt
"i 2
.
V .j e J/,eat, i e i i, t
(1)
ie I i 0 =0
where
,I/,,,,t
and ij
are the sets of equipment items that can perform heat-integrated
operations, and the heat integrated operations for unit j, respectively. The temperatures, TOo, of the processing fluid at discrete time points 0 relative to the start of heat-integrated task i taking place in unit .j, are assumed to be known and depend only on the elapsed processing time. Also, U,Ii and A(i represent the heat transfer coefficient and heat exchange area, respectively. Variables that should be determined by the optimisation algorithm are: Wiit is the allocation binary variable (c,fi [8]) denoting whether unitj starts processing task i at the start of time interval t; |
Ol.lt
is the temperature of HTM leaving unit.] carrying out heat integrated
task i at the start of time interval t; and | is the temperature of material in HTM storage tank s at time point t. The above constraints are also written at the start and end of each time interval t (c./: [7]). The introduction of fouling considerations introduces extra complexity to the mathematical model. The complexity arises primarily due to the allocation and timing constraints of the cleaning operations. The value of the heat transfer coefficient must change only if a heatintegrated operation takes place. Consequently, if a stand-alone operation (i.e. non heatintegrated task) is performed or the processing equipment remains idle, then the heat transfer coefficient should keep its previous value. The following binary variables are then introduced: g i~ " ! if cleaning takes place in unit./during time period t; 0 otherwise. Z j~ 91 if unitj is either idle or performs stand-alone operation during time period t; 0
otherwise. Then, the corresponding allocation constraints should be modified as follows:
1113
[)i --I
Z g
- z.,
-'-
^ i61 i 0 =0
V .i ~ J heat, t
(2)
Pi --1
Z Ewu,,-o <-z.j,
(3)
v.i,,
i~ Ii 0 =0
where !i is the set of stand-alone tasks for unit j. It is assumed here that the duration of each cleaning activity is one time period. However, the above constraints can be generalised to cover cases where each processing equipment item may have its own cleaning duration (see,
[91t. Based on the discretisation of the time horizon of interest piecewise constant profiles are assumed for the heat transfer coefficient. The heat transfer coefficient for equipment j, U i f , (which is entitled to perform heat integrated operation) at time t after 0 periods of continuous operation since the last cleaning is given by parameter ~.ie according to the following constraints" rj
u j, - ~ r
.x j,o
v i ~ Ju, a,, '
(4)
0=1
l'i
~_~ X jtO - 1- Y.i'
V j ~ J l,eat, t
(5)
0=1
where r.i is the maximum period during which equipment j can perform heat integrated task without cleaning. Usually, "c.i is defined by operating or other limitations. The effect of heating operation on the cleaning requirements must be captured since, according to industrial practice, for long heating periods fluid deposit requires intensive cleaning requirements. In this study, two cleaning costs are used; fixed and variable. The fixed cleaning cost represents fixed expenses over the period of cleaning operation (i.e. periods at which gs = 1 ). On the other hand, the variable cost is a function of the preceding heating period and the deviation of the value of the heat transfer coefficient from its original "clean" value. First, the number of elapsed time intervals, tit , since the last cleaning must be calculated lot each heat-integrated equipment item as follows:
r il > t i,r_l + 1- Z.i,r_l - (rj + 1). Fir
Vj ~ Jhe~,, t
(6)
l-j t Jt - Z
O " X jto
V . I E J heat, t
(7)
0=1
ATi t >tj,t_ 1 - t i t - Z j , t _
1
V j~ Jheat, t
(8)
The AT/I variables are used to capture the elapsed time since last cleaning in the objective function. They take non-zero values only if equipment.j is cleaned at time interval t while it
1114
\va\ operating at time t-1. In all other cases, these variables are forced to zero by the optimisiition algorithm, provided that there is an associated cost within the objective function. 7’Iicn. the deviation of the value of the heat transfer coefficient from i t s original “clean” w l i i c can be captured by introducing an additional continuous variable, A U j f , together with t Iic
l’ollow ing constraints:
The ob,jective function represents a measure of the economic performance of the plant, including the value of products, cost of utilities, pimping and cleaning costs. Due to space I i i i i i l i i t i o i i s , these terms are not described here. 3 . SlJMMARY OF THE FORMULATION AND SOLUTION PROCEDURE
T h e oveixll scheduling problem ( P I ) can be summarised below:
C C(cI,.Y/,+ t j. A/T / , +cj,. A l l , , ) I ~ J , ,,/, , , s l l l ~ ; T~ O~ ~ ~ /
(a) Coiiliiig constraints as presented above (constraints (2) to (9)); (b) m a s s and energy balances on each HTM tank and each heat integrated operation [7]; ( c ) capacity and operability constraints on the HTM; and ((1) ;ill other constraints as in the formulation of [8].
wlicrc S , \ / is the ainount of material in state s held in storage over time interval t;
,fril
fi,,, is the
airiouiit o l utility 11 being used over time interval t; ,j’,,/ and are the flowrates of HTM through i i i i i t , j carrying out heat-integrated task i at the start and end, respectively, of time -
i i i t c i w l /: C,ji is the pumping cost;
Fir is the fixed cleaning cost; and 6 ic , 2:ir are the variable
clcaiiirig cost coefficients.
’Thc above inathematical optimisation problem is a non-convex, MINLP model. In the energy balance constraints on heat-integrated operations (c,f: constraint ( I ) ) introduce lion-linearities due to the product of U and the quantity inside the brackets. We
pi1i-t icular.
ernploy ;in exact linearisation strategy by introducing new variables and constraints, and exploring the special structure of the model thus reducing the size of the linearised problem. T l i c modificd linearised model is an MILP problem which can then be solved using s(:iiidartl braiicli-and-boiiiid techniques. It should be added that although the previously mciitioiied linearisations are exact, the resulting MILP problem i s only a relaxation of P 1 due to [lie approximate linearisation of the bilinear terms involved in energy balances 171. Thci-efore, the temperature, flow rate and holdup profiles obtaining by solving it, may not satisfy thc original energy balance equations. Also, the solution of the MILP will only be an upper bound on the optimal objective function that can be attained by the system. We can also consider solving the restricted NLP by fixing all the binary variables related to the resource
1115 allocation and fouling constraints. The solution of this problem, if feasible, provides a lower bound on the true objective function value. We can then proceed by solving a series of MILP/NLP problems where during each iteration a decreasing upper bound in the objective function of the MILP problem and/or integer cuts excluding previous integer solutions can be imposed. Upon termination, we obtain a range within which the optimal objective function of P l is guaranteed to lie.
4. EXAMPLE PROBLEM We consider a plant manufacturing two products, Prodl and Prod2 according to the recipe described in the work of [7]. Heat integration opportunities are explored between an exothermic task Reaction being performed in unit Reactor and an endothermic Distillation task in unit Column. The time horizon of interest is 24 hours and the cost of externally provided cooling water and steam is 8 and 200 relative cost unites (rcu) per metric ton, respectively. Heat is exchanged via a transfer medium, which can be kept in tanks. Details of the properties of the HTM and the heat-integrated tasks along with other data are given by [7]. Fouling data are expressed in terms of discrete values of the product U j . A j o v e r a maximum operating period of 9 hours without the need for cleaning (see Table 1). The constant cleaning cosl is equal to 8 rcu while the variable cleaning cost coefficients for preceding operating period and deviation of heat transfer coefficient from clean state are 1 rcu/hour and 4 rcu.m:.K/W, respectively. The pumping cost is 0.2 rcu/tonne. Table 1: Discrete Values of U j.Aj Time ( 0 ) Reactor CohHmz
1 18.0 14.0
2 17.5 13.5
3 17.0 13.0
4 16.6 12.5
5 16,2 12.1
6 15.8 11.7
7 15.5 11.4
8 15.2 11.1
9 15.0 10.8
The above example was modelled in the GAMS modelling system [10] using the CPLEX and CONOPT optimisers for the solution of the MILP and NLP problems, respectively. The optimal schedule under fouling considerations in shown in Figure 1. Unit/Time Reactor Filter Column,
1 2
3
4
5
6
7 8
- - ] stand-alone task Fig. 1: Optimal Schedule with Fouling
9 10 1I 12 13 14 15 16 17 18 19 20 21 22 23 24
[
] : heat-integrated task
The total amount of the two products is 410 tonnes and the cost of external utilities is 468 rctt. The value of the objective function is 1311 rcu representing a 65% improvement over the case without heat. It should be noted that two cleaning tasks are performed for each of the Reac'tor and Column processing units. Reactor is cleaned every three heat-integrated batches while Cohtnm is cleaned every single batch apart from the last heat-integrated distillation opeFation. The optimal solution was obtained in one major iteration between the MILP and NLP subproblems with a relatively tight margin of optimality (approximately 5%).
1116
"Fhe flowrate of the HTM through the heat-integrated reaction operation is shown is Figure 2. 10
......................................................................................................................................................................
8 B
E
2 0
i
0
2
i
z
6
1
i
i
,
r
'
i
~
i
8 10 12 14 16 18 20 22 24 26 Time(hr)
Fig. 2: HTM Flowrate Profile through Reactor 5.
CONCLUDING REMARKS
H'his paper has considered the exploitation of heat integration in the operation of ~ullil)url-)ose plants taking into account of its interactions with production scheduling. An ilcrati\'e 1-)rocedure for solving the resulting non-convex MINLP has been proposed, involving lhc solution of a series of MILP and NLP sub-problems. In principle, it cannot guarantee the global optimum, or even a feasible solution to this problem. However, in our experience, the solution procedure generates upper and lower bounds that tend to be within a very tight range within which the true solution lies. A simple example problem illustrated the potential effect of fouling considerations on the production schedule. Although this work has been concentrated on the introduction of fouling considerations in the heat integration to a short lerm scheduling formulation, the approach presented is equally applicable to periodic sc'hcduling. I,~EFERENCES 1. Epslein, N., The Canadian Journal q[ Chemical Engineering, 57 (1990) 559. 2. Casado, E., Hydrocarbon Processing., 69 (1990) 71. 3. Sheikh, A.K., S.M. Zubari, M.U. Haq and M.O. Budair, Transactions of" the ASME, 118 (1996) 306. 4. Georgiadis, M.C., L.G. Papageorgiou and S. Macchietto, Comput. & Chem. Eng., $23 (1999) $203. 5. Vaselenak, J.A., I.E. Grossmann and A.W. Westerberg, h~d. Eng. Chem. Process Des. l)e~,., 25 (1986) 357. (~. ('olominas, J., A. Espuna and L. Puigjaner, Comput. & Chem. Eng., S17 (1993) S15. 7. Papageorgiou, L.G., N. Shah and C.C. Pantelides, Ind. Eng. Chem. Res., 33 (1994) 3168. g. l
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
1117
Development of Batch Process Operation Management Platform Atsushi Aoyama*, Isao Yamada+, Rafael Batres*, Yuji Naka* *Tokyo Institute of Technology, Research Laboratory of Resources Utilisation, Nagatsuta Midori-ku Yokohama 226-8503 Japan, Tel: +81-45-924-5271, Fax: + 81-45-924-5270, Email: [email protected] [email protected] +Yamatake Corporation, 1-12-2 Fujisawa-shi Kanagawa 251-8522 Japan, Tel: +81-466-20-2430, Fax: +81-466-20-2431, Email: [email protected] The chemical and biochemical industries face an intense pressure to improve the efficiency, product quality, and human safety, whilst reducing the environmental impact of their operations. Under these circumstances, the batch processing 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 this research, a batch process operation management platform is designed based on ANSI/ISA-S88 and the multi-dimensional object oriented model (MDOOM). The proposed scheme shows a number of advantages such as a systematic and automatic generation of control recipes, design rationales usable throughout the lifecycle of batch process operation, a guaranteed feasibility of control recipes, improved modularity and transparency of auxiliary operations and an easy integration with a simulator. 1. Introduction The chemical and biochemical industries face an intense pressure to improve the efficiency, product quality, and human safety, whilst reducing the environmental impact of their operations. Under these circumstances, the batch processing 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. It is well known that the operation management of batch processes is much more difficult than that of continuous processes and current ways of batch process operation management are tedious and error prone. ANSI/ISA$88 [1] tried to achieve a better batch process operation management through establishing standard models, terminology, data structure and guidelines for language used by batch process control. In this paper, the expansion and modification of ANSI/ISA-88 have been done and a batch process operation management platform has been proposed. The proposed platform is based on the multi-dimensional object oriented model (MDOOM) framework where the plant structure, operation, process behaviour and operation management are modelled independently. This characteristic makes the clear categorisation and separate handling of plant information reconfiguration management, process management and unit management possible and supports a better operation management. The next section describes the differences between continuous and batch process management and problems of current batch process operation management. Section 3 explains the models used by the operation management platform, the correspondence between models and the structure and the function of operation management platform. Section 4 explains the advantages of proposed platform. Section 5 briefly summarizes the achievement done in this research.
1118 2. Problems of current batch process management
It is generally agreed that the operation management of batch processes is much more difficult than that of continuous processes. But why is the batch process operation management so difficult? In the continuous process operation, one-to-one correspondences between processes and process units are established and the relationship is fixed and does not change over time. For example, if an operation is sequenced as Reaction -> Separation -> Purification -> Pelletizing then the corresponding units are also sequenced as Reactor-> Distillation Column -> Membrane -> Pelletizer. It is not necessary to separate the process management and the unit management. On the other hand, in the batch process operation, operations and states of operations carried out in the plant are changing continuously due to scheduling. In order to carry out batch process operations, the plant information has to be reconfigured dynamically so that it corresponds to the current operation status. This requires the management of plant information reconfiguration as well as the separate handling of process management and unit management. Those requirements make the batch process operation management complicated and difficult and lead to the necessity of clear modelling of operation and operation management. Current ways of batch process management are not satisfying those requirements and causing a number of problems. For example, the operation is usually expressed as a sequence function chart (SFC) that elaborates the manipulations of individual valves, pumps etc. in which both the process operation information and the plant structure information are embedded implicitly. It is often the case that a configuration of SFC from the scratch is easier than the modification of it when a change occurs in the plant structure or the operation procedure. Also, a resulting SFC is basically a one-dimensional sequence of manipulation and lacks the ability to hold design rationales used at the operation deign stage. Therefore it is not possible to use that kind of information at the stage of real-time operation and operation analysis. For example, even though the real goal of operation is 'increase the tank level while keeping the temperature steady, the operation expressed in the SFC is something like 'open valve 1 for 30 minutes and turn on the heater a~er 4 minutes and the original rationale of operation is lost when it is executed. In order to model operations clearly, ANSI/ISA-S88 defines three kinds of recipe (general recipe, master recipe, control recipe) corresponding to their levels of abstraction. The general recipe is an operation model not depending on any specific plant, the master recipe is an operation model assumed to be carried out at the specific plant and the control recipe is an operation model after each operation is assigned to the specific unit and used for the manufacturing of the batch. ANSI/ISA-S88 also defines a layered structure of operation; procedure, unit procedure, operation and phase. The management function is also modeled as a multiple layered structure in ANSIflSA-S88. One of the shortcomings of ANSI/ISA-S88 is that the separation of plant information reconfiguration management, process management and unit management is still ambiguous. For example, the specification of operation procedure in each recipe is not defined elaborately enough and how to relate the layered structure of operation procedure and the plant structure topology are not clearly defined in ANSI/ISA-S88. 3. Modelling based on MDOOM
It is important that the batch process operation management platform is constructed based on the internationally accepted model structure, terminology and data structure. In that sense, ANSI/ISA-S88 is very important because it defines the standard of models, terminology, data
1119 structures and guidelines for language of batch control. The models, terminology, data structure of the proposed system are defined strictly in accordance with ANSI/ISA-S88. At the same time, the analysis in the previous section shows that the expansion and modification of ANSI/ISA-S88 is necessary to facilitate the separate modelling of plant structure and operation. This section describes the modelling based on the multi-dimensional object oriented model in accordance with ANSI/ISA-S88. The multi-dimensional object oriented model (MDOOM) provides a conceptual and generic framework for supporting concurrent engineering of chemical products, their production processes and their associated plants [2]. The MDOOM framework allows the identification of the physical, the operational and the behavioral information as separate dimensions. This is particularly important in the batch process, in which operations carried out in physical units change dynamically. The model in behavioral dimension is considered to be no different to the one in the continuous process. So, in this section only the operation and the plant structure models are explained 3.1. R e c i p e
The operation model (recipe) specifies recipe procedures, formula and equipment requirements. Three kinds of recipes are defined. The general recipe contains the information to produce a unit of the specific product at yet unspecified plant (cell). The master recipe contains the information to produce a unit of the specific product at the specific plant (cell). The equipment requirements of master recipe only specifies equipment types to which operations are applied. 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 also has layered structure, Procedure, Unit procedure, Operation and Phase as shown in Figure 1. 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. A procedure (Procedure, Unit procedure, Operation and Phase) has a precondition and a post-condition. A precondition establishes the triggering criteria and a post-condition establishes the termination criteria. The activation of procedure changes the state of the structure (such as opening a valve) and the behaviour comes out from such changes. !
Pr~
J I
Procedure2 I
1 "'" I ,
ProcedureL
)
I
[ Unit procedure l I [ Unitprocedure 2 I --- [UnitprocedureM I I
I erati~
1[ Operation2 1
1 [
Phasel
[
I ,
J I
Phase2
I
Operation N
J
I
j
... [
Phase O
1
Figure 1 Recipe procedures 3.2. P l a n t s t r u c t u r e
model
The plant structure belongs to the physical dimension and refers to the description of the
1120 components of which the plant is built as well as their topological representation (Figure 2). 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. Please note that unlike for continuous plants the CGUs and Units are reconfigured continuously and dynamically during the operation of batch plants. The equipment information is expressed following the ISO-10303 AP221 standard. This provides an interface to an unlimited number of computer aided drafting and design tools and makes it possible to utilise the equipment design information for the control recipe generation and real-time operation. ,
Cell I
,
[ Unit(CGUswithacommonmainECGU) )
-
......I
[ Unit ] ( ,Unit )
[ i i (CGU(AmainECGU+peripheralECGUs) I [ CGU '] ~CGU ] I
'
[
,
I
,
"
,I
,
I
","'l
I
Equipment (e.g. valves, tanks, pumps, pipes etc.) Figure 2 Plant structure model
]
3.3. Correspondence between recipe procedure and plant structure
One of the unique contributions of this research is that the exact correspondence between the recipe procedure and the plant structure is defined (Figure 3). A procedure is an entire process necessary to manufacture a batch corresponding to the Cell. A unit procedure is a process
[ ProcedureA ] ~
Unit31) "q,"-fUnit5]
Unit 3 . [ Unitprocedure A2 ]
.
Main ECGU 3 L_
.
.
+ ( ECGUsusedinA2 1 ,
CGU 32 ( OperationA22 '] IiMainEeGU3"] +[ ECGU2 ] +[ ECGU3 11 [
Phase221 1 ( A specific equipmentin CGU 32 J Figure 3 Correspondence between procedures and plant structure
1121 1121
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. 3.4. Operation management platform As described in the previous section, one weak point of ISA-S88 is that the separation of the plant information reconfiguration management, the process management and the unit management is incomplete. Here, those three kinds of management are defined as functions of the operation management platform. The lower part of 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 multibatches through the management of procedure execution and executes procedures through the management of unit procedures. This function is corresponding to the process management. 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 explained as 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. Another function of unit supervision is to generate Units and CGUs, it is considered to be a part of the plant information reconfiguration management. Finally, Phase execution executes phases by sending commands to DCS. Some equipment such as model predictive controller can be handled as black boxes. In such a case, the equipment receives the command from Process management or Unit supervision directly.
4. Advantages of platform In this section, the advantages of the batch process operation management platform proposed in this paper are explained. First, the layered structure of batch process operation has a complete correspondence to the structure of plant following the expression of STEP (STandard for the Exchange of Product model data) ISO-10303. This makes the systematic generation of control recipes possible and a simple transformation logic can generate control recipes automatically. A control recipe is generated based on a master recipe by the following simple steps. 1 Calculate formula with batch specific information 2 Search necessary equipment 3 Search a charge route 4 Open all valves on the charge route 5 Isolate the charge route by closing necessary valves 6 Operate the pump on the charge route If an operation “Charge 20*(Batch Size kg) of A to Reactor” in a master recipe is given for a plant structure shown in Figure 4, each step will generate following procedures. 1 Charge lOOkg of A (when Batch Size = 5kg) 2 Charge Tank A to Reactor R 3 A route from Tank A to Reactor R 4 Open V l l , CVlO, V18 5 CloseCV20 6 Operate Pump 101 And when changes in plant structure occur, a corresponding procedure is automatically changed appropriately. For example, if Valve V11 in Figure 4 is removed from the plant, a corresponding phase is automatically changed to “Open CV10, V18” instead of “Open V11,
1122 CV10, V18" without changing above described generation steps. Secondly, the design rationales used to generate control recipes are expressed in the control recipe as the layered structure of procedure and pre and post-condition of each procedure. That information can be used at the stage of real-time operation. Also when a control recipe is generated the operations as elaborate as the phase level are always assigned to equipment including utility equipment. Therefore, the feasibility of control recipe is guaranteed before the execution of recipes. Another advantage is that the auxiliary operation such as the equipment cleaning, safety protection and exception handling is expressed as a procedure, unit procedure or operation. Because the operation procedure has a layered structure, when a fault occurs, it is easy to identify that in which layer it is taking place. It improves the modularity and transparency of auxiliary operations. Finally, in the proposed platform, not only the operation but also the operation management itself is expressed as a model therefore it can work as a part of dynamic simulator. Such a simulator constitutes a simulation environment in which different ways to operate a plant can be explored and evaluated with the same models. CV10
CVl0
Pump201 Figure 4 Plant structure examnle 5. Conclusion
In this research, a batch process operation management platform is designed. The proposed platform is compliant to ANSI/ISA-S88 but superior to the original ANSI/ISA-S88 specification because of a better modelling scheme based on the MDOOM, the clear definition of management structure and the well defined correspondence between operation procedure and plant structure. It is shown that the proposed scheme improves the batch process operation management task in a number of points. The analysis of applicability of the scheme to a larger and more realistic batch process operation is underway on the testbed in G2 object oriented modelling environment [3]. Reference
1. ANSI/ISA-S88.01-1995 Batch Control Part 1: Models and Terminology 2. R. Batres, Y. Naka and M. L. Lu, 'A Multidimensional Design Framework and its implementation in an Engineering Design Environment presented at ISPE International Conference on Concurrent Engineering, Tokyo, 1998 3. Gensym Corporation Inc. 'G2 Version 5.1 Reference manual
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
Synthesis, experiments
and simulation of a heterogeneous
1123
batch distillation
process Rodriguez-Donis 1.1,2, Pardillo-Fontdevila E. 2, Gerbaud V. 1, Joulia x. 1 Laboratoire de G6nie Chimique (LGC, UMR CNRS 5503), I N P T - 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 The presence of azeotropes in multicomponent mixtures complicates the design of batch distillation separation processes widely used in pharmaceutical and speciality chemical industries. Most of those processes include the use of homogeneous entrainer to ease the separation. We describe novel methods to break azeotropes using an entrainer that is partially miscible with one of the initial binary mixture components. We depict some of the advantages of heterogeneous batch distillation processes: more design alternatives for the separation of an azeotropic binary mixture than with homogeneous batch distillation, simplified distillation sequences as a consequence of less distillation tasks. Three examples based on the separation of non-ideal azeotropic or close boiling point binary mixtures through heterogeneous batch distillation are given. Experiments and simulations are performed for the separation of a minimum boiling azeotropic m[ixture. 1.
INTRODUCTION
Batch distillation is a common solvent recovery technology in pharmaceutical and speciality chemical industries and help deal with increasing economic incentives and environmental regulations. Azeotropes in waste-solvent streams complicates both synthesis and design steps of batch distillation processes. Azeotropic ternary diagram may display distillation boundaries that outline distillation regions. Hence, batch distillation products sequence depends on the initial feed in the still 1 and separation of a binary mixture into its pure components may require several batch distillation tasks. The complexity of azeotropic batch distillation synthesis and design steps has implicitly restricted the choice of the entrainer added to the azeotropic binary mixture to homogeneous one and thus bounded the number of process alternatives. The use of a heterogeneous entrainer introducing a phase split with one of the binary mixture component will likely increase process choices and even achieve azeotropic separations impossible with homogeneous systems. The scarcit3T of published work on heterogeneous batch distillation (HBD) 2,3 and industrial interests prompted us to address the feasibility of batch distillation using a heterogeneous entrainer for the separation of azeotropic and close boiling binary mixtures. Entrainer choice and residue curve maps analysis under hypotheses of total reflux ratio and infinite theoretical stage number aims at finding the product sequence and selecting the batch tasks needed. Then batch process parameter optimal values are looked at. But they depend on the entrainer choice because it determines largely the azeotropic distillation process efficiency. Thermodynamic 4, product sequence 5 and batch tasks 6 assessment tools are already available for batch distillation process synthesis. The entrainer screening issue for homogeneous batch azeotropic distillation has been looked at 1,2 but was incomplete until a recent paper addressed the topic and that of HBD 7. In this paper, we intend to demonstrate a few advantages of HBD processes through several examples which feasibility is assessed and demonstrated through experiments and simulation with a batch process simulator, ProSimBATCH 8.
1124
2.
I N T E R E S T OF H E T E R O G E N E O U S
BATCH DISTILLATION
Rodriguez-Donis et al. set up HBD process synthesis key points 7: 9 A complete set of rules for the selection of entrainers can be defined from the analysis of all feasible ternary diagrams. As a general rule, a potential entrainer is defined as a component partially miscible with one of the initial mixture component and which generates liquid- liquid tie lines that cross the batch distillation boundaries dividing the regions where the original components are. 9 Comparison of heterogeneous versus homogeneous systems shows that the number of ternary diagrams that enable feasible batch distillation tasks is more than 3 times superior for the heterogeneous case. Hence, design alternatives are increased. Besides, the liquidliquid phase split may reduce the distillation tasks number. 9 Assuming a complete phase split, the separation of the original mixture components can always be carried out with less than four batch distillation tasks involving either one column configuration (rectifier or stripper) or a combination of both column types 6. A typical feasibility assessment is presented below. It especially helps select the batch distillation region where the feed should be. 3.
EXAMPLE OF HETEROGENEOUS
BATCH DISTILLATION
Feasibility of HBD is illustrated with the separation of the water - acetonitrile azeotropic binary mixture with the addition of acrylonitrile as a heterogeneous entrainer. Experiments and simulations are carded out to validate the product sequence and the still path predicted. A w a t e r - acetonitrile minimum temperature azeotrope is to be separated using acrylonitrile as an entrainer (figure 1). Acrylonitrile forms a heteroazeotrope with water. Singular points stability and boiling temperature are shown on the ternary diagram on figure 1. [sn] stands for stable nodes; [sa] for saddle and [un] for unstable nodes. Two batch distillation boundaries join the heteroazeotrope to the acetonitrile vertex and to the homoazeotrope. Three batch distillation regions overlapping the L-L-V zone are defined where the product sequence is unique. The feasibility analysis tells us that two batch column configurations - stripper or rectifiercan be thought up to separate the w a t e r - acetonitrile azeotropic mixture. 9 The stripper column is fed in the decanter with mixture A (usually the water-acetonitrile homoazeotrope) plus organic phase L1 and acrylonitrile. The resulting ternary feed F rests in the batch region containing the entrainer vertex. While processing, pure acetonitrile is completely removed as bottom product P1. At the end of the single batch task, two liquid phases stay in the distillate drum. Phase L2 containing no acetonitrile but water (molar purity = 95%) and acrylonitrile is removed as a product P2, to undergo further purification if needed. The entrainer-rich phase L1 may stay in the decanter for a new cycle of separation water acetonitrile. Entrainer make-up could compensate losses during P2 removal. 9 The rectifier column is fed in the still with a ternary mixture located in a batch region different from the stripping one. Aqueous phase L2 is removed as product P1 from the decanter whereas the entrainer rich phase L1 is refluxed. After the single batch task, the still contains pure acetonitrile taken as product P2. The phase L1 is then put back in the still for a new cycle. The lowest amount of entrainer added to the w a t e r - acetonitrile mixture is obtained for the rectification case as seen on figure 1 diagrams: point F position is closer to the entrainer-free side in the rectifier case. Notice that using a homogeneous entrainer inducing the same residue curve diagram is impossible. Distillation would leave an entrainer- water azeotropic mixture after the first distillation task that would have to be separated again. This proves that HBD increases the number of process alternatives for the separation of azeotropic mixtures.
1125
Figure 1. Heterogeneous batch distillation feasibility (water- acetonitrile- acrylonitrile system). Batch experiments are carried out in a small laboratory column made of 50 plates (including still and condenser). Liquid plate hold-up amounts to 1.25 ml. and to 50 ml. in the decanter. Condenser is subcooled to 25~ A phase split can occur in the column and in the decanter where the aqueous phase is removed while the organic phase is refluxed into the column. Pressure equals 1 atm. and a 0.05 atm pressure drop is estimated. Boiler heat duty equals 205 W. After one hour of total reflux operation, distillation starts. It ends after 2.5 hours as overhead condensed vapour becomes homogeneous. Overhead vapour temperature stays around 72-74~ The decanter heavy phase is poured into the aqueous distillate then analysed. Decanter, reflux pipe and condenser pipes light phases are mixed together and analysed. Distillation is then continued without reflux to recover as much entrainer as possible in shortest operation time. This new distillate cut is related to an overhead vapour temperature between 74-81~ and is obtained in less than 5'. Finally, the still content is weighted. Analysis of the still composition is performed every 15 rain. Analysis of the decanter content is made at the end of the distillate operation. Gas chromatography is used to measure the acetonitrile and acrylonitrile contents. Karl Fisher technique is used for the water content determination. Compositions results are in molar units and quantifies are in mass units. The mean values from three experiments obtained are shown in table 1. Figure 2 shows the still composition evolution with time during the first distillation step. Water is quickly removed from the still and almost pure acetonitrile remains. A more detailed analysis of the results is given below in comparison with the simulation results.
tO
".~
o
0.9 0.8 0.7 0.6
~_ 0.5
~
0.,t7 0.3 0.2
0.1 0
0
0.5
water
. 1 1.5 Time (h)
~ 2
2.5
Figure 2. Time evolution of the still molar liquid compositions during the heterogeneous batch rectification of a w a t e r - acetonitrile- acrylonitrile ternary mixture. Calculated values: ~ Experimental points: [] acetonitrile zx acrylonitrile o water
1126
SIMULATION RESULTS quantity (g)
molar fraction
EXPERIMENTAL RESULTS quantity (g)
molar fraction
,
INITIAL STILL CONTENT 672.85
H20 C2H3N C3H3N
0.262 0.673 0.065
672.85
H20 C2H3N C3H3N
0.262 0.673 0.065
96.2
H20 C2H3N C3H3N
0.948 0.033 0.019
108.3
H20 C2H3N C3H3N
0.918 0.051 0.031
60.2
H20 C2H3N C3H3N
0.174 0.204 0.622
70.3
H20 C2H3N C3H3N
0.076 0.356 0.568
38.6
H20 C2H3N C3H3N
0.032 0.600 0.368
35.6
H20 C2H3N C3H3N
0.014 0.663 0.323
42.76
H20 C2H3N C3H3N
3.10 .8 0.994 0.006
435.1
H20 C2H3N C3H3N
3.10 .5 0.994 0.006
HEAVY AQUEOUS PHASE (P2) Distillate + Decanter head= 72 -- 74~
T
....
LIGHT PHASE (REFLUX) Decanter (+ pipes during experiments) T head = 7 2 - 74~ ....
Distillate 2 nd cut T . . . . head = 74 -- 81~
PLATES HOLD-UP Not measured
FINAL STILL CONTENT
(PRODUCT Pt)
C2H3N mass recovery yield
83.2%
408.6
H20 C2H3N C3H3N
0.0010 0.9985 0.0005
78.7%
Table 1. Experimental and calculated molar compositions and quantities during heterogeneous batch rectification of a water- acetonitrile - acrylonitrile ternary mixture. Batch distillation simulation is done using ProPhyPlus (properties server) and ProSimBatch (batch process simulator) 8. Column technological features and operating conditions described above are input in the simulator. The UNIQUAC model is used to represent phase equilibrium with binary parameters taken from the DECHEMA tables. Calculated equilibrium consistency is checked with ProPhyPlus against experimental data available in the DECHEMA tables. A slight curvature of the heteroazeotrope - homoazeotrope boundary is noticed, in particular near the homoazeotrope point. The temperature dependent liquid - liquid - vapour envelope holds on the column plates. The liquid- liquid envelope at 25~ applies for the subcooled condenser and for the decanter. 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 8. Numerical treatment of the Differential Algebraic Equation (DAE) system and discrete events handling is performed with DISCo, a numerical package for hybrid systems with D A E solver based on Gear's method 9. Reflux ratio is set to 6 at the beginning of the distillation step. Afterwards, it gradually increases to keep a constant overall level in the decanter, as set in the experiments. Simulated operating conditions are: total reflux operation step duration is set to l h like in the experiments. Then, heterogeneous distillation step is undertaken for 2.5h during which aqueous phase is removed. Later, a second batch homogeneous distillation step is performed without reflux to recover the remaining acrylonitrile into the column. It is ended when the overhead temperature reaches 81~ The calculated step duration equals 3 minutes and is related to a final acetonitrile composition in the still of 0.994.
1127
Figure 3. Calculated liquid composition profiles for the heterogeneous batch rectification of a w a t e r - acetonitrile- acrylonitrile ternary mixture. Simulation results are displayed in table 1 and on figures 2 and 3. Figure 3 displays the main features of the ternary diagram and the evolution of the column liquid profile during the distillation step operation. As predicted, the still path moves from its initial position towards the acetonitrile vertex. At t = 0 h, several plate compositions are inside the liquid - liquid - vapour envelope. Condensed vapour composition lies inside the 25~ liquid- liquid envelope but close to the entrainer rich side. Distillate is almost pure water. Applying the lever rule, the aqueous phase volume in the decanter is always lower than the entrainer rich phase one. The condensed vapour path tells us that this feature is more pronounced as distillation proceeds. Then, with the condition of constant decanter volume, the distillate flowrate decreases while the reflux flowrate increases along the distillation. The peculiar twist of the condensed vapour path is related to a overhead temperature that decreases slightly from 72.4~ to 72.1~ during the first 2 hours of distillation when the acetonitrile that filled the plates after the total reflux operation moves down to the still. Then overhead temperature increases to 74.2~ Comparison of Table 1 and figure 2 demonstrates that the experimental and simulation results agree well and that no major discrepancy is noticed. The following remarks are made: 9 Experimental and simulated product streams compositions exhibit the same features after equivalent duration. The experimental acetonitrile recovery yield also agrees to the calculated value by simulation. This validates the feasibility predictions that acetonitrile and water are split in two high purity phases using a heterogeneous entrainer. It also proves the capacity of the simulation model to describe the batch distillation process. 9 Experimental and simulated quantities and compositions of the heavy aqueous phase removed as product Pl are similar. It validates the thermodynamic representation of the liquid - liquid equilibrium prevailing in the decanter. 9 The experimental quantity of light phase obtained in the decanter at the end of heterogeneous batch distillation and the cut of distillate rich in entrainer are similar to estimated values by simulation. One notices also that less than 10% of the initial acrylonitrile leaves the column with the heavy phase Pl and none with the product P2. It indicates the amount of entrainer make-up that should be added before each stage of a cycle of batch water - acetonitrile separations performed sequentially in the column. 9 The acetonitrile recovery yields is far from 100% because the 2 nd distillate cut rich in acrylonitrile takes some acetonitrile from the still and acrylonitrile is replaced on the plate by
1128
acetonitrile. It will likely improve if a cycle of batch water - acetonitrile separations is performed sequentially, as the plates are not emptied at the end of each cycle stage. In conclusion, the simulation and the experiments validate the predictions of the feasibility study and shows that for this particular example, acetonitrile and water can be separated in two product phases with a significant purity by rectification. The amount of entrainer needed for this operation is low and the make-up for a cycle of w a t e r - acetonitrile separation is reasonable. A similar feasibility study and simulation was conducted for the separation of the maximum temperature azeotrope w a t e r - formic acid using diisopropyl ether as an entrainer. A single stripping operation was needed to obtain greater than 98% grade formic acid (88% molar recovery yield) as product 1 and water (with diisopropyl ether impurity) as product 2. Again the amount of entrainer needed for the operation was low (x < 0.1 in the feed F). Heterogeneous batch distillation is also convenient for the separation of the close boiling point binary mixture w a t e r - acetic acid by adding ethyl acetate. The feasibility analysis shows that a stripper configuration can be used to obtain pure acetic acid as bottom product. At the end of the process, two liquid phases stay in the distillate drum. Phase L2 contains no acetic but mainly water (molar purity -- 93%) with ethyl-acetate as an impurity. It is removed as a product P2, to undergo further purification if needed. The entrainer-rich phase L1 is recycled in a new cycle of separation water - acetic acid. Entrainer make-up should be supplemented to compensate the loss during the aqueous phase removal. Notice that if the entrainer was homogeneous, the batch distillation could be performed with a single stripper configuration but with two distillation tasks instead of one for the HBD process. 4.
CONCLUSIONS
The use of a heterogeneous entrainer increases the interest, the number of alternatives and the profitability of batch distillation processes. An application to the separation of solvent-waste mixtures by using the HBD is presented. It concerns the separation of the w a t e r - acetonitrile azeotropic binary mixture with the addition of acrylonitrile as a heterogeneous entrainer. Experiments and simulations are carried out. They are both well in accordance and validate the predictions of the feasibility study. They show that both water - acetonitrile mixture components can be separated in two product phases with a significant purity during a single distillation task. The amount of entrainer needed for this operation is low and can be recycled in a separation cycle. All these features help keep the economics of the heterogeneous batch distillation process quite attractive. REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9.
C. Bernot, M.F. Doherty and M.F. Malone, Chem. Eng. Sci., 46 (1991) 1311. R. Dtissel and J. Sfichlmair, Comp. Chem. Eng., 19 (1995) $113. J. K6hler, H. Haverkamp and N. Schadler, Chem. Ing. Tech., 67 (8) (1995) 967. H.N. Pham and M. F. Doherty, Chem. Eng. Sci., 45 (1990) 1823. B.S.Ahmad, Y. Zhang and P. I. Barton, AIChE J., 44 (1998) 1051. B.T. Safrit and A. W. Westerberg, Ind. Eng. Chem. Res., 36 (1997) 1841. I. Rodriguez-Donis, V. Gerbaud and X. Joulia, submitted to Ind. Chem. Eng. Res. (1999). ProSim SA (France). http://www.prosim.net. A. Sargousse, J.M. Lelann, X. Joulia and L. Jourda, Proceedings of MOSIM 99, (1999) 61.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
Robust Mixed Stochastic Sequencing Problems
Enumerative
1129
Search
Technique
for
Batch
Mois6s Graells, Jordi Cant6n and Luis Puigjaner Department of Chemical Engineering. Universitat Polit6cnica de Catalunya E.T.S.E.I.B., Diagonal 647, 08028-Barcelona, Spain e-mail: [email protected], [email protected], [email protected] This paper addresses the sequencing problem in the case of batch chemical processes. This is a kind of Asymmetric Travelling Salesman Problem (ATSP) commonly characterised by the fact that the distance or cost matrix can not be defined. Such problems are usually treated using simulation, and simulated annealing (SA) results a convenient optimisation technique. This work revises SA and proposes a mixed search technique that combines random search for fast down hill moving with enumerative search for local optima identification, the latter leading to the strictly necessary up hill moves. For batch sequencing problems (ASTP-O(n!)) local optimum identification has proven to be affordable - O(n2). Finally, different case studies, from non-constrained examples to complex cases subject to storage constraints, are presented. Results obtained show a good performance of the proposed technique, which features also greater simplicity and robustness as well as deeper knowledge of the problem. 1. INTRODUCTION The sequencing of jobs in multiproduct batch chemical plants is a blind ASTP, for which the changeover matrix is not available. The reason for this is mainly due to the same structure of operations involved in each job, which may involve the simulation of each pair of consecutive batches to obtain the corresponding changeover time (including clean-up times). Moreover, this situation worsens when such changeover times depend not only in the pair involved but also on other jobs in the series. This is the case when jobs do not share all the same equipment units or they consume the same limited general utilities (steam, manpower, etc). Hence, Simulated Annealing has shown to be a good option for solving this kind of blind ATSP, since it is based on the simulation of each one of the sequences proposed so that they can be validated, evaluated and eventually accepted. Ku and Karimi (1991) first applied SA to multiproduct problems (unconstrained job sequences). More recently, Graells et al. (1996) used SA for solving a kind of multipurpose problems (constrained mini-job sequences). Despite its many advantages, SA has some remarkable drawbacks, the main one being a technique too cautious and conservative. The key idea of SA is to allow up hill moves just in case that a local optimum is reached. However, up hill moves are still allowed even if this local optimum is not reached simply because such a situation is never detected by SA algorithms. Hence, experience shows that, when the probability of falling trapped in a local optimum is very low, the most practical option and the one leading to the best results is often down hill random search (zero temperature). This is the case, for example, when very little computing time is available for improving a bad starting point. It is not until the situation becomes riskier
1130 (as better solutions are attained) that the need for higher temperatures allowing up hill moves seems to arise. Several approaches have been proposed to accelerate the annealing procedure. The NonEquilibrium Simulated Annealing (NESA) proposed by Salcedo et al. (1995) addresses specifically the excessive caution of SA but still from a probabilistic point of view and assuming a temperature decreasing scheme. In the NESA approach "thermal" equilibrium is not necessarily attained at each contraction of the "temperature" control parameter, since the cooling schedule is enforced as soon as an improved solution is obtained. However, the point is still why allowing backward moving when the search is progressing in the desired direction. Solution to this problem could be a combined "cooling/heating" scheme as proposed by Dowsland (1993). However, the problem is still when start heating and how (heating schedule) In any case, such solution approaches give up the original simplicity of SA and better performances are attempted at the expense of increased strategy complexity. On the other hand, this should be considered in addition to the parameter-tuning problem. The determination of a good cooling scheme (Laarhoven et. al. 1987), which is based on previous sampling, may be not affordable in many common situations that require practical solutions within a limited computing time for the every day changing problem (model changes such as parameters, constraints, objective function, etc.). 2. LOCAL AND GLOBAL OPTIMA Global optimum definition is absolute: it is the best solution among all possible solutions. The existence of local optima, however, is a consequence of the limitations of the optimisation techniques and depends on the search procedure employed and the step-size adopted. A local optimum is the best solution among all possible surrounding solutions, attainable from a certain starting point given a search technique and a step-size. Hence, the change of the search procedure may likely result in the change of the local optima obtained for the problem considered (Fig. 1).
Fig.1. Local optima are consequence of the Fig. 2. Sub-spaces explored by the MSES limitations of the search procedure and the technique. step-size (s) employed. Thus, the strategy presented in this work for the TSP is based on stochastic search, but considers the exhaustive search of the surroundings at each step in order to determine if
1131 progress is no longer possible with the current searching procedure (local optimum). When such a situation is detected, this procedure (any procedure) must obviously be replaced. 3. M E T H O D O L O G Y The step-size adopted for the Travelling Salesman Problem (TSP) is usually a change in the sequence defining the problem, which is given by the permutation of two of its elements or by the reallocation of one of the elements in the sequence to another position. The latter is the one producing the least modification in the sequence and the one adopted in this work. Evolutionary search techniques as SA need short step sizes. Otherwise, the larger the stepsize, the lower the chance of finding local optima, but the closer the search to pure random. For a sequence of n elements this means a solution space f2 of n! possible sequences, which cannot be exhaustively explored (Fig. 2). Given the shift of one element of the sequence as the basic step-size, the Mixed Stochastic Enumerative Search (MSES) presented proposes the exhaustive enumeration of the surroundings of each point attained, which correspond to the subsets (Di in figure 2. The enumeration of n(n-1) states in o31 is affordable for the search algorithm proposed (Fig.3). Furthermore, the enumeration is usually not completed because as soon as a better solution is obtained it is proposed as a new candidate for local optimum. Hence, this point is taken as the centre of a new sub-space (092) to be exhaustively explored.
i Initial sequence:n = 1, 2..
14 I
,N
!
[" Setj=lind k=2 I Moveelementj to positionk [
~ U~omo~vel '
~ ~
J
Acceptnew I sequence
no
I no
I Recordbestsequencefound I
~ M o v e ~[no
twice l randomly i__~
I END:Bestsequencefound I Fig. 3. Mixed Stochastic Enumerative Search (MSES) algorithm for unconstrained sequences.
1132 Otherwise, if the enumeration is completed the result is the detection of a local optimum. Therefore, no more single step moves make sense and a farther move is necessary. This is given, for example, using two random moves as illustrated in figure 3. The algorithm in figure 3, however, corresponds to the non-constrained case, in which all sequences are feasible. For the constrained problem the same scheme is still effective, but a checking stage must be included. In such a case, considering pre-ordering constraints (Pinto and Grossmann 1996, Rodrigues et. A1. 1999) may result in increasing the efficiency and speed of the procedure, as the sub-spaces to be explored may be significantly reduced. However, constrained situations may lead to non-convex problems, for which some subspaces (c03) may not be reachable from the currently explored sub-space (6o2). In order to cope with this situation, as well as for increasing flexibility, the implementation of this strategy includes the option for running several times the procedure starting from different random seeds. Put into thermal terms, the strategy proposed consists of fast "freezing" of the system considered with heating when necessary. This is given by the exhaustive exploration of the surroundings of each point, which allows both, the acceptance of every improving change as well as the confidence that the procedure will always be able to escape of local optima. 4. CASE STUDY The following case study is an unconstrained ATSP with 20 cities whose distance or cost matrix has been randomly determined in the range 0+1000 (average circuit distance is 10000). Additionally, the distances for the pairs in the sequence 20-19-18...3-2-1 have been set to 1 so that the optimum solution be known (minimum distance 21). Results are summarised in figures 4a and 4b.
Fig. 4. Average results obtained for different random search procedures following Metropolis criterion compared with the MSES procedure, a) Random search returns to previous solution. b) Random search returns to best solution found. Several computational experiments have been carried out to compare the performance of the MSES approach with random search under different conditions. Figure 4 corresponds to the plot of final objective function attained, OF, after 2.10 6 iterations versus the probability, P, for accepting a 10% positive change of such objective function, A(OF), following Metropolis criterion:
1133
P : ex~-A(OF)/T
(1)
Thus, each column in the plot corresponds to an "isothermal bath" performed at temperature: y
-0.1
. _ .
lnP
(2)
When discarding a change, the procedure may return to the previous solution or to the best solution found during the search. Figures 4a and 4b correspond respectively to these situations. In both cases, six executions at each given probability are compared with other six MSES executions for which the mean value is plotted. Each execution was limited to 2.0E6 iterations (in front of 20! = 2.4E18 feasible solutions). Results show how the MSES approach led readily to the known optimal solution while stochastic search following Metropolis criterion may lead to close performance only if accurately tuned. 5. INDUSTRIAL APPLICATION The MSES approach has been used in the solution of the scheduling problem at an ABS polymer manufacturing plant. The problem addressed is the scheduling of jobs in three semicontinuous pelleting lines at the end of the process. The process scheduling of polymerisation, blending and drying sub-plants under technical criteria, establishes the availability profiles for the intermediate materials required by the pelleting lines. Finally, duedates and colour sequencing are the other main elements to consider in this scheduling problem. The problem has been formulated as an ATSP supported by simulation and has been addressed using SA to improve an initial feasible sequence given by a simple heuristic rule. However, the earlier first prototypes were tested by the pelleting plant managers, the sooner they learnt that zero temperature was the option leading to the most satisfactory solutions.
Fig. 5. User interface for running a Mixed Stochastic Enumerative Search (MSES).
Fig. 6. Square performance profile obtained withthe execution of a MSES procedure.
Of course, this is because the lack of computation time and the help of interactive tools provided in the prototype (Electronic Gantt Chart for manual changes, run, pause, reset buttons to control the search procedure, a graphical display for the evolution of the procedure - figures 5 and 6). Thus, the MSES algorithm was introduced in the prototype in order to enhance the
1134 efficiency of such a working practice. This solution approach allows the fast practical answer required and guarantees that the search procedure will not be stopped in an undetected local optimum. Figure 6 shows the evolution profile of the cost objective function corresponding to a case. The system also provides the number of local optima found during the search and certainly, SA is not deactivated, but kept as another tool within a flexible system. 6. CONCLUSIONS The Mixed Stochastic Enumerative Approach (MSES) has been introduced as an alternative procedure for stochastic optimisation. This technique has proven to be very useful for complex problems that are currently addressed via simulation and decision-making based on trial and error. This is the case of job sequencing in batch chemical processes that can be formulated as blind ATSP for which the changeover matrix is unknown. In opposition to SA, the procedure presented does not seek to elude local optima but searches for them, thus leading to improvement and providing information on whether or not further improvement is possible. For such a reason local optima meant no risk for the search. This approach is general enough, although local optima identification may be strongly related to the procedure adopted for moving through the solution space in each particular kind of problem. The procedure is very practical and robust because no parameter tuning is required and feasible solutions are always attained. Industrial application has also proven the good performance of the procedure implemented into and interactive decision-making tool. Finally, from theoretical point of view this procedure provides the number of local optima found during the search, which results in useful information providing a deeper knowledge of the problem. 7. ACKNOWLEDGEMENTS The support of the European Community (BRPR-CT98-9005) and CICYT (QUI99-1091) is fully appreciated. 8. REFERENCES Cardoso, M.F., R.L. Salcedo, S.F. de Azevedo, "Nonequilibrium simulated annealing: a faster approach to combinatorial minimisation", Ind. Eng. Chem. Res., 33, 1908-1918 (1994). Dowsland, K.A., "Some experiments with simulated annealing techniques for packing problems", European Journal of Operational Research, 68, No. 3, pp. 389-399 (1993). Graells, M., A. Espufia, L. Puigjaner, "Sequencing Intermediate Products: A Practical Solution for Multipurpose Production Scheduling", Comput. Chem. Engng., $20, Sl137-Sl142, (1996). Ku, H. and I.A. Karimi, "An evaluation of simulated annealing for batch process scheduling", Ind. Eng. Chem. Res., 30,163-169 (1991). Laarhoven, P.J.M. Van, E.H.L. Aarts. Simulated Annealing; Theory and Applications. D.Reidel, Dordrecht, Holland, 1987. Pinto J.M., Grossmann I.E.; "A continuous time MILP model for short term scheduling of batch plants with pre-ordering constraints", Comput. Chem. Engng., $20, Sl197-S1202, (1996) Rodrigues M.T.M., Gimeno L., Rodrigues L.A., "Time Windows for Short Term Planning in Multipurpose Batch Plants", Comput. Chem. Engng., 23, $551-$554, (1996)
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000ElsevierScienceB.V. All rightsreserved.
1135
Systematic Assessments of Uncertain Demand Effects on Multiproduct Batch Process Design Using a Multi-Objective Optimization Technique Heung I1 Park a, Catherine Azzaro-Pantel b, Pascal Floquet b and In-Beum Lee c aIntemational Cooperation Laboratory, ENSIGC INPT 18, Chemin de la loge, Toulouse, 31078 Cedex, France bLaboratoire de G6nie Chimique - UMR 5503 CNRS, ENSIGC INPT 18, Chemin de la loge, Toulouse, 31078 Cedex, France CAutomation Research Center, Dept. of Chemical Engineering, POSTECH 790-784, San 31, Hyoja-dong, Namgu, Pohang, Kyungbuk, Korea The objective of this paper is to assess systematically the uncertain demand effects on multiproduct batch process design. Based on the work of Straub and Grossmann (1992) on SPC (Single Product Campaign) multiproduct batch process design, the extension to MPC (Mixed Product Campaign) case is presented without complex numerical integration techniques. The SPC and MPC design problems for the discrete volume case are also treated. The trade-off between maximizing stochastic flexibility and minimizing investment cost is tackled using a goal programming technique. For each case of SPC/MPC with continuous/discrete volumes, the optimal solution set is obtained using GAMS. The computational results showed that the proposed formulations can provide a variety of useful guidelines when designing multiproduct batch processes. 1. INTRODUCTION In chemical process industry, the interest in the batch production, especially specialty chemicals, food and pharmaceutical products, has been continuously increasing. This can be traced to the industrial demand for batch plants with high flexibility, where a large number of products can be produced in relatively low amounts in a same facility. In general, a new batch plant is designed on the basis of prediction of future market demands, which can not be known accurately. In the past, the demand uncertainty is accomodated by overdesign or manipulation of the operating variables, partly intermediate storage. In order to take this uncertainty into account in design, several methodologies have been proposed, which can provide the optimal batch plant design (Ierapetritou and Pistikopoulos (1996), Petkov and Maranas (1998)). aE-mail : [email protected] bE-mail : [email protected],[email protected] CAuthorto whom correspondence should be addressed. E-mail : [email protected]
1136 However, in practical plant design situations, the demand uncertainty effect on the optimal design may be more meaningful than the optimal design solution itself. With this view point, a stochastic flexibility (SF) is evaluated and optimized in a multiproduct batch plant design problem in Straub and Grossmann (1992). They had taken the availability of equipment into account with the expectation of SF. But, it was confined to only single product campaign (SPC) production modes and continuous volume unit cases. The aim of this paper is to systematically assess the demand uncertainty effect on the optimal batch plant design which is obtained from trade-off between stochastic flexibility and investment cost. The work of Straub and Grossmann (1992) is extended to the design of multiproduct batch processes operating with a mixed product campaign (MPC) production. The formulations for SPC/MPC for the discrete volume case under demand uncertainty are also presented. It is assumed that the product demands are mutually independent and modeled with normal distribution function. A goal programming technique is applied to tackle the multi-objective optimization problem, i.e., both maximizing stochastic flexibility (i.e., probability measure of the designed plant ability to tolerate uncertainty) and minimizing investment cost. For each case of SPC/MPC with continuous/discrete volumes, the optimal solution set is obtained using GAMS. The mathematical models will be presented and computational results will be discussed in the subsequent sections. 2. MATHEMATICAL MODEL The stochastic flexibility (SF) is defined as an integral of the joint distribution over the feasible region. Let us consider a two-uncertain demands (Q1 and Q2) case. The SF can be defined as : In Straub and Grossmann (1992), it has been evaluated based on the probability of meeting the uncertain demands :
SF= Prt~-'r,Q,<_ HI M
-
-
where/i is a time conversion parameter for product i
(2)
J
It is equivalent to one dimensional integration (as defined in Eq.(3)) which greatly simplifies the problem, regardless how many products are involved. Otherwise, in order to calculate the multiple integrals of Eq.(1), a numerical integration technique (i.e., Gaussian Quadrature Integration) should be introduced, which would make the optimization problem expensive to solve due to the introduction of a large number of variables. H
SF = I f (HA)dHA
(3)
-oo
Here, f(HA) is a normal distribution function with the following mean and variance, which are the linear combination of that of product demands, respectively.
JU,A = ~_,/ijuo.i and a~A = ~"r~a~i i
(4)
i
The Eq.(3) can be transformed into a normalized variable z with 0 mean and 1 standard deviation.
f(z)dz
SF= -co
H -/..IHA where z - - - ~ O'H A
(5)
1137 Now, SF is only a function of z and also a monotonic function of z. Therefore, maximizing SF is equivalent to maximizing z, where no integration is involved as follows : H -/IMA maxz = ~ (6) 0"H A
This is a transformation of probabilistic constraints to the deterministic forms. A multi-objective programming technique is introduced in order to tackle the trade-off problem, i.e., both maximizing SF and minimizing investment cost. Especially, in this study, a goal programming approach (e-constraint method) is used, which minimizes one objective function while constraining the remaining objectives to be less (in minimization, or more in maximization) than the given target values (see Lim et al. (1999)). 2.1. Single Product C a m p a i g n with Parallel Units 9continuous volume case
Let us consider first the multiproduct batch plant design with parallel units per stage operating in a SPC under the assumption that the unit volume size is a continuous value. Applying the transformation of SF into a probability to the formulation of Grossmann and Sargent (1979), the following MINLP problem is solved by Straub and Grossmann (1992) : min Z
O~j exp(r]j +
(P1)
~jvj)
J
s.t. bi < vj - ln(S~i) , tts > ln(t~i)- rlj
i = l,...,Np , j = I , . . . , M
where /~MA=~--'exp(~,.)/~O;, 0-2 MA= Zexp(2~/)0-Qg , ~/ -tLi-bi
z<(H--/~MA)/0-MA -
i
ln(B~) < b i < ln(By)
i = 1,...,Up,
/'tHA , O'2HA -->0, --~ _<~. _<~ rlj : Z Y j r l n ( r ) ,
ZYjr
r
i= 1,...,Np
i
ln(VjL) < vj < ln(V~)
j = 1,...,M
i = l,'.',N p
: 1, Yjr ~{0,1}
j:
1,...,M,
r: 1,...,Ny
r
As shown above, the exponential transformation (Kocis and Grossmann (1988)) is applied to partially convexify the problem (i.e., Bi = exp(b/), Vj = exp(v/), Nj = exp(r/j), TLi =exp(tLi)). In this study, the investment cost is considered as an objective function and the minimum of normalized value is constrained with z, which is vice versa of Straub and Grossmann (1992). 2.2. Single Product C a m p a i g n with Parallel Units 9discrete volume case
The formulation (P I) can be easily modified into a discrete volume size case using the work of Voudouris and Grossmann (1992). Let us define binary variables Yjsn, which are equal to 1 if stage j has n units in parallel of equal size s, 0 otherwise. The design problem can be formulated as the following MINLP 9 min~-'~-'~-'%nyj, j
s.t.
s
exp(-bi)
.
>
ZZ s
n
ZZ, ,n :1, yj,. s
where % , = n a # ~ j
j= 1,...,m,
s = 1,...,nsj,
(P2)
n= 1,...,N~
n S~ i Y j s n , v J sn
e x p ( _ t L i ) _>
ZZ s
n
--
Np
i = 1,...
n Yjsn
,
j = 1,...,M ,
j=I,,M, s-l, ,ns ,
n
z<-(H-PMA)/0-MA
2 . 2exp(2~.)0-~i, where /~M, =~--'~exp(~i)/~0,, an, . . . ~/ . tLi . b i i
i
i
1,. ,Np
1138 Because the uncertain variables are only product demands, the other constraints are all the same as in the deterministic case except for the time horizon constraint. 2.3. Mixed Product Campaign with Single Unit per Stage : continuous volume case
When designing the multiproduct batch process operating in a MPC production mode, several scheduling constraints should be involved (Birewar and Grossmann (1989)). For a ZW policy, the NLP formulation (P3) is obtained under uncertain demands : min Z aj exp(flj vj ) J s.t. b i <_vj - ln(S0)
(P3) i = 1,...,Np, j = 1,...,M
s
k = ,t.t0, exp(-bi) , k Z~t/Np/k -"/dOk exp(-bk) ' i Zli <--
-1--(lUNqi -- ,r
~0.~r
Z 0.2~k = ~2Q;exp(-2bi) k Z 0.2p/k : 0.20k exp(-2bk) i
exp(-bi))
+ 0.~i exp(-ibi)
i= 1,... , Np k = 1,...
, Np
n - Z [,t/Q/try exp(-bi ) + Z/INPik SZikj ] i k
i = 1,... N p , ZZj <_ ' I~ i
[4t,j2 e x p ( - 2 b i ) + Z 0 . 2 p i k S L ~ k J
j = 1,'", M
]
k
Due to demand uncertainty, the number of product batches will be uncertain. Thus, the number of pairs of products i and k, NP/k, will also be uncertain. In (P3), it is assumed that NP/k, is normally distributed with mean, tUNPik and variance, 0 "2NPik . The constraints from the
second to fourth represent the linear summation of mean and variance of N~k, which correponds to those of number of batches, respectively. Therefore, the distribution of NP~kwill be optimized by determining PNnk and 0.2Ne,k 9The sixth constraints denote the probability of constraints satisfaction for eliminating one product subcycle. Time horizon satisfaction is represented by the seventh constraints. The formulation (P3) can be also applied to UIS policy with non-zero clean-up time, where slack variables SLikj are used as clean-up times. 2.4. Mixed Product Campaign with Single Unit per Stage 9discrete volume case
From the formulations (P2) and (P3), multiproduct batch processes operating under a MPC production mode with discrete volumes can be easily derived as a MINLP formulation (P4): min Z Z CjsYj s j s
s.t. Z y j s v j s
j = 1,'.', M , s = 1,.. 9,nsj
where Cjs = ctjvj,~J
> S~iBi
i= l , . . . , N p , j = I , . . . , M
s 20.NPik._..2Qi/Bi2
Z].lNPik __j.lQi/Bi ' Z k k
(P4)
= 1, Yjs ~{0,1}
ZYj, $
j = I,...,M, s= l,...,nsj
i = 1,'",Np, k = 1,...,Np
i
i
H-
-1 - (~lNl~i -- itlQ/ /Bi )
Zli <
~0.2pi i + 0.~. / B :
i = 1,'", N p,
Z2j <--
tij /8. +
SLi, ]
i
I Zi t4 'o/# 2
k
+ Z o'N~k 2 SLiky ] k
j=I,...,M
1139 3. C O M P U T A T I O N A L RESULTS With the proposed formulations, an example involving three stages and two products is solved for each case. The detailed data are presented in Table 1. The MINLPs ((P 1), (P2) and (P4)) and NLP ((P3)) are formulated and solved using GAMS. Fig. 1 shows the trade-off curves between investment cost and SF for each case. In discrete volume cases ((P2) and (P4)), there is no change in investment cost for the region where SF is lower than around 0.8. For the whole SF range, the lowest investment cost is required in (P3) case. The obtained optimal discrete values are presented in Table 2. For the MPC with continuous volume cases (Fig. 2), while stages 1 and 3 are critical in the region where the value of SF is lower than 0.5, stages 2 and 3 are limiting ones in higher region. From this, it can be noted that the uncertainty in demands influences the design of stage 3. The mean and standard deviation value of N~k are presented in Fig. 3. The standard deviation values of (AB) and (BA) are larger than those of (AA) and (BB), which means that the cleaning-time or setup-time for different product sequences will have more effect on the optimal batch process design as the level of demand uncertainty satisfaction arises. The differences between the given limiting lower bound (z) and its actual value ( z ) at the optimal design are calculated for subcycle-eliminating constraints and time horizon constraints, respectively, as follows : ,Sz~i = z l i - z,i (for product i), &2j = Z z j - z2j (for stage j) (7) These represent the relative tolerance to the variation of demand satisfaction probability, which means that if the value is close to zero, the product or stage is a limiting one on the optimal design. In this example, the &li showed the high postive values in all the considered cases. In Fig. 4, around the region of SF equal to 0.5 and 0.95, the values of t ~ 2 for all the stages have the relatively high positive values, which means that the optimal batch process design has less variability to demand uncertainty in those regions. 4. C O N C L U S I O N S In this study, the uncertain demand effects on multiproduct batch process design is assessed systematically using a goal programming. The SPC multiproduct batch process design by Straub and Grossmann(1992) is extended to the MPC case without complex numerical integration techniques. The computational results showed that the proposed formulation can provide a variety of useful guidelines when designing multiproduct batch processes. Table 1. Detailed data Size factor, Processing time Product demands Cost coefficients Stage Stage a /3 Prod. 1 2 3 /~Q O'Q 1 250 0.6 A 2, 8 3, 20 4, 8 40,000 4,000 2 250 0.6 B 4, 16 6, 6 3, 3 20,000 2,000 3 250 0.6 No. of discrete volume size : n s j = 6 ; vjs n ~ {250n, 500n, 1000n, 1500n, 2000n, 2500n} Lower bounds
and z2j : 13 levels from-3 (Pr = 0.001) to 3 (Pr = 0.998), w/interval size = 0.5 Maximum number of parallel units in (P 1) and (P2) : N~ = 3 z,
z~i
1140 ACKNOWLEDGEMENT This research was supported by Korea Science and Engineering Foundation. (Grant No. KOSEF 995-1100-005-2) LITERATURE CITED 1. M. G. Ierapetritou and E. N. Pistikopoulos, Ind. Eng. Chem. Res., 35 (1996) 772. 2. S. B. Petkov and C. D. Maranas, AIChE J., 44, No. 4 (1998) 896. 3. D. A. Straub and I. E. Grossmann, Computers Chem. Engng, 16, No. 2 (1992) 69. 4. Y. I. Lim, P. Floquet, X. Joulia and S. D. Kim, Ind. Eng. Chem. Res., 38 (1999) 4729. 5. I. E. Grossmann and R. W. Sargent, Ind. Eng. Chem. Process Des. Dev., 18 (1979) 343. 6. G. R. Kocis and I. E. Grossmann, Ind. Eng. Chem. Res., 27 (1988) 1407. 7. V. T. Voudouris and I. E. Grossmann, Ind. Eng. Chem. Res., 31 (1992) 1315. 8. D. B. Birewar and I. E. Grossmann, Computers Chem. Engng., 16, No. 1/2 (1989) 141. Table 2. Optimal discrete values (a" number of parallel units, b 9discrete batch size) (P1)a (p2)a, t) (p4)D
N 1= N 2 = N 3 =
(for all z range)
Y231 =
(for
z =-3.0-
Y331 =
-- 9 (P1) . . . . (P2) I *
?
/
......... t::
300000.0
...... .
(for
..
-
Y13 =
(for
Y23 =
z2j
Y23 -
Zli -- Z2j
1
Y34 = -
'
-
3.0)
2.0-
I7. 9 -- stage1 -4-- stage2 stage3
0.15 ~" u~
1
Y33 =
= - 3 . 0 - 1.5)
0.20
/I
::)
~
zu =
0.25
j
(P3) ~,
---,--(P4)J' / . /
42000 ~ 4oooo ~ 38000 s 36000 - j l ~ ~ - I ~ a4000 2 32ooo !
Y12 =
1
1.0)
Y]31 = Y231 = Y341 = 1 (for z = 1.5-~ 3 . 0 )
52000 50000 ~ 48000460OO~ ~
Y121 =
1
~= I 7
/
0.10 0.05 0.00
012
014
016
018
0.0
0.2
0.4
SF
0.6
0.8
1.0
SF
Fig. 1 Trade-offcurve between investment cost and SF
Fig. 2 Relative tolerance to the variation of demand satisfaction probability (P3)
2.O-
220
l
- = - stage1 I
'
---o..
1.5.
A,
9
stage2 stage3
180
1.0. *
0.0
,
,
012
.....
9 <....
014
.......
0'.6
%
9.......... 9 018
0.5.
-e .It
110
SF
Fig. 3 Mean and standard deviation of number of pairs (P3)
o.o ~. 0.0
_
.
_
,
0.2
.
014
015
018
'.~-
,
1.0
SF
Fig. 4 Relative tolerance to the variation of demand satisfaction probability (P4)
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rights reserved.
1141
A Mixed Integer Model for LPG Scheduling Jos6. M. Pinto a'* and Lincoln. F. L. Moro a'b aDepartment of Chemical Engineering, University of Silo Paulo 05508-900 Silo Paulo SP Brazil - e-mail: [email protected] bpETROBRAS, Petr61eo Brasileiro S/A - Rod. Pres. Dutra Km 143. 12220-840 Silo Jos6 dos Campos SP Brazil - e-mail: [email protected] This work addresses the problem of LPG production and inventory management of a real world refinery. The problem concerns decisions related to LPG and LPG byproduct production strategies and decisions related to the selection of storage facilities that are used to receive these products and to feed the product pipeline. We develop a general framework for the modeling of similar scheduling problems. The result is a mixed integer optimization model with non-uniform time slots that can generate a short-term schedule for refinery LPG management. This schedule, spanning a horizon of approximately one-week, takes into account volume constraints as well as operational rules. 1. I N T R O D U C T I O N In the past 20 years, the implementation of advanced control in oil refineries has allowed significant improvements in these plants. These results have created a growing interest for systems that take into account more complex production objectives. Despite that, the field of production scheduling has received considerably less attention from the software providers than production planning and process optimization, resulting in low availability of commercial scheduling tools. Additionally, the literature presents very few optimization-based formulations for the scheduling of continuous multiproduct plants, as opposed to batch plants (Reklaitis, 1992; Pinto and Grossmann, 1998). In this paper we propose a mixed integer optimization model with variable length time slots that can generate a short-term schedule for refinery LPG management. 2. P R O B L E M DEFINITION In the refinery studied the liquefied petroleum gas, LPG, a mix of hydrocarbons with 3 and 4 carbon atoms, is separated in a distillation column, known as depropanizer, into a stream rich in 3 carbon-atom hydrocarbons, referred to as propane, and another rich in 4 carbon-atom hydrocarbons, called butane. When operating in high purity mode, this column produces propane for petrochemical purposes, known as intermediate propane, which is a very * Author to whom all correspondence should be addressed. The authors acknowledge support received from FAPESP under grant 96/02444-6 and Petrobras under grant 001/ABAST-REF/EPUSP-DEQ.
1142 profitable product and thus usually maximized. The butane produced is usually fed to the MTBE unit, which produces methyl-tert-butyl-ether, and a byproduct known as raffinate, a stream rich in butane. The storage farm comprises 8 spheres for LPG or propane and 4 spheres suitable for butane storage. Figure 1 depicts the LPG production system. LPG and C3 storagef--"~, ('-'-,,,.
Columnb assand reprocessingpiping
~T J ~ j [ ,...... ~ .~, ........ ,
K: set of all time-sl0ts 8
I 1 [ 2 131 4 15161 71 1
19 !10 1
IrA:set of time slots' defined a p/iori
(hara time s:ots)'i~,,
i...... ~ [
.,'-rff
I~: set of time slots free to 'be defined (soft tim~ slots) 1 i 2 i3[ 4 iJ[ T--'~
I II
I
I IMi"BE~
iI'C
.~Tq v ~
I t
Fig. 1. Refinery LPG system
I
LPG delivery operations scheduling horizon
Fig. 2. "Hard" and "soft" time slots
3. O P T I M I Z A T I O N M O D E L The optimization relies on a mixed integer linear programming-MILP model. When building such a model, one is confronted with two main decisions, which are the time representation and the model structure itself, involving the representation of continuous and discrete variables and their relationship. In such MILP models, the time horizon is divided into a series of subsequent intervals (time slots) and decisions are allowed to occur only at the slot boundaries. The division in slots of the time horizon can be basically performed in two approaches: uniform and non-uniform. The uniform approach or discrete formulation, defines a number of evenly-spaced time slots. For large time horizons, this approach generates a large number of time slots, making problem solution a difficult and time-consuming task, if not unattainable altogether. On the other hand, these are in general tight formulations, which means that they present a relatively low integrality gap (Xueya and Sargent, 1996). In the nonuniform approach (continuous formulation) the horizon is divided into a fixed number of time slots of unknown duration. For some time slots the initial or final time instant may be previously fixed due to operations whose precise time is known in advance, known as "hard" time slots. The remaining slots are entirely free ("soft" time slots), whose duration is defined by the algorithm. These ideas are illustrated in figure 2. The basic aspects of the formulation used in this work were described by Moro et al. (1998) for a planning problem. The extension for the scheduling problem, where decisions must be sequenced and the time is a crucial issue was presented by Moro and Pinto (1998) for unequally spaced time-slots. The current formulation is a generalization of that extension applied to a specific refinery problem. There are two kinds of units in the formulation: processing units and storage units. Processing units continuously transform the feed into one
1143 or more products. On the other hand, storage units do not transform the feed, but merely store it. In this case, mass balances must account for the accumulation of material. Figure 3 shows the representation of a typical unit. The model of a typical unit u is represented by equations (1) to (5). The feed flow is the sum of all streams from all other units directed to the unit under consideration, as seen in equation (1):
QFu, k -
Z
Z Qu',s,u,k
u~Uu s~USu,,u
VusU, ksK
(1)
where K is set of time slots, U is the set of units included in the model, Uu is the set of units whose destination is u and USu, u denotes the set of streams from u' whose destination is u. In addition, QFu,k represents the unit u feed flow during time slot k and Qu',s,u,k the flow of stream s from unit u' to unit u during time slot k. The flow of each stream from processing unit u is a function of unit feed flow:
QSu,s,k = fu,s ( QFu,k )
Vus Uproc, ss Su, ks K
(2)
In (2), Su is the set of unit u streams, fu,s represents a general function and QSu,s,k denotes the flow of s from u at slot k. The total flow of product streams from storage units is calculated through a material balance, which must include the non-stationary accumulation term:
mWu, k =
QFu,k - ~ QS .... k
V u s Us, ks K
(3)
s~Su In (3), AVu,k = Vu,k- Vu,k-I is the volume change in unit u during time slot k and Vu,k is the product volume in storage unit u at the end of k (m3). Us represent the storage units (spheres). The product stream flow rate is calculated by a splitter equation: QS .... k--
~ Q ..... 'j,
V u s U , ssSu, k s K
(4)
u~SUs,u where SUs,u is the set of units fed from stream s from unit u. Moreover, the product stream flow rate only exists if the appropriate decision is selected:
Q ..... ,,k-QU-YQ ..... ',k
Vus U, ss Su, u' s SUs,u,ks K (5)
The binary variable YQu,s,u',k represents the decision to send stream s from unit u to unit u' during time slot k and QU the maximum value for Qu,s,u',k.
1144
4. L P G O P T I M I Z A T I O N M O D E L Figure 4 depicts the main relationships among the units used to model the LPG system. The LPG model is constituted by the unit equations as well as the following equations: LPG and C3 storage Stream i ! /Splitter"P~
QS~; k'4 I :":~
Feed 1
9
Feed F
TQF~,~
l~Iodel of Unit u
9
Stream s"
9
' '
Fig. 3. Typical process unit.
C4 storage Fig. 4. Units used in the LPG model.
The objective function maximizes the production profit (P), defined as the sales revenue and deducting the cost of operating each unit. P- Z ( k
Z Cpu'QFu,k ueUprod
~CFu'QF.,k )
(6)
ueUproe
In (6), Cpu and CFu are the price of the product represented by pool unit u and the cost of feed, respectively ($/m3). The sets Uprod and Uproc represent product pools and processing units (non-storage units). The product stream flows are calculated through a material balance around the unit, similar to the one shown in equation (3). A constant yield model is assumed for the processing units. QS .... k -- rl .... "QFu,k / 100
Vks K, us Uproc, ss Su
(7)
In (7), qu,s denotes the volumetric yield of product stream s from unit u (% vol.). After a sphere receives any product, it is necessary to wait some time until it can be delivered to customers. This allows water separation and draining as well as product analysis. This is known as resting time, and is calculated as follows: Tru,,k <__Tr u . [ 1-YQu,s,u',k ] Tru'k <--Tk-Tk-i + Tru',k-I
V u ' e Us, us Uu,, se USu,u,, ke K Vu' 9
keK
(8) (9)
Tru'k is the time elapsed since the last time sphere u' received any product before time slot k. Tr u is the upper bound for the resting time and is equal or greater than the time horizon.
1145 Finished products can be drawn from a sphere only after the minimum resting time has elapsed: Tru',k-I -->YQu',s,u,k. Tr m
V u ' s Us, us Uprod, ss USu,,u, ks K
(10)
Additional operating rules must be enforced, such as the ones that establish that only one sphere may receive product at any time and that LPG to customers can be drawn from at most two spheres at the same time. In addition, it is possible for the butane spheres to receive butane and feed the MTBE unit simultaneously but product delivery and butane receiving cannot be performed, as is the case for all the remaining spheres. 5. R E S U L T S We show an example closely related to the actual refinery situation. The total time horizon spans 108 hours, during which propane, LPG and butane ought to be produced, sampled, analyzed and delivered to customers. The objective is to maximize product deliveries and the available inventory of intermediate propane.
Fig. 5: Gantt chart for LPG and C3 spheres
Fig. 6. Gantt chart for butane spheres.
In order to solve this problem, a total of 12 variable-size time slots were defined and the modeling system GAMS version 2.50 (Brooke et al., 1992) was used to implement the optimization model, which contains 536 discrete variables and 3294 equations and was solved with OSL (IBM, 1991). The solution of such a problem in a typical PC (Pentium II 300Mhz) may take about 20 minutes. Figure 5 shows the operations performed during the scheduling horizon in the propane and LPG spheres. This figure makes clear that the main operating rules are satisfied, e.g., that finished product can be drawn from a sphere only after the minimum resting time of 24 hours has elapsed. In this example, only 5 spheres were necessary to accomplish the operations. The results for the butane spheres reveal a similar behavior. Only 3 butane spheres were necessary to perform the operations (figure 6).
1146 6. CONCLUSIONS In this work we present a scheduling model, developed from a planning model through the inclusion of the time domain and operational rules. We used a non-uniform approach for the representation of the time-horizon in slots, since this is necessary for the implementation of the results in an industrial environment. Real time problems can be adequately solved as shown for the LPG area in an oil refinery. NOTATION Cpu K QFu,k Qu',s,u,k QSu,s,k Su SUs,u Tk Tru'.k Tdk U Uproe Uprod
UU Us Usl USu,,u Vu,k
qU,S
price of the product represented by pool unit u ($/m 3) set of time slots. unit u feed flow during time slot k (m 3) flow of stream s from unit u' to u during slot k (m 3) flow of product s from unit u during time slot k (m 3) set of unit u streams. set of units fed from stream s from unit u time at the end of time slot k (h) time elapsed since unit u' received product before k (h) values of event times known in advance (h). set of units included in the scheduling model. set of processing units (non-storage units). set of product pools. set of units (u') whose destination is u. set of storage units (spheres). subset of Us (spheres that store LPG or propane). set of u' streams whose destination is u. product volume in unit u at the end of time slot k (m3). volumetric yield of product stream s from unit u (vol%).
REFERENCES
Brooke, A.; Kendrick, D.; Meeraus, A. (1992) GAMS - A user's guide (release 2.25). The Scientific Press, San Francisco (CA). IBM (1991) OSL guide and reference (release 2), Kingston (NY). Moro, L.F.L; Zanin, A.C.; Pinto, J.M. (1998) A planning model for refinery diesel production. Comput. chem. Engng., 22 (Suppl.), S 1039- S 1042. Moro, L.F.L.; Pinto, J.M. (1998) A Mixed-Integer Model for Short-Term Crude Oil Scheduling. In: AIChE Annual National Meeting, paper 241 c, Miami Beach (FL). Pinto, J.M.; Grossmann, I.E. (1998) Assignment and Sequencing Models for the Scheduling of Chemical Processes. Annals of Operations Research, 81,433-466. Reklaitis, G.V. (1992) Overview of scheduling and planning of batch process operations. NATO Advanced Study Institute- Batch Process Systems Engineering, Antalya (Turkey). Xueya, Z.; Sargent, R.W.H. (1996) The optimal operation of mixed production facilities. Comput. chem. Engng., 20(6/7), 897-904.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
Simulation-aided Batch
Implementation
of Supervisory
1147
Control
of Industrial
Reactors
K. Preu[3 ~', M.-V. Le
Lannb, M. Cabassud ~, G. Anne-Archard ~ and G. Casamatta ~
:qNPT - ENSIGC - CNRS UMR 5503 18, Chemin de la Loge - 31078 Toulouse, France, E-mail: [email protected] bLAAS/INSA- CNRS UPR 8001 7, Av. du Colonel Roche - 31400 Toulouse
Csanofi-synthelabo 195, route d'Espagne - 31036 Toulouse
Temperature control of batch reactors equipped with a multifluid type heating/cooling system containing an intermediate thermal fluid (pressurised water) is presented. By means of this intermediate fluid the heating/cooling system gains monofluid type behaviour over a wide temperature range. Hereby operation of the reactor is improved. The control strategy consists in using a GPC algorithm to calculate the thermal flux that has to be transferred between the heating/cooling system and the reaction mixture. The thermal flux is then introduced into a model based supervisory algorithm that automatically chooses on-line the appropriate utility. Testing the intervening software is partially carried out by connecting the original plant control software to a simulator of the process. Hereby the software validation procedure is accelerated and utility consumption is reduced, too. Experimental validation of the presented approach on an industrial 160 litres batch reactor gives satisfying performances. 1. I N T R O D U C T I O N Operation of batch reactors is marked by the absence of a dominating steady state due to their inherent batch nature. Hence, operators have to pay considerable attention to a running batch process. Therefore its automation is of great interest. The main problem that arises from the operation of batch reactors is temperature control. This requires the use of different utilities which are typically: steam, water and a cold utility. An approach was proposed, introducing an intermediate utility based on the monofluid type concept [1 ]. This thermal fluid (pressurised water) is generated by mixing steam and water in a mixing unit added to the batch reactor. The mixing unit is located between the inlet of the utilities and the reactor jacket. Especially when hot water is generated at higher pressures, this intermediate utility can cover a wide temperature range (e.g. from 20 ~ to 120~ Therefore the number of required utility changes is significantly reduced and the batch operation is carried out in a smoother way. Automation of such a batch reactor requires high level supervisory control (managing the utilities) linked to low level control (setting the appropriate valves). A general solution of the supervisory control problem based on the works of [ 1] has been presented in [2]. The reaction mixture temperature is controlled by manipulating the enthalpy flux exchanged between the reaction mixture and the heating/cooling utility. The process is represented by a simple input/output model. This model is then introduced into a Generalised Predictive Control (GPC) algorithm. At each cycle the GPC computes the enthalpy flux that should be exchanged between the reactor reaction mixture and the heating/cooling utility. Then, by means of enthalpy balances between the reactor jacket and the reaction mixture, the maximum and minimum thermal flux that can be exchanged by a utility is calculated for all available utilities. The ap-
1148
propriate utility is chosen by comparing these enthalpy fluxes with the one calculated by the GPC. This choice is made on-line by a supervisory algorithm (section 2). Once the appropriate utility is determined, the required flow of this utility is calculated by means of enthalpy balances between the reactor jacket, the reaction mixture and - if hot water is the current utility the mixing unit. The control actions resulting from supervisory control have to be carried out on a lower control level which is located on the SCADA or PLC. On this lower level the sequences for valve settings, safety routines etc. are programmed. It has to be verified that, for every possible state of the SCADA or the supervisory controller, the automated system exhibits the desired behaviour and remains operational. Checking these sequences might be time consuming or limited by practical constraints when carried out on the plant. But a main part of the necessary testing can be carried out using a simulator of the process which will be explained in section 3 of this paper. Experimental results of the described approach for temperature control of a 160 litres industrial batch reactor are presented in section 4.
2. STRATEGY FOR TEMPERATURE CONTROL 2.1 Heating/Cooling System Let us consider a chemical batch reactor equipped with a jacket for a thermal fluid (heatng/cooling utility). For temperature control of the reaction mixture two main approaches exist ( ) the monofluid system and (ii) the multifluid heating/cooling system [2]. To avoid the drawbacks of these two approaches an alternative system was proposed in [2]. Its main characteristic is the use of an intermediate utility, pressurised water. It is generated at a constant flow rate in the thermal environment of the reactor unit by mixing steam and water (Fig. 1). The ratio of steam and water is I I Ot tlet adjusted at each sampling period so that the desired - time varying - temperature ( ~ l y c o l \vale~ Proportional valve is reached. Pressurised water circulates in 0/1 - v a l v e a closed loop through the reactor jacket, a Fig. 1: Batch reactor with its thermal environment. pump and a mixer where the required amount of steam or water are added. At a pressure of 3 bars the pressurised water covers a temperature range from 20 to 120~ Within this range the heating/cooling system exhibits a monofluid type behaviour. If a greater heating / cooling capacity is required, the system switches to a lnultifluid behaviour and, after a purge of the reactor jacket, steam or a cold utility are directly injected into the reactor jacket. am
II I II
M,e,
2.2 Predictive Control A GPC algorithm [3] was adapted to this specific process [4]. A simplified energy balance o1: the reaction mixture gives:
17-7,.:~: cp,. , __U---7(t) dT,. - q(t-(nr
+ 1)* at)
( 1)
The associated input/output model used for the GPC is as follows: Tr(k) = - a I * Tr(k-1) + bo * q ( k - n r - 1 )
(2)
1149 where al = -1 and b0= l/(mr*cpr). The model parameters a 1 and b 0 are recalculated on-line at each sampling time by a recursive least squares method. A fixed value was chosen for the time delay nr. The predictive control algorithm is receiving a setpoint T c for the temperature ol; the reaction mixture T r. The manipulated variable (controller output) is the thermal flux q that has to be transferred fiom the thermal fluid in the reactor jacket to the reaction mixture. Therefore the predictive control algorithm contains a process model linking the control variable T r and the manipulated variable q. 2.3 Supervision Strategy The value of the controller output q is then introduced into the supervisory algorithm. This routine computes - by means of enthalpy balances on the reactor jacket and the reaction mixture - the maximum/lninimum thermal flux qi, min / qi, m a x that can be exchanged by the utility i, for all available utilities. The appropriate utility is chosen by comparing these enthalpy fluxes with the one calculated by the predictive control algorithm. If this enthalpy flux is in the range given by the lnaximum and minimum thermal flux of the current utility, this utility is kept. Otherwise a utility with a higher/lower maximum/minimum enthalpy flux is chosen respectively. Once the appropriate utility is determined, the required flow F of this utility is calculated by means of enthalpy balances between the reactor jacket, the reaction mixture and if pressurised water is the current utility - the mixing unit. -
3. S I M U L A T I O N - A I D E D I M P L E M E N T A T I O N An important step is the validation of the automation algorithms. The validation can be subdivided into different tasks, depending on the type of automation algorithm concerned: actuator settings, operation and safety routines, additional software tools and interactions between algorithms. Checking these algorithms might be time consuming or limited by practical constraints when carried out on the plant. But a main part of the necessary testing can be carried out using a simulator. Hereby the automation algorithms and in particular interactions between the involved software tools (SCA, SCADA) are tested without the need to work on the real plant. Therefore time demand and the use of technical resources can be reduced by simulation-aided implementation. In the case of the presented example of supervisory temperature control of a batch reactor (Fig. 1), the sequences for actuator settings and operation/safety routines are implemented in a SCADA (DeltaV, Fisher-Rosemount). The superviso U control algorithm (SCA) is an additional software tool containing the control and supervision algorithm described in the preceding section. Information is transferred between SCA and SCADA at each sampling period. The SCA determines the appropriate utility to use and the required flow as well and transfers this information via the OPC interface to DeltaV which then carries out the actuator settings in the thermal environment of the reactor. Measurements from the process are taken up by DeltaV and fed back, together with the current values of discrete process states, to the SCA via the OPC interface. The control actions initiated by the SCA are carried out on a lower control level, which is located on the SCADA. They require sequences of actuator (valves, pumps) settings which involve time delays and the fulfilment of several conditions with respect to process state and in particular to the state of the reactor .jacket. Checking these sequences by simulation requires a simulation model that contains the concerned actuators and also comprises the physical effects affected by the actuators (e.g. mass flow and pressure in a tube connected to a valve). Consequently very detailed models are
1150
required. Operation and safety routines are understood to be customised algorithms, allowing to parameterise the software for a specific batch run and to correspond to security requirements that are not taken into account by the commercial software. Operation routines (e.g. choice of the active plant equipment or of controller modes) will affect variables of the SCADA or SCA and will not directly lead to actuator settings. Safety routines detect critical situations and choose the appropriate reaction. A reaction might be an actuator sequence. In this case the reaction can be tested independently as described above. Due to this separation, safety and operation routines can be tested using a first principle simulation model. A model as detailed as in the case of actuator settings is not required. It is the same if an additional software tool like in the present case (SCA) is concerned. It is important to test interactions between the different algorithms. In the present case algorithms implemented in the SCADA and the SCA affect their execution mutually for example during switchover from one utility to another (Fig. 2). SCA~7
Interactionsignal S
I Chooseutility l 1 Stop controller Stop adaptation Stop data saving J .... l I StartC~176 l Start data saving
Changeutility ,11
Time delay ]
I Start Adaptation t
Apply utility I l
9
[
Changingutility
Carry out utility change Time delay
Utility change finished I 9 Apply utility Apply utility flow
I
"1
'q Reactorjacket filled I "-,
I
SCADA
"",,_............................ The action following this symbol can only be carried out after the corresponding interaction signal had been received. ..............................
] El,,~.~..~v S C A D A / S C A interactions
The utility flows F,., F ..... and F~, are calculated by the calibration curve of the corresponding valve as a function of the valve opening degree u (manipulated variable of the SCA). This simulator was implemented as independent software application. By means of an OPC software interface it can be connected to the SCADA which then receives simulated values instead of the real plant measurements. Either the real plant or its simulator is connected to the SCADA, performing read/write operations on the same variables of its on-line database. This offers the advantage that the tests in simulation can be carried out with exactly the same configuration of SCADA and SCA as it will be in real operation. Figure 3 shows a test of the algorithms described previously (SCADA, SCA) by simulating a 160 litres batch reactor equipped with the heating/cooling system described in section 2. The imposed setpoint profile drives the supervisory algorithm to initiate a switchovex from pressurised water to glycol water
after 1.5 hours, triggering the interactions shown in figure 2 . By means of this test it was verified that, t:or this type of switchover, the SCADA and SCA interact correctly. It has to be verified that both algorithms use the same signal for a specific interaction, that the signal is set at the right time fox an appropriate duration and that a dead-state is never reached so that the system always recovers an operational state. This verification can only be carried out if both algorithms are working together. For this aim a simulator of the batch reactor and its thermal environment was connected to the SCADA. The simulator is described in detail in [5]. The thermal environment of the batch reactor is modelled by a static mass balance containing all
1151 flows entering or leaving the thermal environment. Dynamic behaviour of the thermal environment is not taken into account and it is supposed that changes take place instantaneously. 0 - F.,,,,,, - F.,,<.,,,v<' - F,. ( u ) - F . (u ....) - F, ( u , )
An enthalpy balance is given by:
0Steam [kg] 138
Time [11] 14
Water [kg] 1030
Solvent [kg] 100
Table I: Reduction of resource consumption by simulating test batch runs. 50 i 40
.
.
3o
U
/
Pressurised water
<_ .............. .
' Glycol water
-,................
.
i.
.
.
.
.
.
_,,.................
.
, . . . . . . . . . .
--!
121/
7 . . . .
.........
..................
1 0.75
~
i .....
"%.: 1 ....
10
i--
Tct~
......
, ~o/0
i
_,,....... _>
~----
m
o
........
-10
~
-0.25
'-20 .
.
.
.
.
0.5
Fig. 3: Test of SCADA and SCA in simulation.
The simulation of 3.5 hours process time was carried out in 20 minutes. This speed allows to follow on-line the events occurring during the test by means of the visualisation facilities of the SCADA. Besides time, further savings concern solvents (as reaction mixture) and utilities. Supposing that all permitted combinations of switchover were tested in simulation the resources listed in table 1 could be saved compared to tests on the real plant.
4. E X P E R I M E N T A L
Tile experimental results shown in the following have been obtained on an industrial 160 litres reactor that is a part of a production unit for pharmaceutical compounds. The unit was equipped with the heating/cooling system and the software described in sections 2 and 3. The reactor was filled with 100 litres DMF. ~
~
10 120
120
i |
1O0 i --q---Tc ___Tr r [kCal [.C]~ ["C] ls] [[:~l ; ............ ----Y....y..,.//~,..-.b-'7-"--)-"-~t{_i .... i 8O . i --,U '
-
,,
/
iO.~~ 10080~1~" ~' 60
tl
~"\
~
~
,
,
t
TcTr[~ ~
tJ 0.8 t 0"4
'- ~'.... .... ~'t'\k-~- ;i................. . . . --
/~11 Vl//,k ~
\i
~ q
1.2
[kCal/@
--- \,-\ ~-.....
,....
~0 40 _.L__
-
20 0 i'-'---J ............... 0.5 ...... 0
-i .
;
_
Time
[h]
1.5
.
.
.
/
.
.
_
~o
0 -20
__
__
---i .......... :_v: _
~
r
)
-0.4
-0.8 1 2
Fig. 4: Temperature control of the batch Fig. 5: Temperature control using pressurised reactor using steam, water and glycol water. In figure 4 steam was chosen as a utility by the supervisory routine because the maximum setpoint temperature can not be reached using pressurised water. Low frequent oscillations
1152
appear in the Tr-curve, leading to an average control error of 2.7~
This is due to the fact that
the process model supposes direct heat transfer between the utility and the reaction mixture. Heat conduction through the reactor wall is not taken into account. The resulting structural model mismatch leads to the observed oscillations. For the batch run depicted in figure 5 the supervisory algorithm chose pressurised water as utility which allowed to heat and cool the reactor in a temperature range between 25~ and l l0~ without changing the utility. Only in the final phase a switchover to glycol water was initiated by the supervisory algorithm in order to cool tile reactor to -5~ The observed average control error is 0.8~ when pressurised water is used and 2. I~ in the case of glycol water. 5. CONLUSION This paper described which tasks concerning control and supervision appeared when temperature control was implemented on an industrial reactor and how these tasks were distributed among the intervening software. The testing of these software was partially carried out using a simulator of the process. The simulator was connected to the plant control software by an OPC software interface for on-line data exchange. This approach led to substantial savings in terms of time and utility consumption compared to tests on the real plant. Unfortunately these tests were limited to a subset of functions of the process control software due to a lack of sufficiently detailed process models. The proposed strategy for temperature control of batch reactors was applied to an industrial batch reactor of 160 litres size. By means of the model based supervisory algorithm the appropriate utility was automatically chosen on-line with respect to process state and the desired control objective. Despite the fact that the heating/cooling system was based on the multifluid type, the intermediate thermal fluid, introduced by the presented approach, allowed for smooth - monofluid t y p e - behaviour of the heating/cooling system in a temperature range from 20~ to 120~ enabling heating and cooling in this range without the need to switch from one utility to another. Using this intermediate fluid, precise control of the reactor temperature for time variant setpoints was achieved. In the case of steam or glycol water, the control error increases but still remains satisfying. As far as the manipulated variable is concerned, oscillations were observed to some extend. It is desirable to decrease these oscillations in order to increase the lifetime of control valves and to reduce utility consumption. Solving this problem will be subject to ti~rther work, that will focus as well on improving precision of control when steam or glycol water are used as utilities.
REFERENCES [1] [2] [3] [4]
[5]
Louleh Z.: Moddlisation et conduite des rdacteurs discontinus par analyse des.flux thermiques. PhD Thesis, Institut National Polytechnique de Toulouse, France (1996). Louleh Z. et al: A New Strategy Jbr Temperature Control of Batch Reactors. Chem. Engng. J. 75 (1999). Clarke D.W., Mohtadi C., Tufts P.S.: Generalized Predictive Control. Automatica 23, No. 2 (1987). Le Lann M.-V., Cabassud M., Casamatta G. (1995): Adaptive Model Predictive Control In: Methods of Model Based Process Control, R. Berber (Ed.), Dordrecht: Kluwer Academic Publishers Xaumier F., Ettedgui B., Le Lann M.-V., Cabassud M., Casamatta G. ESCAPE-9 Supplement to Comp. Chem. Engng. Vol. 23 (1999)
European Symposiumon ComputerAidedProcessEngineering- 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
1153
Dynamic modeling of batch distillation: comparison between commercial software Laureano Jim6nez a, Marta Basualdo b, Luis Toselli c and Miguel Rosa c aChemical Engineering and Metallurgy Department, University of Barcelona, Martf i Franqu6s 1, 08028 Barcelona, Spain* bGIAIQ-Rosario Regional Faculty, Technical National University, Riobamba, 326, 2000 Rosario, Santa Fe, Argentina CVilla Maria Regional Faculty, Technical National University, Av. Universidad 450, 5900 Villa Maria, Argentina. 1. INTRODUCTION The current increasing production of small volume, high added value products has called attention to batch production technologies. Although batch distillation typically consumes more energy than continuous distillation, it provides more flexibility and involves lower capital investment. Thus, since energy costs are not too significant in the separation of small volume of high profit products, batch distillation is often attractive. Available batch distillation models have appeared in the literature, ranging from very simple approaches to more detailed. Commercial software packages such as BATCHFRAC TM (Boston et al. 1981), BASIS (Simulation Sciences Inc., 1989) and ProSimBatch, (1992) are available only to model batch distillation operation. Luyben W. (1992) implemented an inferential control using a rigorous quasi-dynamic model of a batch distillation column. Recent progress in nonlinear model-based control techniques have made the practical applicability of nonlinear controller a true. Many of these techniques use a nonlinear dynamic process model directly in the control law development phase (which is performed off-line). These features are very attractive for practical implementation. Henson and Seborg (1997) list a number of simulation and experimental studies in which this feedback controllers were used. In this context, a nonlinear model-based strategy implemented with commercial software is analyzed here. Therefore BATCHFRAC TM, HYSYS | and CHEMCAD | performances were tested to evaluate whether their implementation in an inferential control structure and/or soft sensor could be attractive. Hence, using the worldwide commercial available tools, the dynamic behavior of a batch distillation column was modeled. 2. P R O B L E M STATEMENT The experimental data were taken from Nad and Spiegel (1987), from a distillation column of 162 mm i.d. filled with structured packing Sulzer Mellapak 250Y. A packing height of 8.0 m * Present address: Chemical Engineering Department, University Rovira i Virgili, Carretera de Salou s/n, 43006 Tarragona, Spain. Tel.: 34-977-559617; fax: 34-977-559667/21; e-mail: [email protected]
1154 was used, and the number of theoretical stages was measured with a standard method (chlorobenzene and ethylbenzene at total reflux). The whole column, including the reboiler and the condenser, has 20 theoretical stages. The volumetrically measured liquid hold-up of the column and condenser were respectively 0.015 m 3 and 0.005 m 3. A ternary mixture of cyclohexane + n-heptane + toluene was selected in order to test the system with a well-known mixture. The initial charge, in molar basis, is 40.70% for cyclohexane, 39.40% for n-heptane and 19.90% for toluene. The reflux ratio sequence recommended by Nad & Spiegel for the production period was followed in all the cases performed. Checking the accuracy of the physical properties' model is the key factor to succeed in process simulation, obtaining realistic results (Carlson, 1996). To compare the results from different process simulators, the same physical property model and parameters were used. For the high tendency to avoid the use of any group contribution method (i.e. UNIFAC or DISQUAC) for process simulation, UNIQUAC method was selected. UNIQUAC parameters were modified to be temperature-dependent (Table 1) as defined in Eq. 1 and Eq. 2.
,i:expEU.Uii1 u ; i - uii = a j i + - T
Table 1 Interaction parameters aii, aii, bii and bii (UNIQUAC), Eqs. (1-2). Toluene Toluene ciclohexane n-heptane aij 0.1775 -0.8597 aji -0.0238 0.7788 bij/K -54.66 178.89 bji/K -30.49 - 192.14 Tlower/K 293.1 303.1 Tupper/K 383.7 383.9
(2)
Ciclohexane n-heptane 0.6985 -0.8744 -80.76 90.41 298.1 371.6
Distillation line diagrams and residue curve maps (RCM) provide insights in a different way than process simulation. On one hand, simulation results are snapshots of equipment performance for selected design and operating conditions. On the other, these graphical tools provide many fundamental insights similar to the traditional McCabe Thiele diagrams for binary mixtures, however, much more applicable to complex multicomponent separations. The analysis for the whole composition map, reveals that the binary system n-heptane + toluene has a high purity binary azeotrope (0.985 mole fraction in n-heptane). This azeotrope is non-sensitive to pressure (0.951 molar at five atmospheres), and therefore pressure swing distillation is not a good approach for this particular separation. Following the rules formulated by Doherty and Perkins (1979), no ternary azeotrope exists in the system. The singular point detected acts as a saddle point in the RCM (Figure 1) giving rise to a distillation boundary running from the azeotrope to the low-boiling component (ciclohexane) and thus leading to two zones in the ternary diagram. In the fight region, where feed composition is
1155
located, all distillation lines begin at ciclohexane and end at the high-boiling toluene, indicating that removal of toluene as a bottom product is feasible. In the small left region, all lines go from ciclohexane to n-heptane.
_s Distillation / boundary \ . . . / /
~~ ~ \,,
1st column /'// ~. profile ~ i : " ~. . . . . .
/ Fig. 1. Residue curve map for cyclohexane + n-heptane + toluene at 101.3 KPa.
/
cyr
--~ [ F~n-heptane
/ 2ndcolumnprofile
Fig. 2. Column profiles removing the light component in the first column.
3. C L A S S I C A L A P P R O A C H In general, it is not possible to cross distillation boundaries using only distillation columns and recycle. It may be possible to cross distillation boundaries if unit operations other than distillation (i.e. liquid-liquid separation, reactor, membrane, etc.) are also included in the recycle or if the distillation boundary is strongly curved. For all practical purposes, we can state that for homogeneous distillation sequences lie in the same distillation region. Two design methods were used for the synthesis and preliminary design of the distillation columns. Boundary value design (BV) procedure to assess separation feasibility and to calculate minimum reflux/reboil ratio for a fixed component recovery achieved by starting calculations at both ends of the column and work towards the middle. The second method, omega design (~) procedure, computes the number of stages and investigates the effect of feed stage location on the design, determining the optimum feed location. The last method is only applicable for separations in which one of the products is a node (stable or unstable). With continuos distillation n-1 columns are necessary to separate a mixture of n components. The design results for the two different approaches (remove first the light component or remove first the heavy component) are approximately the same applying BV and ~ methods. Figure 2 shows the composition profiles for the second strategy. 4. P R O C E S S M O D E L I N G HYSYS | and BATCHFRAC TMuse rigorous calculations, using the Euler method to integrate the differential equations, while CC-BATCIT M uses a pseudo-stationary model. BATCHFRAC TMhas a rigorous model for batch procedure. 4.1. Initialization with H Y S Y S |
Two feeds are introduced to the reboiler and condenser, with the same temperature and composition. Once the condenser is filled, the reflux valve is opened until the stages had the
1156 experimental steady state hold-up; hence, the initial conditions can be reproduced. To model the startup period, the feeds are set to zero and the heat duty is fixed, the bottom flow rate is neglected and the reflux ratio is set to infinite. The startup duration is 2.66 hours.
4.2. Initialization using BATCHFRAC TM Initial conditions were fixed as the total reflux stationary values. No time-considerations were implemented for the start-up period, and therefore the batch steady state was reached instantaneously. To model the whole production period in a single run, the accumulator type, the feed loading time and the reflux ratio policy (operation steps and stop criteria) was fixed. The total number of operation specifications was nineteen. Dump action for the different products was continuous, and the different cut time switches were introduced between specifications.
4.3. Pseudo-stationary model CHEMCAD | models the batch operation using the CC-BATCIT M module. This module simulates the batch operation as a sequence of pseudo-stationary states, in which each new step begins with the final conditions of the previous one and the calculation is done as a steady state procedure The option of same composition in still pot, plates and condenser is adopted for the initialization period. During startup, the conditions were set to total reflux. The startup conditions were fixed as the total reflux ones. The production period was solved using a pseudo-stationary model. Each new step begins with the final conditions of the previous one and the calculation is done as a steady state procedure. For the example analyzed here a set of thirty independent operating steps were necessary. Batch distillation was integrated with continuous operations with the use of tanks and time switches between certain operation steps. 5. S I M U L A T I O N R E S U L T S The results for top and bottom compositions (Figure 3) for the three models are in good agreement with experimental data, especially CHEMCAD, which incurs in less erroneous predictions. Thus, they are good candidates for acting as soft sensors either on or off line. This strategy would be extremely useful to replace any in-line analyzer, which is an expensive and high maintenance piece of equipment with a significant time lag. However if the models are to be used as ad hoc observer or estimator for control purposes, the temperature measurements have lower time lag (around 30 seconds) and are the best economic way to estimate the composition. In all cases, the essential factor is interface between process simulators and plant data. The experimental temperature data confronted with the prediction for each software are in a good correlation (Figure 4). Therefore additional considerations about commercial elements of available software and hardware should be done.
1157 1.0
i
i
Proauct'onPer'o
:::
1 .o
...................
............................................................................................................................................................................ ~.
Startup period
g 0=
0.8
... ,.-.... .2'
,~" /
i
---
_,/._~'.
Hysys
•
,&,';~...."i .... ;
Production period
i
i
0.6 Hy=y, ..... C.Batch
/
2
4
_ _ _
0.4
~
~
i
.......
0.2 m
.. 0.0
6
0.0
8
time/h
....
0
2
4
6
8
Time/h
Fig. 3. Dynamic behavior of top and bottom composition. ..................................................................................................................................................................................... i
j
Batchplus
___ Hysys ..... CC-Batch
o = 95 4.,
l:" - -
Batchplus
105
-
-
-
.//
I'2';=
E
, , ~ "'.-, , ~ ,
E ~ i-.
" "
,llr"~
.=
o
as
.......i:V
m 95 ..~'/J
75
2
4
6 Time/h
8
90
2
4
6
8
Time/h
Fig.4. Dynamic behavior of top and bottom temperatures. 6. I N T E R F A C E B E T W E E N S O F T W A R E AND PLANT D A T A On-line applications of process simulators can cover a wide range: from specific property estimation or yield calculation to a rigorous dynamic model executing in parallel with the plant. Hence when it is necessary to exchange, store or collect data between the simulator and the process (DCSs and PLCs), an interface tool is needed. Usually, a generic implementation of a Model based Process Controllers (MPC) can be dynamically linked to the interface, and therefore any variable from the process simulator can be exported as a process variable into any interface tag and any interface tag can be imported as a process simulator controller setpoint and/or output. For the batch distillation case the process simulator can be used as a composition soft s e n s o r to write a calculated composition value on the operator's console. The sensor could be configured for inferring composition by confronting real temperature data with the simulated values. In this section an specific application to interface HYSYS-PI | with MATLAB | 5.1 data is presented (Basualdo and G6mez, 1999). HYSYS's capability to transfer data was used to obtain the data. According to the well-known expression for the state-space representation (Eq. 3 and Eq. 4) to reduce the model order in the state-space,
1158 (3)
XK+l = A'XK + B'UK YK = C'XK + D'UK
subspace-based identification method was explored, and the following state-space model was obtained
A=
C=
0.9963 0.0036 0.0003 0.0006
-0.0049 0.9935 0.0005 0.0004
-0.2866 -0.0261 0.9936 0.0061
0.0284 -0.2783 -0.0438 0.9272
E oo4 , 0.2910
- 0.0452
0.4622
- 0.0662
,
B=
-0.0010 ] 0.0005 0.0000 0.0002
(4)
o,,oo I o:EOol '
7. CONCLUSIONS In this work a case study was considered in order to test the capabilities for modeling and interfacing experimental plant data with estimated values from the three worldwide commercial software packages available (BATCHFRAC TM from Aspen Technology | CCBATCI-ITM from Chemstations Inc. and HYSYS | from AEA Technolgy). From the computational efficiency point of view, HYSYS | and BATCHFRAC TM have the best performance. In particular, the use of InfoPlus.21 | in conjunction with its layered application (BATCHFRACrWBATCHPLUS TM) and as interface with plant data has an important synergic effect. However from the user-friendly point of view, CC-BATCH | and BATCHFRAC | offer a simple' way to simulate for less trained users. REFERENCES
Aspen Split Reference Manual, Aspen Technology Inc., Cambridge, MA, USA, (1998). Basualdo M. and Gomez J. C. "Subspace-Based Identification Of Multicomponent Batch Distillation Processes" Computer-Aided Design For The 21 st Century Colorado-USA, (1999) Batchfrac Reference Manual, Aspen Technology Inc., Cambridge, MA, USA, (1998). Boston J.F., Britt H.F., Jirapongphan S. and Shah V.B., "An advanced System for the simulation of batch distillation operation". FOCAPD, 2, 203, (1981). Carlson, E. C., "Don't Gamble with Physical Properties for Simulations". Chem. Eng. Prog., 35-46, (1996). Doherty M. E and Perkins, J. D. "On the dynamics of Distillation Processes-Ill. The topological structure of Ternary Residue Curve Maps", Chem. Eng. Sci., 34, 1401-1414 (1979). Henson M. A. and D. Seborg "Nonlinear Process Control". Prentice Hall PTR. London (1997). Luyben W. "Practical Distillation Control" VNR (1992) Nad M. and Spiegel L., Simulation of batch distillation with experiment. Proc. CEF'87, p 737, Taormina, Italy (1987). ProSimBatch. Mannual Utilisateur. Prosim SA.Toulouse, (1992). Renon, H.; Prausnitz, J.M. Local compositions in thermodynamic excess functions for liquid mixtures. AIChE J., 14, 135-144, (1968). Simulation Sciences Inc., BASIS user Manual, Fullerton CA, (1989).
European Symposiumon ComputerAidedProcess Engineering - 10 S. Pierucci(Editor) 9 2000 ElsevierScienceB.V. All rightsreserved.
1159
Design, Synthesis and Scheduling of Multipurpose Batch Plants via an Effective Continuous-Time Formulation Xiaoxia Lin and Christodoulos A. Floudas * Department of Chemical Engineering, Princeton University, Princeton, N.J. 08544-5263,USA Design, synthesis and scheduling issues are considered simultaneously for multipurpose batch plants. Processing recipes are represented by the State-Task Network. The proposed model takes into account the trade-offs between capital costs, revenues and operational flexibility. Both linear and nonlinear examples are studied, resulting in MILP and MINLP problems, respectively. Comparisons with another approach are presented. Introduction
In multipurpose batch plants, a wide variety of products can be produced via different processing recipes by sharing available pieces of equipment, raw materials and intermediates, utilities and production time resources. In order to ensure that any resource incorporated in the design can be used as efficiently as possible, detailed considerations of plant scheduling must be taken into account at the design stage. All formulations for design and scheduling of batch processes can be classified into two groups based on the time representations. Examples of discrete time formulations are found in Grossmann and Sargent (1979); Barbosa-P6voa and Macchietto (1994). Grossmann and Sargent (1979) solved the problem of optimal design of sequential multiproduct batch processes as a mixed integer nonlinear programming (MINLP) problem. Barbosa-P6voa and Macchietto (1994) presented a detailed formulation of multipurpose batch plant design and retrofit based on the State-Task Network (STN) description and equally-spaced fixed event time representation proposed by Kondili et al. (1993). More recently Xia and Macchietto (1997) presented a formulation based on the variable event time scheduling model of Zhang and Sargent (1996) and used a stochastic method to solve the resulting non-convex MINLP problems. Ierapetritou and Floudas (1998a,b) proposed a novel continuous-time mathematical model l:or the general short-term scheduling problem of batch, continuous and semicontinuous processes. In this paper, we extend the formulation to address the problem of integrated design, synthesis and scheduling of multipurpose batch plants. Problem D e f i n i t i o n Given: 9 Product recipes (i.e., the processing times for each task at the suitable units, and the amount of the materials required for the production of each product); 9 Potentially available processing/storage equipment and their ranges of capacities; 9 Production requirement; *Aulhor to whom all correspondenceshould be addressed
1160
?. f--~.
[
I
r-,,.A
Product 4
I
1 IntAB
I.x.f'~
~"" |
~,-p,,,-~~: ,
/ ....
l
I
....
m
t
I
/ ....
O Feed C
Figure 1: STN Representation
Figure 2: Plant Superstructure
9 The time horizon under consideration; Dctermine: ~ The number, type and size of equipment items; ~ A feasible operation schedule: - The optimal sequence of tasks taking place in each unit; - The amount of material being processed at each time in each unit; - The processing time of each task in each unit;
So as to optimize a performance criterion, for example, the minimization of the capital cost or the maximization of the overall profit. Process R e c i p e a n d P l a n t S u p e r s t r u c t u r e R e p r e s e n t a t i o n s We use the concept of State-Task Network(STN) proposed by Kondili et al. (1993). Figure 1 gives an illustration of the STN description of a batch precess named KPS process. In addition to the product recipe, the information of potentially available pieces of equipment and their suitability for different tasks is used to construct a superstructure of the plant under consideration that includes all possible designs. For example, based on the KPS process recipe in Figure 1 and equipment data in Table l, we are able to establish a superstructure of the KPS plant, as shown in Figure 2. Full connectivity of the processing units/storage vessels network is assumed. Mathematical
Formulation
To l:orlnulate the mathematical model, we require the following notation: Inclices."
tasks: .7 units; <s states; /~, event points representing the beginning of a task; Setx.
I tasks: /7 tasks that can be performed in unit (j); 1,2 tasks that either produce or consume slate (s): lj, processing tasks; I~ storage tasks;
.1 tlilils:-]i units that can perform task (i); ,It storage units; ~\: event points within the time horizon; 2Vz the last event point; ,%' all involved states; S~, states that can only be stored in storage units;
1161
Unit Heater
20-50
Reactor 1 50-70
Reactor 2 Still Vessel Vessel Vessel Vessel
Cost Task Time Model KPSLIN KPSLIN KPSNON Heating 1.0+0.0067b 1.0+0.0067b 2~176 100.0+0.2s Reaction 1 2.0+0.0267b 2.0+0.0267b ~'25 Reaction 2 2.0+0.0267b 2.0+0.0267b ~'25 150.0+0.5s Reaction 3 1.0+0.0133b 1.0+0.0133b ~~ Reaction 1 2.0+0.0167b 2.0+0.0167b 1~5 Reaction 2 2.0+0.0167b 2.0+0.0167b ~~5 120.0 Reaction 3 1.0+0.0083b 1.0+0.0083b ~~~ Separation 2.0+0.0033b 2.0+0.0033b i5~ 150.0+0.3s I1 (Hot A) 30.0+0.1 s I2(IntBC) 15.0+0.Is I3(IntAB) 10.0+0.1 s I4(Impure E) 20.0+0.2s
Capacity Suitability
70
50-80 4 10-30 5 10-60 6 10-70 7 50-100
Model KPSNON 100.0+0.2s 150.0+0.5s ~5
120.0 150.0+0.3s 15 30.0+0. ls 15.0+0.Is ~5 10.0+0. I s 12 20.0+0.2s 1"5
_
,
.
Table 1: Equipment Data for KPS Plant Parameters:
r~ market requirement for state (s) at the end of time horizon;
[ si, Psi proportion of state (s) produced, consumed from task (i), respectively; aij,/_3~j, 7~j constant term, coefficient and exponent of variable term of processing time of task (i) in unit (j), respectively; H time horizon; p.~ price of state (s); V,,,i,,. minimum and maximum size of unit (j), respectively 9 j , -lj,.,,,,,x j &j, ~j, "~j constant term, coefficient and exponent of variable term of capital cost of unit (j), respectively;
Variables: e(j) s(j) wv(i,n) yv(j,n) b(i,j,n) d(s, n) st(s,n)
binary variables to determine if unit j exists; size of unit j; binary variables to assign the beginning of task (i) at event point (n); binary variables to assign the utilization of unit (j) at event point (n); amount of material undertaking task (i) in unit (j) at event point (n); amount of state (s) being delivered to the market at event point (n); amount of state (s) at event point (n);
t~(i, j, 'n) starting time of task (i) in unit (j) at event point (n); t f (i, j, n) finishing time of task (i) in unit (j) at event point (n). Then the mathematical model involves the following constraints: Existence Constraints
yv(j,n) <_ e(j), Vj E J, n E N
(1)
Unit Size Constraints vmin
,j
e(j) < s(j) <_ Tvjrmax e(j), Vj E J
(2)
<(
"Q/~
H5
~,~
~m [-,~
<
~
~~ +
m
H3
m
<2
m
~.
m m
<~ ,
~
(9,, j ' )
but
-'
~.~.
.
~
~.~.
9
+
-~
~
~.
~
4-
9
::j-
--'-~.
0 "~
m
IV
o
,.-.,
.~
,...,.
~.
IV
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II
~ m
+
I
2M
~
~.
IA
IA
,--. ,..,.
II o 9
m
on
C~ C) CJq
,...,
~2 ~I) _. ,.-,
_
_
Tlic..;e coiistraints are written for different tasks (?,i ' ) performed in different units ~ a k cplace consecutively according to the production recipe. "Zero-wait" condition
+ 1) 5 t , f ( ' i ' ? , / ' , i ? , )
~
~ ( 2w , u ( x , ~ I , +1) - w u ( , i ' , 7 i , ) ) ti,, i' E I , j t ,J,>j' E c J 7 t >n, E N , 'Ti, # N ~
~..~.
-
,~
t'(/;;j.~i
(15)
1163 Combined with Constraints (13) and (14), these constraints enforce that task (i) at event point (n+l) starts immediately after the end of task (i') at event point (n) if both of them are activated. Time Horizon Constraints
t.f(i,j,n) <_ H, Vi C I , j E J~,n E N t s ( i , j , n ) <_ H, gi C I , j c J~,n c N
(16) (17)
Objective: Minimize
.j
S
7"/,
The ol!jective is to minimize the capital costs of units minus profits due to product sales. Other perl:ormance criteria also can be incorporated easily. We should note that due to the nonlinear models of processing time and capital cost, the resulting mathematical programming model is a nonconvex MINLP problem, which needs deterministic global optimization methods to determine the global optimal solution.
Computational Study The above mathematical formulation is applied to an example taken from Xia and Macchietto (1997). The process recipe, equipment data and plant superstructure are those we visited in a previous session (see Figure l, Table 1 and Figure 2, respectively). The production requirements for Product I and Product 2 are 40 and 60 respectively in the linear case of KPSLIN, while 20 and 30 respectively in the nonlinear case of KPSNON. The prices of Feed A, Feed B, Feed C, Product 1 and Product 2 are 0.001,0.002, 0.0015, 0.02 and 0.03, respectively. The time horizon under consideration is 12 hours. MINOPT, an advanced modeling language and algorithmic framework proposed by Schweiger and Floudas (1997), is used to establish and solve the resulting MILP/MINLP mathematical programming problems. The MILP problems are solved using CPLEX, a branch and bound method. Table 2 shows the results of the proposed formulation compared with the results found in literature. Xia and Macchietto (1997) transformed the formulation they presented into an alternative one without giving the necessary details of the transformation. In addition to the reported data of the transformed formulation, the corresponding data we obtain according to their original one are also presented here. It is shown that the formulation proposed in this paper has the following advantages: (i) It gives rise to a simpler mixed-integer optimization problem mainly in terms of a smaller number of binary variables. (ii) The optimal solution obtained corresponds to a better objective function value and consequently a better integrated design and scheduling strategy. (iii) The computational efforts required are significantly reduced, which makes it very promising to solve large-scale industrial problems. Conclusions In this paper, a continuous-time formulation is proposed for the design, synthesis and scheduling of multipurpose batch plants. A computational study is presented to demonstrate the effectiveness of the proposed formulation. The computational results are compared with those in literature and show that the proposed formulation results in smaller size MILP/MINLP mathematical models primarily in terms of binary variables and better objective values can be accomplished with significantly less computational efforts.
1164
Case KPSLIN
KPSNON
Formulation Xia and Macchietto (1997) This Work Xia and Macchietto (1997) This Work
Cost ($10 a) 585.62 572.898 495.11 490.433
Integer Variables 62 t 288 ~ 128 62 t 288 ~ 108
Continuous Variable 34 t 201 ~ 341 34 t 201 ~ 287
Constraints 122 t 425 ~ 877 122 t 425 ~ 722
CPU (sec) 24(i)7.62" 22.49** 7849.23* 7.31"*
Table 2: Results and Comparisons (t: reported based on transformed formulation; o: recounted based on original formulation; *: Sun Ultra station-1 ; **: HP-C160 workstation) Acknowledgments The authors gratefully acknowledge support from the National Science Foundation, the Mobil Technology Company, and Elf-Atochem Company.
References Barbosa-P6voa A. and Macchietto S., 1994, Detailed design of multipurpose batch plants. Coral). Chem. Engng. 18, 1013-1042. Grossmann I. and Sargent R., 1979, Optimal design of multipurpose chemical plants. Ind. Eng. Chem. Process Des. Dev. 18, 343-348. lerapetritou M. and Floudas C., 1998a, Effective continuous-time formulation for short-term scheduling: I.multipurpose batch processes. Ind. Eng. Chem. Res. 37, 4341-4359. Ierapetritou M. and Floudas C., 1998b, Effective continuous-time formulation for short-term scheduling: Ii. continuous and semi-continuous processes. Ind. Eng. Chem. Res. 37, 43604374. Kondili E., Pantelides C., and Sargent R., 1993, A general algorithm for short-term scheduling of batch operations -i. milp formulation. Comp. Chem. Engng. 17, 211-227. Schweiger C. and Floudas C., 1997, MINOPT : A Software Package for Mixed-Integer Nonlinear Oi)timization, User's Guide. Computer-Aided Systems Laboratory, Dept. of Chemical Engineering, Princeton University, NJ. Xia Q. andMacchietto S., 1997, Design and synthesis of batch plants - minlp solution based oil a stochastic method. Comp. Chem. Engng. 21, $697-$702. Zhang X. and Sargent R., 1996, The optimal operation of mixed production facilities - general formulation and some solution approaches for the solution. Comp. Chem. Engng. 20, 897904.
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
i165
A Novel S u p e r s t r u c t u r e and O p t i m i s a t i o n S c h e m e for the S y n t h e s i s of R e a c t i o n - S e p a r a t i o n P r o c e s s e s Patrick Linke § Vipul Mehta # and Antonis Kokossis + + Department of Process Integration, University of Manchester Institute of Science and Technology, PO Box 88, Manchester, M60 1QD, United Kingdom # Bayer AG, Corporate Technology, ZT-TE 7.1, 51368 Leverkusen, Germany A design tool is presented that allows simultaneous consideration of reaction and separation in isothermal and non-isothermal multiphase systems. The approach incorporates previous achievements in the area of multiphase reactor network synthesis with the representation being described in form of a superstructure of generic reactor/mass exchanger units and multi-purpose separators. The functionalities provided by the synthesis scheme are exploited by stochastic optimisation techniques. In contrast to past methods, the synthesis tool is applicable to general systems involving reaction and separation and is guided by the basic phenomena of reaction, mass transfer, and phase equilibria. Implementation of the methodology enables the development of targets and screening procedures that help the engineer to assess the system performance and review promising design options. The steps to generate the superstructure along with its modelling components are explained and two examples illustrate the efficiency of the approach. 1. INTRODUCTION Simultaneous consideration of reaction and separation in design can in many cases lead to a superior process performance. For most industrially relevant systems, the synthesis problem is characterised by a larger number of feasible design options and the models describing the phenomena are highly non-linear. Due to the complex design trade-offs, heuristics and graphical design approaches are highly likely to yield under-performing processes. Performance guarantees can be delivered by optimisation-based approaches. To date, such approaches have been proposed for special cases of the synthesis problem, including the synthesis of homogeneous, isothermal reactor-separator-recycle systems 1'2'3, the design of a reactive distillation column 4, and homogeneous reactor network synthesis considering intermediate separations 5'6. In order to address a more general synthesis problem, Papalexandri and Pistikopoulos v suggested the use of generic heat/mass exchange units to generate superstructures that are subsequently formulated as MINLPs. However, their representation of the reactor units poses a severe limitation to the approach if applied to multiphase reacting and reactive separation systems, as common mixing patterns cannot be represented by the units or combinations of these. Moreover, previous approaches make use of deterministic optimisation techniques and are hence severely limited by the nonconvexities in the mathematical formulations.
1166 A compact representation for multiphase reaction systems has been developed by Mehta and Kokossis 8'9 that allows to simultaneously address all possible mixing and contacting patterns between streams of different phases as well as temperature policies. The network superstructures are synthesised using a robust stochastic optimisation framework. Following this philosophy, a targeting and design tool for general reaction-separation systems is developed, which is not restricted to conventional process configurations but additionally facilitates all possible novel design options embedded in the network superstructures. 2. S Y S T E M R E P R E S E N T A T I O N The proposed system representation accounts for multiphase reaction and separation operations by introducing (i) generic multiphase reactor / mass exchanger units and (ii) multi-purpose separator units. The multiphase reactor / mass exchanger unit builds upon the compact representation of multiphase reactors introduced by Mehta and Kokossis a. It can either take the functionality of a reactor, a mass exchanger, or a combined reactor/mass exchanger unit by introducing decision variables indicating the existence of diffusional links between the different phases and reactions. If representing a reactor, one such unit consists of a homogeneous reactor compartment per phase present in the system. A reactor compartment of one phase exchanges mass via diffusional links with reactor compartments belonging to the same generic reactor unit but to a different phase. Temperature effects are incorporated by imposing temperature profiles on the reactor compartments or through a unit-based representation with all possible intermediate direct and indirect heat exchange 9. Mass exchanger units are realised by not considering reaction inside the compartments. By default, the mass exchanger units are modelled using a rate-based approach. As for the reactor units, all different mixing and contacting patterns can be realised by a single mass exchanger unit, e.g. for the case of a membrane unit, a single mass exchanger can represent well-mixed, co- and counter-current units as well as cross flow. Equilibrium based models such as equilibrium stages can easily be adopted for the mass exchangers. The mass transfer links between the different phases can be established or deactivated. For two-phase systems, this allows to account not only for multiphase reactors but also for homogeneous reactors to be present in the network. If more phases are present, more combinations exist, e.g. for the case of a gas-liquid-liquid system with one reacting liquid phase, a single multiphase unit can represent (i) a three phase reactor, (ii) a homogeneous reactor and an absorber/stripper, (iii) a reactive extraction unit, (iv) a gas-liquid reactor, or (v) an absorber/stripper. The multi-purpose separator units split existing streams of one phase into a number of streams of the same phase but of different compositions. These units allow to represent the many different unit operations that are commonly used to separate process streams into the desired products or intermediates. Additionally, by not restricting the feasibility of the separations, the units can be employed to identify the most beneficial separations and guide the choice of proper unit operations or give incentives for development of new separation technology such as solvent design. When representing unit operations, the tasks that are performed by the separator units are chosen according to the separation order of the mixture and the components present in the inlet streams. One such unit is associated with a single unit operation, e.g. distillation,
1167 crystallisation, or absorption. The individual tasks and the number of tasks performed by a unit are decision variables. If more than two tasks are performed, a single separator unit represents a sequence of simple separators featuring one feed and two products. The sequences are represented by task vectors and all different sequences can be realised by partitioning the vector. The separator units perform sharp splits between key components; however, additional degrees of freedom for the sloppiness of the splits can easily be introduced. It should be noted that the main aim here is to screen and scope beneficial interactions between reaction and separation. In this context, the sharp splits are in most cases sufficient to investigate the trade-offs present in the system. To allow comparison of design alternatives on a common basis, costs need to be associated with the separator units. Real costs can either be realised through regressed cost models or short-cut equipment sizing procedures. Approximate costs of splits between ke~ components can be calculated as a function of feed flowrates and compositions ~. Alternatively, if short-cut sizing methods such as the Underwood method for distillation design exist for a particular unit operation, these models can directly be employed in conjunction with approximate equipment cost models. The search space for the separations can alternatively be extended to include all combinatorially possible separations between the components present in the system. This allows to search for the most beneficial separations in terms of the system performance enhancement regardless of feasibility considerations. The designer can introduce biases so that separations known to be unrealistic can be excluded from the search. If the identified most beneficial separations can effectively be carried out by conventional technology, e.g. by distillation, this particular unit operation can confidently be employed for subsequent synthesis studies, associating one type of the multi-purpose separator units with the separation orders of the unit operation. Although the identified separations might not be known to be achievable by conventional technologies, they can be used to give incentives for the development of novel separation techniques such as the design of novel solvents. 3. R E A C T I O N - S E P A R A T I O N S U P E R S T R U C T U R E A superstructure of multiphase reactor/mass exchanger units and multipurpose separator units is generated featuring all novel and conventional structural alternatives that can be realised by combinations of generic units. In each phase (or state) present, a network of intraphase streams exists that realises all feasible connections between the different units employed in the representation. Convective links between different phases are established via an interphase stream network of all feasible connections between units of different phases. The interphase streams are associated with equipment used for converting the state of streams such as reboilers, condensers, pumps, compressors, throttles, or turbines. Two of the many possible designs that can be obtained from the superstructure are illustrated in Figure 1. A layout of a reactive distillation system integrated with a separation sequence is shown in Figure la. A sequential arrangement of reactors and separators is shown in Figure lb. It should be noted that the representation is not restricted to any particular system, but instead can be applied to a variety of reaction, separation or combined processes such as extraction and reaction, reactive crystallisation, and membrane networks amongst others.
1168
Fig. 1 Special cases embedded in the network superstructure
4. NETWORK OPTIMISATION The network representation outlined above is used to formulate a mathematical model that is subjected to optimisation. This model facilitates the balances for each compartment in each phase as well as all separator units, stream mixers and splitters. The equilibrium relationships, which are required to model the mass transfer links, the kinetic models as well as the general regression models for costs or hydrodynamics may result in a highly nonlinear model. Optimisation variables include the component flows, the volumes of the reactor units, the temperatures within the network, the existence of units and streams, the type of reactor units, the relative flow directions of the phases, the types and tasks of the separator units, the sharpness of the splits performed. Any expression of these variables as well as the stream compositions can be used as an objective function for the optimisation problem. Profit, annualised cost, network yield or selectivity are examples for possible objective functions. The network is optimised using stochastic optimisation techniques in the form of simulated annealing which has previously been successfully applied in homogeneous and multiphase reactor network synthesisS'9'l~
5. EXAMPLES 5.1. Multiphase Denbigh reaction system The following series/parallel reactions occur in the liquid phase of the gas-liquid system: A+F A + 2F B+F B + 2F
----, ~ --, --,
B + G; D + 2G; C + G; E + 2G;
rl
- kl
r2
-- k2 CA CFCF
CACF
r3 = k3 CA CF r4
= k4CACFCF
(1) (2) (3) (4)
Component A is fed to the liquid phase where it reacts with component F to form the desired product B and by-product D. Product B further reacts to by-products C and E. The gas feed consists of reactant F and insoluble species H. The liquid phase components A, B, C, D, and E are non-volatile, whereas the solubility of by-product G is low. A multiphase
1169 reactor network synthesis study has been carried out for this system ll, establishing a target yield of 65%. The obtained designs feature co- and counter-current PFRs and a complex network of high flowrate recycle streams in the gas phase. The yield of the system is optimised allowing any combinatorially possible separation to be performed in (i) the gas phase only and (ii) in both phases present. In the former case, an improvement in terms of the performance target can not be observed for this system as compared to the results from the reactor network optimisation. The optimal separations aim at low concentrations of reactant F inside the reactor modules. However, the same effect is achieved via the excessive recycle flowrates observed in the reactor design study. In terms of structural complexity, less reactor units are generally observed when separators are present in the gas phase and the recycle flowrates are significantly reduced. As in the reactor design study, the reactor units consist of co- and/or counter-current PFRs. A significantly better system performance is observed when separators are additionally considered in the liquid phase. The yield for the system is consistently found to be above 95%, i.e. almost complete conversion of reactant A into the desired product. A typical design that achieves the target is shown in figure 2. Separations in the liquid phase aim at removing the desired product B from the network and completely recycling reactant A to the reaction zones. The by-products are found to be either partly recycled or completely removed. The reaction sections feature co-current and counter-current PFR units with side streams in the gas phase. CSTRs are occasionally present but always occur in conjunction with the PFR units. The separator units in the gas phase are not observed in most designs and a number of structures do not exhibit gas recycles and separators at all. This suggests that by choosing the proper process structure in the liquid phase, the previously identified bottleneck in the gas phase vanishes.
5.2. Production of Ethylbenzene The alkylation of benzene with ethylene is investigated. Gaseous ethylene (E) reacts with liquid benzene (B) to desired product ethylbenzene (EB). Four transalkylation reactions involve production of by-products diethylbenzene (DEB) and triethylbenzene (TEB). All reactions are assumed to be of first order with respect to each reactant. The multi-purpose separator units assume the functionality of distillation (separation order: B/EB/DEB/TEB) and the profit is to be maximised for a benzene feed of 10kmol/hr. Optimisation of the homogeneous system under the assumption that the liquid phase is saturated in ethylene yielded a profit target of around $730k/yr and sequential designs featuring a reactor, a direct separation sequence, and recycles of B, EB, and TEB. Reactor designs involve PFRs and in some rare cases CSTR/PFR combinations. These structures have been identified in earlier studies of this process 2'3. However, no information is gained on the interactions between liquid and gas phase. Optimisation of the multiphase system without consideration of the inter-phase stream network yields a slightly improved profit target. The optimal designs feature reactor-separator-recycle arrangements with intermediate reactant, by-product, and off-gas recycles as well as a direct separation sequence. Reaction zones are consistently found to be counter-current PFRs. Network optimisation of the general reaction-separation superstructure including the interphase stream network yields a target profit of around $860k for this process, an increase of around 20% as compared to the previous case. Optimal designs again feature
1170
I,
<
"- Liquid feed
,.
-~ g
R1
as0 Sl
EB
L
DEB
Liquid product (B)
Liquid product (ABCDE)
Fig. 2. Typical structure from the network optimisation for example 1
> EB
E > TEB
Fig. 3. Typical structure from the network optimisation for example 2
counter-current PFRs and a direct separation sequence; however, the reactor units are interconnected via intra- and inter-phase streams and show similarities with reactive distillation. A typical design is shown in figure 3. The improvements in profit stem from low component flowrates through the separation sequence with the separation of benzene being performed inside the reactor units. 6. CONCLUSIONS A general design tool for the synthesis of reaction-separation systems has been presented. The proposed network representation results in a rich and inclusive structure that embeds various mixing, phase distribution, and separation options. The stochastic optimisation scheme employed for network synthesis can handle the complexities resulting from kinetic expressions, phase equilibrium and mass transfer relationships. The proposed methodology can be used to analyse the problem trade-offs and to suggest incentives for the development of novel process structures. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Pibouleau, L., P. Floquet, and S. Domenech, AIChE J., 1988, 34, 163. Kokossis, A.C. and C.A. Floudas, Chem. Eng. Sci., 1991, 46, 1361. Smith, E.M.B. and C.C Pantelides, Comp. Chem. Eng., 1995, 19, $83. Ciric, A.R. and D. Gu, AIChE J., 1994, 40, 1479. Balakrishna, S. and L.T. Biegler, Ind. Eng. Chem. Res., 1993, 32, 1327. Lakshmanan, A, Biegler, L.T., Ind. Eng. Chem. Res., 1996, 35, 4523. Papalexandri, K.P. and E.N. Pistikopoulos, AIChE J., 1996, 42, 1010. Mehta, V.L. and A.C. Kokossis, Comp. Chem. Eng., 1997, 21, $325. Mehta, V.L. and A.C. Kokossis, Comp. Chem. Eng., 1998, 22, S 119. Marcoulaki, E. and A.C. Kokossis, Comp. Chem. Eng., 1996, 20, $231. Mehta, V.L., Ph.D. thesis, UMIST, UK, 1998.
European Symposiumon ComputerAided Process Engineering- 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1171
Short term Product distribution plan for multisite batch production, warehousing and distribution operations: solutions through SupplyDemand network and Resource-Task network optimisation approaches B. P. Das a*, N. Shah a and A. D. Dimitriadisb Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, London, SW7 2BY, UK. b i2 Technologies Ltd, Burnham, UK. c Unilever Research Division, Port Sunlight, UK. ABSTRACT An industrial production-distribution planning problem encompassing multisite batch processing plants, warehousing and distribution operations, containing a large number of process operational constraints has been investigated using both Supply-Demand Network and Resource-Task Network approaches. Both approaches have been able to solve the problem optimally and generated an integrated product distribution plan; but the distribution patterns are found to be different. Detailed analysis indicate that the Resource-Task Network approach offers a higher potential in solving varieties of integrated planning problems including complicated batch process operations.. 1. I N T R O D U C T I O N The work reported here is based on a short term multisite production and distribution scheduling problem posed to the Centre for Process Systems Engineering by a multinational manufacturing and distributing company. A literature survey showed that the most common approach in solving such problem is through the mathematical formulation of a network system [ 1-3] which requires specifying the exact "capacity" of the plants as well as the constraints associated with those plants so that an achievable plan can be developed. However for batch process plants it is fairly difficult to specify the exact capacity, particularly due to their capability of producing varieties of products with different processing times and process routes. A survey of the literature indicates that these facts are not considered while developing the production and distribution plan for such plants. Generally, overall static numbers such as machine hour, labour hour, tons of items etc. are used to represent the capacity of these plants in the formulation. This may be misleading. Perhaps the inclusion of more detailed/dynamic capacity restrictions of the plant in the formulation will be more appropriate. * Author to whom all correspondence should be addressed.
1172 Therefore, a slightly different concept is adopted in solving the problem. In this concept, all the operational activities and the constraints of the batch process plants are captured and linked with the warehousing and distribution activities of the system. Then an integrated product distribution plan is developed through an integrated mathematical formulation. The novelty of this concept is the inclusion of the dynamic capacity of the plant in the formulation as it changes with operational activities such as product type change over, product size change over etc. Based on this concept, two different approaches in solving the problem have been investigated. In the first approach, a Supply-Demand Network methodology is used and in the second approach the Resource-Task Network [4] technique is used. For both approaches, production and distribution plans from the sources to destinations are developed by optimising appropriate mathematical models. Section two contains a brief description of the problem which is used for this investigative work. Section three deals with Supply-Demand Network approach. Section four deals with the Resource-Task Network approach for solving the planning problem. In section five, the whole process has been compared using the results obtained so far. Finally, in section six, some conclusions are made highlighting the areas where further investigations are required. 2. PROBLEM DESCRIPTION The organisation has three highly flexible batch processing plants which can make varieties of products. The capacities of these plants are different, but each plant can make all the products. Briefly, the process operations include Mixing, Intermediate storage and Packaging in a range of equipment. The packaged units are despatched to distribution outlets via three warehouses. Considering the type, size and variants each of these plants can make at least 288 varieties of products in terms of final stock keeping units (SKU). The company has set up eighteen distribution outlets at various locations for servicing the local distributors. Based on weekly demand of the distribution outlets, the company is keen to know the best production and distribution plan so that the three batch-processing plants can develop feasible schedules to make these finished products and despatch them. Some of the important data relating to the problem are given in Tables l a and lb. 3. SUPPLY-DEMAND NETWORK A P P R O A C H - Product distribution plan In this approach, the entire system from the source (including process operations) to destinations has been treated as a Supply-Demand Network (SDN) system. Figure 1, shows a simplified network system of the problem, which contains three sets of nodes: Process plants (P), Warehouses (W) and Distribution outlets (D). Process plant nodes include a number of sub-processing nodes representing various operations of a batch plant such as mixing, packing etc. Each node of the system performs either the function of supplying materials or receiving materials or both. Due to limitations of space, the mathematical formulation of the problem is not included in this paper. Briefly, the basic objective function of the mathematical formulation (LP model) is to minimise the transportation cost of final end product (SKU) from the source to destination while satisfying the distribution outlet's demand. The formulation considered: (1) Capacity constraints of each process plant which included the process route, product types, sizes, variants etc., and the process equipment specifications as shown in Table la, (2) Material
1173 balance constraints across the entire network system and (3) Demand constraints of distribution outlets. The methodology applied in deriving the solution is very similar to the Transshipment Model reported in the literature [1,3]. Based on this formulation, an application program was developed using the GAMS modelling language [5] and the CPLEX solver [6] which ran successfully on an Ultra workstation and gave an optimum solution within reasonable time. The distribution plan of a typical product from process plants to the warehouses as obtained from this solution is shown in Table 2a. In Table 2b the distribution plan of the same product from warehouses to various distribution outlets is given.
4. RESOURCE-TASK NETWORK APPROACH- Product distribution plan In the second approach, the entire system from the source (including process operations) to destinations has been treated as a R e s o u r c e - T a s k N e t w o r k (RTN) system - a framework recently presented by Pantelides [4] for a range of process scheduling problems. Figure 2 shows a simplified network system of the problem under consideration, where a rectangular box represents a task or activity and a circle with a directed arc represents a resource. The node of this network system is the task and all the inputs to the task and outputs of the task are treated as resources including the unit / equipment item in which this task is carried out. Again, due to limitations of space, the mathematical formulation of the problem is not included in this paper. Detailed works on the RTN system are now available [7-8]. The objective function of the formulation is to maximise the total profit while satisfying the distribution outlet's demand. Briefly the formulation considered: (1) Excess resource balance constraints (2) Excess resource capacity constraints and (3) operational constraints. Based on this objective function and constraints, a first order RTN aggregate model was set-up and a program was written in GAMS modelling language. The program used the OSL solver [9] and ran successfully on an Ultra workstation which gave an optimum solution within reasonable time. The distribution plan of a typical product from process plants to the warehouses as obtained from this solution is also shown in Table 2a. Again, in Table 2b the distribution plan of the same product from warehouses to various distribution outlets is given. 5. C O M P A R I S O N Having developed two optimised production and distribution plans using two different approaches, it is now possible to compare the whole process. Compared to the SDN approach, the RTN approach requires a longer solution time due to the presence of integer variables. Very interesting observations can be made from the product delivery plans presented in Table 2a and 2b. First of all, it is clear from Table 2a, that the plan generated by SDN approach, all the plants will deliver this product to their local warehouse, instead of distributing them to various warehouses, due to the lower transportation cost. On the other hand, in the distribution plan generated by the RTN approach, Plant 1 will deliver this product to all the three warehouses and incidentally, Plant 3, will not deliver any of this material to the three warehouses. But both approaches have satisfied the demand. This variation may be due to more rigorous formulation of RTN system, which has taken into account of product changeovers and storage facilities in detail. Next, from Table 2b, it appears that compared to the RTN approach, the spread of delivery service provided by each warehouse in the SDN approach is very limited. For example, Warehouse one will distribute the same material only to two distribution outlets using the SDN approach, whereas using RTN approach, the
1174 Warehouse one will distribute the same material to five distribution outlets, and therefore will have more distribution routes to cover. Similar observations can be made for other two warehouses. 6. CONCLUSION In this paper, two methodologies have been introduced and compared for developing optimised production- distribution plans encompassing multisite batch process operations, warehouses and distribution outlets using a novel concept where dynamic capacities of batch process operations and other associated constraints are included in the formulations. Both approaches have been able to solve the problem requiring transportation of large number of variants of SKU. However, on the whole, it appears that the RTN approach is more realistic and offers higher potential for handling complicated process events. It is therefore necessary to continue further research work with the RTN approach to prove its value. The SDN approach may be difficult to manage for complicated batch process operations involving large number of process equipment, because the formulation relies mainly on the material balance of all the streams at each node. Although both formulations can generate optimised plans by taking into account of dynamic capacities of batch process plants, which should to a large extent ensure that these plans are achievable, still there is no guarantee that they are fully feasible at the plant level. Therefore there is still a need for confirmation of the feasibility of these plans at the batch process plant level which is usually done by performing detailed short term scheduling of the proposed plan. To this extent, the Centre for Process Systems Engineering has already developed a short term production scheduling package "gBSS" [10] which is ideal for such feasibility analysis. The end results which are obtained from this work can be easily interfaced to the gBSS software package. Preliminary results of this analysis with this package are very encouraging.
REFERENCES 1. Anderson, D.R., Sweeney, D.J., and Williams, T.A., An Introduction to Management Science Quantitative Approaches to Decision Making, Eighth edition, West Publishing Company, USA, (1997). 2. Heizer, J and Render, B., Production and Operations Managemnt- Strategies and Tactics, Third edition, Allyn and Bacon, USA, (1993). 3. Taha, H.A, Operations Research An Introduction, Fourth Edition, Macmillan Publishing Company, USA, (1987). 4. Pantelides, C.C., Unified frameworks for optimal process planning and scheduling. In Rippin, D.W.T. and J. Hale, editors, Proc. Second Conf. On Foundations of Computer-Aided Operations, pages 253-274 (1994). 5. GAMS, Release 2.25, A User's guide, GAMS Development Corporation, Washington, DC, USA, (1996). 6. CPLEX Linear Optimizer 6.5.1, ILOG Inc, USA, (1999). 7. Wilkinson, S.J., Aggregate Formulations for Large Scale Process Scheduling Problems. PhD thesis, University of London (1996). 8. Dimitriadis, A.D., Algorithms for the solution of Large-Scale Scheduling Problems. PhD thesis, University of London (1999). 9. OSL solver, Release 2, IBM Inc, USA, (1998).
1175
10. gBSS, version 1.1., User manual and language reference, Process Systems Enterprise Ltd, London, (1998). Table la
Table lb
Process equipment specifications
Transport cost per ton
Plant
Plant 1
Plant 2
Plant 3
Item
Mixer 1 Mixer 2 Line A Line B LineD Mixer 1 Mixer 2 Line 5 Line 7 Line 8 Line 10 Mixer 1 Mixer 2 Line 2 Line 3 Line 4 Line 6 Line 7 Line 8
Avl. hr
Bat. size
Bat. time
Eric. %
per week
ton
(min) D C
112 112 105 105 105 112 112 120 165 120 120 112 112 120 105 105 150 105 105
15 15
52 0 52 75
85 85
15 15
52 52
85 85
0 75
Rate ton/
36 36
66 66
Locn.
hr x x 3.15 6.3 10 x x 4.5
1.26
15 15
Suitable for
85 85
6.48 2.25 x x 4.2 14.4 4.2
1.47 7.74 1.5
W1
W2
W3
0
71
64
Plant 1 D
C
Y Y N Y Y
N Y Y Y N
Y Y Y Y Y N Y Y Y Y N N Y Y
N Y N Y N Y Y Y N N N Y N N
Plant 2 71 0 81 Plant 3 60 6! 0 D1 0 0 0 D2 0 0 0 D3 140 60 135 D4 43 56 17 D5 0 0 0 D6 0 0 0 D7 41 54 18 D8 41 46 36 D9 46 50 36 D10 70 4 61 Dll 39 49 11 DI2 0 0 0 D13 23 104 79 DI4 0 75 63 D15 0 0 0 D16 0 0 0 DI7 0 0 0 D18 0 0 0 Note: D = Distribution outlet
Table 2b
A typical weekly product delivery plan from Warehouses to Distribution outlets: Dilute material 1 Outlet
Demand (ton)
Supply-Demand Network approach (SDN)
W3
Resource-Task Network approach (RTN) W1 W2 W3
W1
W2
D1 D2 D3 D4 D4 D6 D7 D8 D9 D10 DI 1 D12 D 13 D 14 DI5 DI6 DI7 DI8
0 0 67.04 30.92 0 0 141.25 82.96 386.51 257.66 54.61 0 43.64 932.66 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 43.64 932.66 0 0 0 0
0 0 67.04 0 0 0 0 0 386.51 257.66 0 0 0 0 0 0 0 0
0 0 0 30.92 0 0 141.25 82.96 0 0 54.61 0 0 0 0 0 0 0
0 0 25 0 0 0 55 0 0 0 27 0 36 232 0 0 0 0
0 0 44 27 0 0 10 84 249 177 23 0 9 448 0 0 0 0
0 0 0 5 0 0 72 0 140 83 5 0 0 255 0 0 0 0
Total
1997.25
976.3
711.21
309.74
375
1071
560
Grand total (ton) 91997.25
Table 2a
Grand total (ton) 92006
1176
A typical weekly product delivery plan from Process Plants to Warehouses: Dilute material 1 Process Plant
Supply-Demand Network approach W1 976.30 0 0 976.30
Plant 1 Plant 2 Plant 3 Total
W2
Resource-Task Network approach
W3
W1 340 40 0 380
0 0 711.21 0 0 309.74 711.21 309.74 Grand total (ton) : 1997.25
W2 W3 151.93 560 919.06 0 0 0 1070.99 560 Grand total (ton) : 2010.9
L
Process Plants
Warehouses
Distbn. outlets
Fig. 1. A simplified Supply-Demand Network system of example problem
F
IV
FPAP
RTP
T2
FPAPD Mixer
S Tank
Pack line
KEYS: IV= Intermediate variants RTP= Ready to pack FPAP= Final product at plant TPAW= Transported product at warehouse FPAPD = Final product at place of demand F- Feed M= Mixing S= Storing P= Packing TI= Transportl T2= Transport2
Fig. 2. A simplified Resource-Task Network system of example problem: Plant 1
European Symposium on Computer Aided Process Engineering - 10 S. Pierucci (Editor) 9 2000 Elsevier Science B.V. All rights reserved.
1177
A New Generation in ERM Systems: The Tic Tac Toe Algorithm Mariana Badell and Luis Puigjaner Department of Chemical Engineering. Universitat Polit6cnica de Catalunya E.T.S.E.I.B., Diagonal 647, 08028-Barcelona, Spain e-mail: [email protected] [email protected] In this work are briefly reviewed the management achievements and the present challenges to the batch process industry, giving special emphasis to the current difficulties that face up the order management systems. The enterprise resource management systems (ERM) require real time execution and an efficient transaction-oriented approach. The system performance proposed is based on the Tic Tac Toe algorithm, which follows an exact non-combinatorial deterministic approach capable of giving not only the optimum, but all the optimal solutions for a scheduling problem in multiproduct plants in very short time. The resulting tool is appropriate to fulfil the requirements of autonomous order entry systems in integrated real time systems. 1. INTRODUCTION Good management practices are necessary in process engineering during the exploitation phase to preserve the firm during its whole lifecycle. The driving force is not the same as years ago: lowering costs or filling manufacturing capacities. Now low costs are a trivial consequence of other decisive facts, which are shorter lead-time designs of new products and processes, quality, service flexibility, reliable availability. On line schedulers tracking the production events and the financial resources in ERM systems can optimise the planning activities and assure the solvency during the nowadays short lifecycle models of products and processes. These powerful tools must support the challenge of making profitable the process dynamic and product innovation. The ERM system is a decision-making tool for the manufacturing industry, which makes integrated financial/production trade-off planning and optimisation in the supply chain management (Badell and Puigjaner, 1998). It is particularly different to the predecessors: MRP, MRP II extensions, MRP II + DCS, MES or ERP. The ERM system pretends to be a system (because works as a whole) with cybernetic quality (because possesses auto-regulation possibilities), and that is why is defined as an enterprise 'managementsystem'. The application supports management cycles because during its use are available degrees of freedom to make the system work as a decision-making tool. The actual 'ERP' type systems do not fulfil this role because they stay at the integration of information; they do not develop multi-functional integration as ERM does. The production cycle is money~materials~production--,product~ sales~money and this cycle is not considered in its entirety in the more than 350 types of ERP systems available. Although these information-gathering systems are so appreciated, the most important strategic variable - enterprise finance - is not located as just at any hierarchical level. As the time variable controlled production events, the ERM business-controlled variable is money, which requires real time management of data with concern about the state of the process, the finance and the market. At the present ERP systems use web-based environments and XML to manage customer orders. The companies must be turned in make-to-order
1178 operation linking the actual/forecasted orders instantly to the plant floor. Simultaneously plant's logistics, inventory control, maintenance, production quality, accounting, finances and sales functions must be updated. But, could the sales department estimate a precise profitable price and an adequate time of delivery at the order entry? The firm's dilemma in the make-toorder policy is what price and due date to quote when a potential customer requests a bid. When the production costing is given simultaneously with the real plan schedule as ERM does, immediate and reliable decisions regarding market pricing and production plans can be made. However, if the firms have to make quotations within a limited time, the accurate estimation of processing time is not assured. In multi-site companies the situation is worst. Cannot solve this dilemma the sophisticated process control, modelling, and optimisation arsenal available to aid the decision-making process? The repertoires of optimisation techniques to locate optimal delivery times during the firm's planning activities are laborious, not always arriving to an optimum (Sinclair and Zentner, 1999). Many practical problems are too large to the computer and not physically realistic. Quoting standard lead times is not enough if competitors provide answers while the customer is still on line. In the new paradigm the competitive advantage could be determined in the rapid and accurate generation of optimal quotations. In order to take advantage of the possibilities given by the enterprise overall optimisation in real time, new approaches to solve old complex problems must be developed in a new generation of management packages type ERM systems. The Tic Tac Toe algorithm is a deterministic optimisation technique, which can optimally sequence orders in on line web-based order entry systems of multiproduct plants. 2. EXACT OPTIMAL SCHEDULING SOLUTIONS: TIC TAC TOE ALGORITHM In the family of combinatorial problems the travelling salesman problem (TSP) is representative of a large number of important scientific and engineering problems, one of them the scheduling problem. The interest in finding better ways to solve combinatorial problems goes beyond scheduling and is growing, in part due to the integrated systems requirements. No efficient algorithm for solving the TSP has been conceived that is guaranteed to produce a minimum time cost solution giving an exact solution for every case. The theory has gained almost mythical significance because, despite years of effort, no algorithm for the TSP has been found to meet the standard of efficiency. This failure has led to the widely held conjecture that no such efficient algorithm will ever be found, giving the space to a huge repertoire of approximate acceptable solutions to real problems (Miller and Pekny, 1999). As a large number of important problems, the combinatorial scheduling task can be formulated as an asymmetric TSP (ATSP), which has proven considerably more problematic and less attention. Pekny and Reklaitis (1998) remarked that scheduling problem can be formulated as X e S = (F, 1) where X is the schedule from the set S of possible solutions, which can be partitioned in feasible, F, and infeasible,/. The optimal solutions can be symbolised as: optimise objective (X) such that X e S = (F, 1) where objective (X) is the measure of merit of the solution. The most difficult alternative is X e S optimise objective (X) such that X e F, which requires, besides the optimal schedule, the satisfaction of all the engineering constraints. ERM systems must test not only the traditional time-capacity engineering constraints (equipment and resources), but also money-time-capacity constraints to optimally plan the financial and production resources. Pekny and Reklaitis reviewed that the strategies addressed to accommodate NP completeness in combinatorial problems are: making easier the problem; accepting an unreasonable time to search exact solutions; or use a heuristic algorithm that may
1179 get a good/poor solution. With these possibilities the actual above mentioned problematic can not be solved. Nowadays the requirements of integrated enterprise systems are at a higher position: automatic real time scheduling tools giving optimal suggestions with high reliability. The sequencing process problem can be interpreted as an ATSP due to the fact that the sequences of batches do not have equal opposite time 'distance'. The Tic Tac Toe algorithm is based on equivalent matrixes to the inter-city distance matrix, the matrixes of overlapping MoT(i, i ') or not overlapping times MNor(i, i ~). A previous version of this algorithm calculated the optimal sequence by an objective function of maximum overlapping time (minimum makespan) (see Fig. 1). In this work, a best alternative is considered: the minimization of not overlapping times. Notice that the scheduling problem with this approach is not only the ATSP because the solution also must include the process time of the head or first element of the sequence.
Figure 1. Details of the variables considered in the minimisation objective function, OTii' and NOTii' are the overlapping and not overlapping times respectively Once the algorithm is applied the matrix MoT(i, i ') or the MNOT(i,i ') of all the possible binary sequences of a plan is determined using the recipes of products. A Solver determines the optimal sequence for minimum makespan. For example, in a case of 12 products shown in Fig. 2 was found '8-1-9-5-11-7-3-12-6-4-10-2' as optimal sequence. A set of equivalent to optimal cells 1 are marked totalling 34 usable cells for the whole set of optimal sequences (highlighted in Fig. 2). The information structure obtained permits to determine the resting optimal solutions with different sequence order, creating better adaptive optimal plans to the due date preference. The Generator determined 144 optimal sequences.
Figure 2. The test-bed Solver program. Equivalent cells in Mot(i, i') for the Generator (right). A cell is a distance arc of a graph or an element of a matrix of overlapping/not overlapping time.
1180 In this work is considered the minimisation of the not overlapping times using the matrix of not overlapping times, MNor(i,j) which will be considered equivalent to the inter-city distances matrix. Also is considered a quantitative road map matrix. With this approach the problem is decomposed in two different inter-related sub-problems: the selection of the first process of the sequence (head) using the SPT (shortest process time) priority law and the scheduling ATSP. In order to solve the scheduling task using the MNor(i,j) of n batches, which form a n x n square matrix, must be found the sequence, which minimises not only the 'not overlapping times' but also the makespan. A comparative value of dispersion by row or column in the MNor(i,j) set of minimum cells can be initially observed and consists in the difference between the sum of the n minimum values by rows and by columns (only one selection for each row and for each column) for an hypothetical best solution discarding the position constraints, e.g.,
RowMin -
ColMin
(1)
c=l
where r and c represent the row and column respectively; RowMin and ColMin are the minimum not overlapping times per row or per column. A positive result indicates that minimum values are more concentrated in the columns. The row with worst minimum (WorstRowMin) is the first candidate to be the tail of the sequence due to the fact that the production path is not a circuit and hence the last batch of the plan does not overlap with any other batch. Obviously, in every optimal solution this position is 'deserved' by the row with worst minimum. In other words, the tail of the sequence is the only one that does not have a complementary binary. For example, in the sequence '123456', the principal binaries are '12', '34' and '56' and the complementary binaries are '23' and '45' without which the not overlapping time of the sequence can not be calculated. Only the tail does not have complementary binary because having it converts the path sequence in a Hamiltonian graph, which is not the solution of a scheduling problem. The batch head of the optimal sequence of the muor(i,j) matrix coincides with the number of the column that must stay without a selected cell to fulfil the constraint of n-1 selected cells. The first candidate to be head of the optimal sequence is the one with the worst position to the ATSP solution, but at the same time must coincide with the production with shorter processing time (SPT priority law), so a trade off solution is needed during the scheduling task solution. In previous works of the authors a qualitative road map matrix was used as a pointer of the search. In that road map each cell of the MNOT(i,j) matrix was represented by a 'Minimum or not minimum' by row and/or column mark (Mm). In this work is introduced a quantitative road map supplying optimal precision to the search. For each matrix element is calculated the sum of the differences between a cell of the MNOT(i,j) matrix and its minimum by row and minimum by column. A new matrix DM~j, is formed, which cells dm O.are calculated using the not o. elements of mUOT(i,j)by the following equation:
dm i,j
--
[2(not i,j ) - - M r i , k - M r k',j ]
(2)
where its elements represent the deviation of each cell of the matrix MNor(i,j) in row i, column j from its minimal possible distance in the row i, column j. In matrix mUOT(i,j) if a cell was qualified as 'MM' now the value in matrix DM~,j.is zero. In other words, the elements of DM~j represent the "time cost" (assumed as intercity distance) between i and j of producing a product
1181 i before a product j instead of the product located in position k' (minimum value in row i) before the product located in the position k (minimum value in column j). In the Tic Tac Toe algorithm the DM,.j matrix plays the role of a pointer, which rigurously guides the search in a perfect increasing order from the best possible selection SELij in the MNOT(i,j), matrix to the worst. This specific ordering process plays a very important role due to the obvious fact that depending on the search order the constraints of position (only one SEL ij by row and column) will be activated. This ordering method assures the correct search path substituting the qualitative road map utilised by the authors in previous works. With the aid of the DM~j road map the minimum values, considering first the cells with dmo.= 0 (MM cells), are determined. Following the Tic Tac Toe algorithm the principal and complementary binaries represented by the matrix cells are selected for minimum not overlapping times (minimum makespan) avoiding the formation of sub-tours. Some details of the calculation of the M M o road map matrix and the algorithm can be formally represented as: AMM -- { (i,j) [ MM,.j = "MM" }, AtoM = { (i,j) [ MMij = "Mm "} AMm = { (i,j) [ MMi,j = "mM" }, Atom = { (i,j) [ MMi,j = "mm "} ANUL(i,j, XN,N) ={(a,b) ] (a=i v b=j v (a=j A b=i)) A--,(a=i a b=j ) ^ l_~t,b.__~V/x Xa, b--O} ORDERMM ((i,j),(i',j'),Xu, u)
-- (MNOTi,j < MNOTi',j' ) V (MNoTi,j = MNOTi'j' A
#{(a,b)~
ANUL(i,j, XN,N) U (j,i) I MMa, b= "MM"}_<# {(a,b)e ANUL(i',j',XN, N) U (j" i') [ MMa, b= "MM"} W = AMM while W r 0 ^ #{(a,b) ] PATHa, b = SEL} <_N - 2 let (a,b)~ W t.q. V ( c , d ) ~ W" ORDERMM((a,b),(c,d),PATH ) PATHa, b -- SEL
W = W - (a,b)- ANUL(a,b,PATH) V(c,d)~ ANUL(a,b, PATH)'PATHc, d = NUL W = (AM, k.) AtoM ) ('5 {(a,b) [ PATHa, b = 0} ibid... W = Atom 0 {(•,b) [ PATH., b : O} ibid...
where i, j are rows and columns respectively, MNOTi,j is the matrix of not overlapping time; MMij is the qualitative road map matrix (the quantitative road map for this version is DMo.); ANUL(i,j, XN,N) is the annulled matrix zone during selections; XN, N is a special dynamic assignment matrix that masks with zeros the active cells available for selection; and ORDER((i,j), (i ',j '),XN,N) is a Boolean ordering function. This formal representation is applied to Mm, mM, Mm and mm cells (or dm O.elements). In the case of using the DM 0. matrix as a road map the loops are not grouped by quality 'MM', 'mM', 'Mm' or 'mm'. The search is done by ordering in increasing order the dm o. elements. The matrix PATHij is used for capturing the selected cells SELo.. Finally the last step is the backward 'what if' step, which improves the obliged (n-l) th selection by testing all its (n-l) positions through optimising permutations between the already selected cells. The following case study is a discrete manufacturing plant with 10 products and 5 machines (see Table 1). The fight part of Table 2 shows the road map DM0.. The MNOTi,j. optimal solution '6-5-8-2-10-9-3-7-4-1' is highlighted.
1182 Table 1. Recipes of products 1-10 in a manufacturing discrete process.
Machines i=l j=2 j =3 j=4 ./=5 Proc.time, h
i=1 25 25 13 4 1 68
i=2 3 7 19 22 11 62
i=3 3 16 21 5 7 52
i=4 5 13 25 2 22 67
i=5 19 14 15 10 19 77
i=6 1 9 24 15 8 57
i=7 4 15 5 17 16 57
i=8 21 8 9 22 19 79
i=9 23 9 12 19 12 75
i=10 16 13 22 13 10 74
Table 2. The MNor(i,j) matrix of products 1-10, highlighted the optimal solution, and DMij.
A case study of 50 batches for a multiproduct plant of 5 units: 3 semicontinuous and 2 discontinuous was solved in a Pentium II in 1.4 s including the voluminous output data and preprocessing time of the recipes. The same solution was tried with a MILP formulation spending more than 3 h without arrival to optimum. In the Tic Tac Toe algorithm the Solver makes less than n schedules to find the optimal. The Generator obtains all optimal feasible (without subtours) scheduling solutions by a deterministic algorithm using the cells targeted as 'equivalent' ones. Next, the feasibility is considered taking into account the resting engineering and financial constraints. The scheduling tool MOPP, developed at the UPC, is used for the complete solution of the problem with an interactive electronic Gantt chart output. 3. CONCLUSIONS An open, web-based environment for collaborative engineering must exist during the exploitation of plants. A refined Tic Tac Toe algorithm could fulfil the optimisation role in on line web-based order entry systems. Also makes possible the consideration of all optimal sequences to best adapt optimal plans to the due dates preference of customers. The development and generalization of this novel approach could be a step toward the search of a standard solution to TSP formulations by non-combinatorial methods. The support of the European Community (BRPR-CT98-9005) and CICYT (QUI99-1091) is acknowledged.
4. REFERENCES M. Badell and L. Puigjaner, AIChE Symposium Series 94, 320, 217 (1998). M. Badell, J. Cant6n, and L. Puigjaner, Annual AIChE Meeting, 239b, Dallas, USA (1999). D. L. Miller, J. F. Pekny, Science, 25,754 (1991). D. R. Moodie, Production and Operations Management, 8, 2, 151(1999). J. F. Pekny and G. V. Reklaitis, FOCAPO'98 Final Agenda, July 5-10, Utah, U.S.A. (1998). G. Sinclair and M. G. Zentner, Chem. Engng. Progress, 8, 43 (1999).
1183
AUTHOR INDEX
Abreu L. 907 Adjiman C.S. 793 Adler P.M. 667 Adrover A. 451 Agachi $.P. 235, 271,289 Aittamaa J. 811 Akiya T. 661 Aldrich C. 103 Alexopoulos A.H. 43 Alonso A.A. 481 Alonso A.I. 877 Alpbaz M. 187 Alvarenga W. 1039 Amand T. 757 Andersen T. 331 Andersen T.R. 709 Andresen B. 709 Andreussi P. 739 Anne-Archard G. 1147 Antifora A. 859 Antonopoulos D.K. 967 Antonucci V. 295 Aoyama A. 1117 Araujo O.Q.F. 943 Arafijo P.H.H. 565 Arkun Y. 211 Artufel C. 385 Arvela J. 1105 Arzamendi G. 457 Asteasuain M. 559 Asteris G. 967 Asua J.M. 457,565 Aust E. 553 Aziz N. 175 Azzaro-Pantel C. 493, 1135 Badell M. 1177 Bahl V. 163 Bakker R. 883 Balasubramanian J. 79 Baldea M. 271 Baldi G. 697 Balsa-Canto E. 481 Bafiares-Alcfintara R. 703 Bandoni A. 487,559 Banga J.R. 37, 481 Barbosa-P6voa A.P.F.D. 715
Barnard J.P. 103 Barolo M. 1081 Barresi A.A. 697 Bartolozzi V. 199 Barton P.I. 655 Basualdo M.S. 259, 1153 Batistella C.B. 505 Batres R. 1117 Bek-Pedersen E. 955 Bemporad A. 301 Benedetto D. 421 Benz S.J. 805 Bergstedt U. 355 Betlem B.H.L. 1093 Bezerra V.M.F. 415 Biardi G. 763 Bildea C.S. 169 Binder T. 31 Birch J. 673 Bistolfi M. 379 Bitzer B. 925 BjiSrkqvist J. 13 Blank L. 31 Blotto P. 739 Bodizs L. 235 Bogle I.D.L. 157, 523 Bolland O. 331 Boltersdorf U. 433 Bonuccelli M. 739 Borland J.N. 721 Bouhenchir H. 601 Bozinis N.A. 301 Bozzano G. 649 Braga A.S. 361 Brahmadatta M. 547 Brandolin A. 559 Brignole E.A. 319 Brignole N.B. 127 Brusis D. 109 Bua L. 859 Bubbico R. 769 Cabassud M. 349, 601, 1147 Caldi M.L. 763 Calis H.P.A. 883 Canton J. 745, 1069, 1129 Capilla M. 583
Casamatta G. 349,601, 1147 Caussat B. 385 Cavin L. 889 CerdS. J. 1045 Chachuat B. 853 Chakraborty A. 901 Charalambous C. 1087 Chetty S. 961 Chigirinskiy M. 91 Cisternas L.A. 991, 1009 Cittadini M. 697 Coelho M.A.N. 361 Coletti S. 385 Constantinescu C. 841 Corvalfin S.M. 475,877 Couderc J.P. 385 Cristea V.M. 271 da Costa C.A.V. 361 Dahmen W. 31 Dail3 A. 427 Dal Cengio P. 1081 Das B.P. 1171 Dash S. 775 Davin A. 493,973 de la Cal J.C. 565 de Souza E.T.I. 253 Deerberg G. 97,355,433 DempfD. 541 Dente M. 649 Di Cave S. 769 Di Natale M. 913 Diamantis Z.G. 25 Diaz S. 319,487 Dijkema G.P.J. 631,727 Dimian A.C. 169 Dimitriadis A.D. 1171 Domenech S. 493,973 Drozdowicz B. 805 Dua V. 301 Duarte C.G: 403 Ducato R. 199 Duessel R. 109 Diinnebier G. 265 Eckert E. 313 Efthimeros G.A. 25 Eliceche A.M. 475,877
1184 Endo A. 661 Engell S. 1075 Erdogan S. 187 Espinosa J. 1033 Espinosa S. 319 Esposito W.R. 73 Espufia A. 283,745, 907, 1069 Estrada-Villagrana A.D. 157 Falcitelli M. 421 Faravelli T. 859 Farid S. 673 Fernandes F.A.N. 511 Feyo De Azevedo S. 247 Fidaleo M. 451 Fischer U. 889, 931 Floquet P. 493,973, 1135 Floudas C.A. 73, 1159 Fonyo Z. 643 Fox R.O. 427 Fraga E.S. 157, 637 Franchi D. 763 Frassoldati A. 859 Freitas Jr. B.B. 499 FreyT. 109, 115 Friedl A. 847, 919 Fukushima Y. 895 Galfin O. 211 Galinetto R. 739 Galluzzo M. 199 Gfilvez E.D. 1009 Gani R. 7, 157, 949, 955 Geiger G. 781 Gelbe H. 409 Georgiadis M.C. 1111 Gerbaud V. 1123 Gilles E.D. 205,547 Gimeno L. 1039, 1069 Giona A.R. 529 Giona M. 451,529 Giordano M. 295 Glavi~ P. 61 Glismann K. 1099 Goel H.D. 823 Gontarski C.A. 865 Goossens L. 751 Gouveia R. 871 Graells M. 1069, 1129 Gregoritza W. 781 Grievink J. 631
Grossmann I.E. 79 Gruhn G. 1099 Guerrero C.P. 991 Guerrini De Luca P. 607 Guirardello R. 397 Gupta A. 133 Guzmfin-Reyna A. 703 Hagesaether L. 367 Hale A.R. 751 Halim I. 829 Han C. 469 Hapoglu H. 187 Harasek M. 919 Harjunkoski I. 13 Harper EM. 949 Harrison B.K. 799 Heijckers C.P. 391 Henning G.P. 1045 Henriksen J.P. 7 Herder P.M. 823 Hertzberg T. 517 Heyen G. 757 Hillestad M. 517 Hindi K.S. 1087 Hirao M. 895 Hjarbo K. 367 Hostrup M. 955 Houston R.B. 1057 Huang H. 793 Huang K. 661 Hui C.-W. 133 Hungerbiihler K. 889, 931 Hurme M. 811, 1015 Hussain M.A. 175 Iedema P.D. 169 Ierapetritou M.G. 91 Imre-Lucaci ,~. 289 Inserra S. 295 Iordanidi A.A. 49 Iribarren O.A. 1021 Itigin A. 619 Jakobsen H.A. 367 Jankowitsch O. 889 Jeannerot L. 385 Jeong S.H. 55 Jer6nimo M.A.S. 361 Jim6nez L. 985, 1153 Jirfit J. 835 Joly M. 1063
Jorgensen S.B. 709 Joulia X. 55, 1123 Ju S. 469 Julka V. 985 Kabasci S. 355 KalitventzeffB. 679, 757 Karlsson S. 13,463 KarnerW. 919 Katsipou I.G. 25 Keeping B.R. 67 Kesavan P. 655 Kienle A. 619 Kim J.H. 469 Kim K.H. 733 Kiparissides C. 43 Klatt K.-U. 265 Klein E. 619 Kleinendorst F.I. 589 Klose F. 205 Knez Z. 577 Kobayasi M.S. 511 KiShler R. 547 Koiranen T. 1015 Kokossis A.C. 967, 1003, 1165 Kolhapure N. 427 Koller G. 931 Kong M.-T. 613 K~Srner H.J. 355 Kraslawski A. 595,625 Kravanja Z. 979 Krijnsen H.C. 883 Kronberg A.E. 49 Kubi~ek M. 667, 835 Kuipers J.A.M. 49 Kuppinger F.-F. 109 Kvamsdal H.M. 331 Lamas E.J. 805 Larrayoz M.-A. 583 Latifi M.A. 853 Lavric E.D. 439 Lavric V. 439 Law V.J. 277 Le Lann J.M. 55 Le Lann M.-V. 349, 601, 1147 Lebl A. 919 Lee I-B. 1135 Lelkes Z. 643 Lemkowitz S.M. 751 LiB. 19
1185
Li H.-S. 145 Li H.Z. 445 LiP. 121 Lim Y.I. 55 Lima E.L. 457, 943 Lin X. 1159 Linke P. 1165 Linninger A.A. 163, 901 Lintomen L. 343 List T. 541 Lombardi L.L.M. 529 Lona L.M.F 1027 USvik I. 517 L~Swe K. 151 Lu M.-L. 145 Luc J.-C. 583 Maciel Filho R. 223,241,253, 325, 415,499, 505 M~ihling F.O. 427 Mancini N. 379,739 Mangold M. 205 Marechal F. 679 Marek M. 667,835 Maria G. 841 Markert A. 1075 Marquardt W. 31,535 Marriott J.I. 523 Martin E.B. 19, 1051 Martins C. 361 Martins F.G. 361 Maschio G. 607 Matko D. 781 Mattedi A. 241 Mazzarotta B. 769 MehtaV. 1165 Meirelles A.J.A. 343 M6ndez C.A. 1045 Meng X. 1051 Mercado E.R.L. 397 Merola G. 913 Miquel J. 583 Mitrovi6 A. 547 Mockus L. 1057 Modigell M. 373,571 Monheim P. 571 Moraes E.B. 505 Moraes Jr. D. 511 Morari M. 301 Mori M. 865
Moro L.F.L. 1141 Morris A.J. 19, 1051 Mouline Y. 445 Mujtaba I.M. 175 Mulder P. 1093 MUller H. 307 Musmarra D. 913 Nagy Z. 235 Naito K. 661 Naka Y. 1117 Nakaiwa M. 661 Nakane T. 661 Nicolais L. 295 Nieto J.P. 385 Nougu6s J.M. 745 Nourai F. 937 Novais A.Q. 715 Nunhez J.R. 397,403 Nystr~Sm L. 595, 625 Oliveira R. 247 Olujic Z. 391 Ordieres J. 139 Oreki S. 61 Ortega F. 139 Ortiz I. 877 Owa M. 661 Ozen S. 187 Ozil P. 841 Ozkan G. 187 Pahor B. 979 Pajula E. 1015 Pakowski Z. 337 Palazoglu A. 211 Paloschi J.R. 229 Pantelides C.C. 67 Papageorgiou L.G. 1105, 1111 Pardillo-Fontdevila E. 1123 Park H.I. 1135 Pasini S. 421 Pasman H.J. 751 Pastor R. 907 Pedrosa S. M. C.P. 403 Pellegrini L. 181 Peres J. 247 Perrier M. 283 Petersen S. 571 Pettersson F. 463 Peuker T. 307 Photeinos D.I. 25
Pibouleau L. 493,973 Picciotto A. 199 Pickartz U. 571 Pinto J.C. 457, 565 Pinto J.M. 871, 1063, 1141 Pinto R.T.P. 343 Pinto T. 715 Pistikopoulos E.N. 301, 1105 Pizzo S.M. 511 Podenzani F. 379, 739 Pokki J.-P. 811 P~Srn R. 1 Prenem L.F. 865 Preul3 K. 1147 Puigjaner L. 283,745,907, 1069, 1129, 1177 Rainoldi R. 127 Raisch J. 619 Ranzi E. 859 Rashtchian D. 937 Ratto M. 181 Recasens F. 583 Reuter M.A. 727 Rev E. 643 Richters J. 925 Roche N. 853 Rodrigues L.C.A. 1039, 1069 Rodrigues M.T.M. 1039, 1069 Rodrigues P.R. 865 Rodriguez-Donis I. 1123 Roffel B. 1093 Romagnoli J. 211 Rong B.G. 595,625 Roque M.C. 1027 Rosa M. 1153 RosliSf J. 13 Ruiz D. 259,745 Russel B.M. 7 Sadr-Kazemi N. 721 Salamon P. 709 Salcedo B.J. 259 Salomone E. 1033 Samsatli N.J. 685 Sand G. 1075 Sayer C. 457 Scali C. 607 Scenna N. 805 Schanz M. 181 Scheffer R. 223
1186
Schliiter S. 97,433 Scholl S. 553 Schr~Sder K. 409 Schulz C. 1075 Schulz E. 487 Schulz R. 1075 Schupp B.A. 751 Schuster G. 847 Serra M. 283 Seuranen T. 1015 Shafiei S. 973 Shah N. 613,685,793, 1171 Shah P.B. 1003 SharifM. 685 Sharratt P.N. 721 Shayegan J. 937 Shin D. 733 Simonin O. 385 Siragusa G. 709 Skerget M. 577 Skrifvars H. 463 Sogaro A. 763 Song J.H. 733 S~Srensen E. 193,523 SouzaV.C. 397 Srinivasan R. 829 Stanbridge D. 391 St~pfinek F. 667 Stichlmair J. 109, 115, 691 Stoy S.F. 7 Sugaya M.F. 325 Svendsen H.F. 367 Swanborn R. 391 Swaney R.E. 991 Sweeney P. 91
Szitkai Z. 643 Tahmassebi T. 1087 Takamatsu T. 661 Tantimuratha L. 967 Teoh H.K. 193 Titchener-Hooker N. 193,673 Tognotti L. 421,859 Tomas E. 253 Tonelli S.M. 559 Toselli L. 1153 Trabelsi F. 583 Traebert A. 571 Tsahalis D.T. 25 Turner M. 193 Ubler C. 553 van den Bleek C.M. 883 van Goethem M.W.M. 589 van Impe J.F. 37 van Leeuwen C. 589 van Leeuwen J.C.M. 883 van Velzen N. 589 Van~k T. 313 Vanni M. 697 Vasconcelos C.J.G. 217 Vasquez V.R. 85 Vassiliadis C.G. 1105 Vassiliadis V.S. 481 Vazquez G.E. 127 VenkatasubramanianV. 775, 787 Venselaar J. 817 Verhoef E.V. 727 Versyck K.J. 37 Vin J. 91 Vinson J. 1057
Viswanathan S. Von Wedel L. Wack T. Wagner I. Wall K. Wanhschafft O.M. Warter M. Washbrook J. Weigl K. Weijnen M.P.C. Wendt M. Weng M. Westerlund T. Westerterp K.R. Westphalen D.L. Whiting W.B. Wolf-Maciel M.R. Wozny G. Wukovits W. Xaumier F. Xin Y. Xu S. Yamada I. Yang A.D. Yoon E.S. You S.H. Zamar S.D. Zeitz M. Zhao J. Zhelev T.K. Zupan J. Zyngier D.
787 535 97 109 721 985 691 673 847 823 121 373 1, 13,463 49 997 85 217, 343, 505, 511,997 121, 151,307 919 349 85 1021, 1033 1117 145 733 469 1021 547 787 961 61 943
